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linear-massiv (empty) → 0.1.0.0

raw patch · 38 files changed

+13840/−0 lines, 38 filesdep +QuickCheckdep +basedep +criterion

Dependencies added: QuickCheck, base, criterion, deepseq, ghc-prim, hmatrix, linear, linear-massiv, massiv, primitive, tasty, tasty-hunit, tasty-quickcheck, vector

Files

+ bench-comparison/Main.hs view
@@ -0,0 +1,340 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- | Cross-library benchmark: linear-massiv vs hmatrix vs linear.+--+-- Compares performance of numerical linear algebra operations across+-- three Haskell libraries:+--+--   * __linear-massiv__ — pure Haskell, massiv-backed, type-safe dimensions+--   * __hmatrix__       — FFI to BLAS\/LAPACK (OpenBLAS on this system)+--   * __linear__        — pure Haskell, optimised for small fixed-size (V2–V4)+--+-- Run with @+RTS -N1@ for fair single-threaded comparison.+module Main (main) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import GHC.TypeNats (KnownNat)++-- linear-massiv+import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1 (dotP)+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvecP)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMulP, matMulPPar)+import Numeric.LinearAlgebra.Massiv.Solve.LU (luSolve, luSolveP)+import Numeric.LinearAlgebra.Massiv.Solve.Cholesky (choleskySolve, choleskySolveP)+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR (qr, qrP)+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric (symmetricEigen, symmetricEigenP, symmetricEigenPPar, symmetricEigenPDC, tridiagonalizeP)+import Numeric.LinearAlgebra.Massiv.Eigen.SVD (svd, svdP, svdGKP)++-- hmatrix+import qualified Numeric.LinearAlgebra as H++-- linear (small fixed-size)+import Linear.V4 (V4(..))+import qualified Linear.Matrix as LM+import qualified Linear.Metric as LMet++------------------------------------------------------------------------+-- linear-massiv matrix generators+------------------------------------------------------------------------++mkMatLM :: forall m n. (KnownNat m, KnownNat n) => Matrix m n M.P Double+mkMatLM = makeMatrix @m @n @M.P $ \i j ->+  fromIntegral (i * 7 + j * 3 + 1) / 100.0++mkVecLM :: forall n. KnownNat n => Vector n M.P Double+mkVecLM = makeVector @n @M.P $ \i -> fromIntegral (i + 1) / 10.0++-- Diagonally dominant for LU+mkDDLM :: forall n. KnownNat n => Matrix n n M.P Double+mkDDLM = makeMatrix @n @n @M.P $ \i j ->+  fromIntegral (i * 7 + j * 3 + 1) / 100.0+    + if i == j then fromIntegral (dimVal @n) else 0++-- SPD: B^T B + nI+mkSPDLM :: forall n. KnownNat n => Matrix n n M.P Double+mkSPDLM =+  let nn = dimVal @n+      b = makeMatrix @n @n @M.P $ \i j ->+            fromIntegral (i * nn + j + 1) / fromIntegral (nn * nn)+  in makeMatrix @n @n @M.P $ \i j ->+       foldl' (\acc k -> acc + (b ! (i, k)) * (b ! (j, k)))+              (if i == j then 1 else 0)+              [0..nn-1]++------------------------------------------------------------------------+-- hmatrix matrix generators (same numerical entries)+------------------------------------------------------------------------++mkMatHM :: Int -> H.Matrix Double+mkMatHM n = (n H.>< n) [ fromIntegral (i * 7 + j * 3 + 1) / 100.0+                        | i <- [0..n-1], j <- [0..n-1] ]++mkVecHM :: Int -> H.Vector Double+mkVecHM n = H.fromList [ fromIntegral (i + 1) / 10.0 | i <- [0..n-1] ]++mkDDHM :: Int -> H.Matrix Double+mkDDHM n = (n H.>< n) [ fromIntegral (i * 7 + j * 3 + 1) / 100.0+                           + if i == j then fromIntegral n else 0+                       | i <- [0..n-1], j <- [0..n-1] ]++mkSPDHM :: Int -> H.Matrix Double+mkSPDHM n =+  let b = (n H.>< n) [ fromIntegral (i * n + j + 1) / fromIntegral (n * n)+                      | i <- [0..n-1], j <- [0..n-1] ]+  in H.tr b H.<> b + H.scale (fromIntegral n) (H.ident n)++------------------------------------------------------------------------+-- linear (V4) data+------------------------------------------------------------------------++linM44a :: V4 (V4 Double)+linM44a = V4 (V4 0.01 0.04 0.07 0.10)+              (V4 0.08 0.11 0.14 0.17)+              (V4 0.15 0.18 0.21 0.24)+              (V4 0.22 0.25 0.28 0.31)++linM44b :: V4 (V4 Double)+linM44b = V4 (V4 0.34 0.37 0.40 0.43)+              (V4 0.41 0.44 0.47 0.50)+              (V4 0.48 0.51 0.54 0.57)+              (V4 0.55 0.58 0.61 0.64)++linV4a :: V4 Double+linV4a = V4 0.1 0.2 0.3 0.4++linV4b :: V4 Double+linV4b = V4 0.5 0.6 0.7 0.8++------------------------------------------------------------------------+-- hmatrix helpers (avoid operator section issues)+------------------------------------------------------------------------++hmGemm :: H.Matrix Double -> H.Matrix Double -> H.Matrix Double+hmGemm = (H.<>)++hmMatvec :: H.Matrix Double -> H.Vector Double -> H.Vector Double+hmMatvec = (H.#>)++hmDot :: H.Vector Double -> H.Vector Double -> Double+hmDot = H.dot++hmLinearSolve :: H.Matrix Double -> H.Vector Double -> H.Matrix Double+hmLinearSolve a b = case H.linearSolve a (H.asColumn b) of+  Just x  -> x+  Nothing -> error "hmLinearSolve: singular matrix"++hmCholSolve :: H.Matrix Double -> H.Vector Double -> H.Matrix Double+hmCholSolve a b =+  let r = H.chol (H.trustSym a)+  in H.cholSolve r (H.asColumn b)++------------------------------------------------------------------------+-- Benchmarks+------------------------------------------------------------------------++main :: IO ()+main = do+  -- Pre-compute all matrices to avoid construction overhead in benchmarks+  let hm4   = mkMatHM 4;   hm10  = mkMatHM 10;  hm50  = mkMatHM 50+      hm100 = mkMatHM 100; hm200 = mkMatHM 200; hm500 = mkMatHM 500+      hv4   = mkVecHM 4;   hv10  = mkVecHM 10;   hv50  = mkVecHM 50+      hv100 = mkVecHM 100; hv1000 = mkVecHM 1000+      dd10 = mkDDHM 10; dd50 = mkDDHM 50; dd100 = mkDDHM 100+      spd10 = mkSPDHM 10; spd50 = mkSPDHM 50; spd100 = mkSPDHM 100+      spd200 = mkSPDHM 200; spd500 = mkSPDHM 500++  defaultMain+    [ bgroup "GEMM"+      [ bgroup "4x4"+        [ bench "linear"        $ nf (linM44a LM.!*!) linM44b+        , bench "hmatrix"       $ nf (hmGemm hm4) hm4+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @4 @4)) (mkMatLM @4 @4)+        ]+      , bgroup "10x10"+        [ bench "hmatrix"       $ nf (hmGemm hm10) hm10+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @10 @10)) (mkMatLM @10 @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf (hmGemm hm50) hm50+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @50 @50)) (mkMatLM @50 @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf (hmGemm hm100) hm100+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @100 @100)) (mkMatLM @100 @100)+        ]+      , bgroup "200x200"+        [ bench "hmatrix"       $ nf (hmGemm hm200) hm200+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @200 @200)) (mkMatLM @200 @200)+        , bench "lm-parallel"   $ nf (matMulPPar (mkMatLM @200 @200)) (mkMatLM @200 @200)+        ]+      , bgroup "500x500"+        [ bench "hmatrix"       $ nf (hmGemm hm500) hm500+        , bench "linear-massiv" $ nf (matMulP (mkMatLM @500 @500)) (mkMatLM @500 @500)+        , bench "lm-parallel"   $ nf (matMulPPar (mkMatLM @500 @500)) (mkMatLM @500 @500)+        ]+      ]+    , bgroup "dot"+      [ bgroup "4"+        [ bench "linear"        $ nf (LMet.dot linV4a) linV4b+        , bench "hmatrix"       $ nf (hmDot hv4) hv4+        , bench "linear-massiv" $ nf (dotP (mkVecLM @4)) (mkVecLM @4)+        ]+      , bgroup "100"+        [ bench "hmatrix"       $ nf (hmDot hv100) hv100+        , bench "linear-massiv" $ nf (dotP (mkVecLM @100)) (mkVecLM @100)+        ]+      , bgroup "1000"+        [ bench "hmatrix"       $ nf (hmDot hv1000) hv1000+        , bench "linear-massiv" $ nf (dotP (mkVecLM @1000)) (mkVecLM @1000)+        ]+      ]+    , bgroup "matvec"+      [ bgroup "4"+        [ bench "linear"        $ nf (linM44a LM.!*) linV4a+        , bench "hmatrix"       $ nf (hmMatvec hm4) hv4+        , bench "linear-massiv" $ nf (matvecP (mkMatLM @4 @4)) (mkVecLM @4)+        ]+      , bgroup "50"+        [ bench "hmatrix"       $ nf (hmMatvec hm50) hv50+        , bench "linear-massiv" $ nf (matvecP (mkMatLM @50 @50)) (mkVecLM @50)+        ]+      , bgroup "100"+        [ bench "hmatrix"       $ nf (hmMatvec hm100) hv100+        , bench "linear-massiv" $ nf (matvecP (mkMatLM @100 @100)) (mkVecLM @100)+        ]+      ]+    , bgroup "luSolve"+      [ bgroup "10x10"+        [ bench "hmatrix"       $ nf (hmLinearSolve dd10) hv10+        , bench "linear-massiv" $ nf (luSolveP (mkDDLM @10)) (mkVecLM @10)+        , bench "lm-generic"    $ nf (luSolve (mkDDLM @10)) (mkVecLM @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf (hmLinearSolve dd50) hv50+        , bench "linear-massiv" $ nf (luSolveP (mkDDLM @50)) (mkVecLM @50)+        , bench "lm-generic"    $ nf (luSolve (mkDDLM @50)) (mkVecLM @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf (hmLinearSolve dd100) hv100+        , bench "linear-massiv" $ nf (luSolveP (mkDDLM @100)) (mkVecLM @100)+        , bench "lm-generic"    $ nf (luSolve (mkDDLM @100)) (mkVecLM @100)+        ]+      ]+    , bgroup "choleskySolve"+      [ bgroup "10x10"+        [ bench "hmatrix"       $ nf (hmCholSolve spd10) hv10+        , bench "linear-massiv" $ nf (choleskySolveP (mkSPDLM @10)) (mkVecLM @10)+        , bench "lm-generic"    $ nf (choleskySolve (mkSPDLM @10)) (mkVecLM @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf (hmCholSolve spd50) hv50+        , bench "linear-massiv" $ nf (choleskySolveP (mkSPDLM @50)) (mkVecLM @50)+        , bench "lm-generic"    $ nf (choleskySolve (mkSPDLM @50)) (mkVecLM @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf (hmCholSolve spd100) hv100+        , bench "linear-massiv" $ nf (choleskySolveP (mkSPDLM @100)) (mkVecLM @100)+        , bench "lm-generic"    $ nf (choleskySolve (mkSPDLM @100)) (mkVecLM @100)+        ]+      ]+    , bgroup "QR"+      [ bgroup "10x10"+        [ bench "hmatrix"       $ nf H.qr hm10+        , bench "linear-massiv" $ nf qrP (mkMatLM @10 @10)+        , bench "lm-generic"    $ nf qr (mkMatLM @10 @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf H.qr hm50+        , bench "linear-massiv" $ nf qrP (mkMatLM @50 @50)+        , bench "lm-generic"    $ nf qr (mkMatLM @50 @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf H.qr hm100+        , bench "linear-massiv" $ nf qrP (mkMatLM @100 @100)+        , bench "lm-generic"    $ nf qr (mkMatLM @100 @100)+        ]+      ]+    , bgroup "eigenSH"+      [ bgroup "10x10"+        [ bench "hmatrix"       $ nf (H.eigSH . H.trustSym) spd10+        , bench "linear-massiv" $ nf (\a -> symmetricEigenP a 200 1e-12) (mkSPDLM @10)+        , bench "lm-generic"    $ nf (\a -> symmetricEigen a 200 1e-12) (mkSPDLM @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf (H.eigSH . H.trustSym) spd50+        , bench "linear-massiv" $ nf (\a -> symmetricEigenP a 500 1e-12) (mkSPDLM @50)+        , bench "lm-generic"    $ nf (\a -> symmetricEigen a 500 1e-12) (mkSPDLM @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf (H.eigSH . H.trustSym) spd100+        , bench "linear-massiv" $ nf (\a -> symmetricEigenP a 1000 1e-12) (mkSPDLM @100)+        , bench "lm-parallel"   $ nf (\a -> symmetricEigenPPar a 1000 1e-12) (mkSPDLM @100)+        ]+      , bgroup "200x200"+        [ bench "hmatrix"       $ nf (H.eigSH . H.trustSym) spd200+        , bench "linear-massiv" $ nf (\a -> symmetricEigenP a 2000 1e-12) (mkSPDLM @200)+        ]+      , bgroup "500x500"+        [ bench "hmatrix"       $ nf (H.eigSH . H.trustSym) spd500+        , bench "linear-massiv" $ nf (\a -> symmetricEigenP a 5000 1e-12) (mkSPDLM @500)+        ]+      ]+    , bgroup "eigenSH-breakdown"+      [ bgroup "200x200"+        [ bench "tridiagP-only" $ nf tridiagonalizeP (mkSPDLM @200)+        , bench "full-eigenP"   $ nf (\a -> symmetricEigenP a 2000 1e-12) (mkSPDLM @200)+        ]+      , bgroup "500x500"+        [ bench "tridiagP-only" $ nf tridiagonalizeP (mkSPDLM @500)+        , bench "full-eigenP"   $ nf (\a -> symmetricEigenP a 5000 1e-12) (mkSPDLM @500)+        ]+      ]+    , bgroup "eigenSH-DC"+      [ bgroup "50x50"+        [ bench "QR"  $ nf (\a -> symmetricEigenP a 500 1e-12) (mkSPDLM @50)+        , bench "D&C" $ nf (\a -> symmetricEigenPDC a 1e-12) (mkSPDLM @50)+        ]+      , bgroup "100x100"+        [ bench "QR"  $ nf (\a -> symmetricEigenP a 1000 1e-12) (mkSPDLM @100)+        , bench "D&C" $ nf (\a -> symmetricEigenPDC a 1e-12) (mkSPDLM @100)+        ]+      , bgroup "200x200"+        [ bench "QR"  $ nf (\a -> symmetricEigenP a 2000 1e-12) (mkSPDLM @200)+        , bench "D&C" $ nf (\a -> symmetricEigenPDC a 1e-12) (mkSPDLM @200)+        ]+      , bgroup "500x500"+        [ bench "QR"  $ nf (\a -> symmetricEigenP a 5000 1e-12) (mkSPDLM @500)+        , bench "D&C" $ nf (\a -> symmetricEigenPDC a 1e-12) (mkSPDLM @500)+        ]+      ]+    , bgroup "SVD"+      [ bgroup "10x10"+        [ bench "hmatrix"       $ nf H.svd hm10+        , bench "linear-massiv" $ nf svdP (mkMatLM @10 @10)+        , bench "lm-generic"    $ nf svd (mkMatLM @10 @10)+        ]+      , bgroup "50x50"+        [ bench "hmatrix"       $ nf H.svd hm50+        , bench "linear-massiv" $ nf svdP (mkMatLM @50 @50)+        , bench "lm-generic"    $ nf svd (mkMatLM @50 @50)+        ]+      , bgroup "100x100"+        [ bench "hmatrix"       $ nf H.svd hm100+        , bench "linear-massiv" $ nf svdP (mkMatLM @100 @100)+        , bench "lm-gk"         $ nf svdGKP (mkMatLM @100 @100)+        ]+      , bgroup "200x200"+        [ bench "hmatrix"       $ nf H.svd hm200+        , bench "linear-massiv" $ nf svdP (mkMatLM @200 @200)+        , bench "lm-gk"         $ nf svdGKP (mkMatLM @200 @200)+        ]+      , bgroup "500x500"+        [ bench "hmatrix"       $ nf H.svd hm500+        , bench "linear-massiv" $ nf svdP (mkMatLM @500 @500)+        , bench "lm-gk"         $ nf svdGKP (mkMatLM @500 @500)+        ]+      ]+    ]
+ bench/Bench/BLAS.hs view
@@ -0,0 +1,104 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Bench.BLAS (blasBenchmarks) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Comp(..))+import Control.DeepSeq ()+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1 (dotP)+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvecP)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMulP, matMulComp)++-- linear library imports for comparison+import Linear.V4 (V4(..))+import Linear.Matrix ((!*!), (!*))+import qualified Linear.Metric as LM+import Linear.V4 ()++-- Massiv matrix generators+mkMatP :: forall m n. (KnownNat m, KnownNat n) => Matrix m n M.P Double+mkMatP = makeMatrix @m @n @M.P $ \i j -> fromIntegral (i * 7 + j * 3 + 1) / 100.0++mkVecP :: forall n. KnownNat n => Vector n M.P Double+mkVecP = makeVector @n @M.P $ \i -> fromIntegral (i + 1) / 10.0++-- linear library 4x4 matrices for comparison+linearM44 :: V4 (V4 Double)+linearM44 = V4 (V4 1 2 3 4)+                (V4 5 6 7 8)+                (V4 9 10 11 12)+                (V4 13 14 15 16)++linearM44b :: V4 (V4 Double)+linearM44b = V4 (V4 17 18 19 20)+                 (V4 21 22 23 24)+                 (V4 25 26 27 28)+                 (V4 29 30 31 32)++linearV4 :: V4 Double+linearV4 = V4 1 2 3 4++linearV4b :: V4 Double+linearV4b = V4 5 6 7 8++blasBenchmarks :: [Benchmark]+blasBenchmarks =+  [ bgroup "gemm"+    [ -- Small sizes: compare linear vs massiv+      bgroup "4x4"+        [ bench "linear-V4" $ nf (linearM44 !*!) linearM44b+        , bench "massiv-P"  $ nf (uncurry (matMulP @4 @4 @4)) (mkMatP @4 @4, mkMatP @4 @4)+        ]+    , bgroup "10x10"+        [ bench "P/Seq" $ nf (uncurry (matMulP @10 @10 @10)) (mkMatP @10 @10, mkMatP @10 @10)+        ]+    , bgroup "50x50"+        [ bench "P/Seq" $ nf (uncurry (matMulP @50 @50 @50)) (mkMatP @50 @50, mkMatP @50 @50)+        , bench "P/Par" $ nf (uncurry (matMulComp @50 @50 @50 Par)) (mkMatP @50 @50, mkMatP @50 @50)+        ]+    , bgroup "100x100"+        [ bench "P/Seq" $ nf (uncurry (matMulP @100 @100 @100)) (mkMatP @100 @100, mkMatP @100 @100)+        , bench "P/Par" $ nf (uncurry (matMulComp @100 @100 @100 Par)) (mkMatP @100 @100, mkMatP @100 @100)+        ]+    , bgroup "200x200"+        [ bench "P/Seq" $ nf (uncurry (matMulP @200 @200 @200)) (mkMatP @200 @200, mkMatP @200 @200)+        , bench "P/Par" $ nf (uncurry (matMulComp @200 @200 @200 Par)) (mkMatP @200 @200, mkMatP @200 @200)+        ]+    -- Representation comparison at 100x100+    , bgroup "repr-100x100"+        [ bench "P" $ nf (uncurry (matMulP @100 @100 @100)) (mkMatP @100 @100, mkMatP @100 @100)+        ]+    ]+  , bgroup "dot"+    [ bgroup "4"+        [ bench "linear-V4" $ nf (LM.dot linearV4) linearV4b+        , bench "massiv-P"  $ nf (uncurry (dotP @4)) (mkVecP @4, mkVecP @4)+        ]+    , bgroup "100"+        [ bench "P" $ nf (uncurry (dotP @100)) (mkVecP @100, mkVecP @100)+        ]+    , bgroup "1000"+        [ bench "P" $ nf (uncurry (dotP @1000)) (mkVecP @1000, mkVecP @1000)+        ]+    , bgroup "10000"+        [ bench "P" $ nf (uncurry (dotP @10000)) (mkVecP @10000, mkVecP @10000)+        ]+    ]+  , bgroup "matvec"+    [ bgroup "4"+        [ bench "linear-V4" $ nf (linearM44 !*) linearV4+        , bench "massiv-P"  $ nf (uncurry (matvecP @4 @4)) (mkMatP @4 @4, mkVecP @4)+        ]+    , bgroup "50"+        [ bench "P" $ nf (uncurry (matvecP @50 @50)) (mkMatP @50 @50, mkVecP @50)+        ]+    , bgroup "100"+        [ bench "P" $ nf (uncurry (matvecP @100 @100)) (mkMatP @100 @100, mkVecP @100)+        ]+    ]+  ]
+ bench/Bench/Eigen.hs view
@@ -0,0 +1,40 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Bench.Eigen (eigenBenchmarks) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric (symmetricEigen, jacobiEigen)+import Numeric.LinearAlgebra.Massiv.Eigen.SVD (svd)++mkSPDP :: forall n. KnownNat n => Matrix n n M.P Double+mkSPDP =+  let nn = dimVal @n+      b = makeMatrix @n @n @M.P $ \i j -> fromIntegral (i * nn + j + 1) / fromIntegral (nn * nn)+  in makeMatrix @n @n @M.P $ \i j ->+    foldl' (\acc k -> acc + (b ! (i, k)) * (b ! (j, k))) (if i == j then 1 else 0) [0..nn-1]++mkMatP :: forall m n. (KnownNat m, KnownNat n) => Matrix m n M.P Double+mkMatP = makeMatrix @m @n @M.P $ \i j -> fromIntegral (i * 7 + j * 3 + 1) / 100.0++eigenBenchmarks :: [Benchmark]+eigenBenchmarks =+  [ bgroup "symmetricEigen"+    [ bench "10x10/P" $ nf (\a -> symmetricEigen a 200 1e-12) (mkSPDP @10)+    , bench "20x20/P" $ nf (\a -> symmetricEigen a 500 1e-12) (mkSPDP @20)+    , bench "50x50/P" $ nf (\a -> symmetricEigen a 500 1e-12) (mkSPDP @50)+    ]+  , bgroup "jacobiEigen"+    [ bench "10x10/P" $ nf (\a -> jacobiEigen a 50 1e-12) (mkSPDP @10)+    , bench "20x20/P" $ nf (\a -> jacobiEigen a 50 1e-12) (mkSPDP @20)+    ]+  , bgroup "svd"+    [ bench "10x10/P" $ nf svd (mkMatP @10 @10)+    , bench "20x20/P" $ nf svd (mkMatP @20 @20)+    , bench "50x50/P" $ nf svd (mkMatP @50 @50)+    ]+  ]
+ bench/Bench/Orthogonal.hs view
@@ -0,0 +1,27 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Bench.Orthogonal (orthogonalBenchmarks) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR (qr, qrGivens)++mkMatP :: forall m n. (KnownNat m, KnownNat n) => Matrix m n M.P Double+mkMatP = makeMatrix @m @n @M.P $ \i j -> fromIntegral (i * 7 + j * 3 + 1) / 100.0++orthogonalBenchmarks :: [Benchmark]+orthogonalBenchmarks =+  [ bgroup "qr-householder"+    [ bench "10x10/P"  $ nf qr (mkMatP @10 @10)+    , bench "50x50/P"  $ nf qr (mkMatP @50 @50)+    , bench "100x100/P" $ nf qr (mkMatP @100 @100)+    ]+  , bgroup "qr-givens"+    [ bench "10x10/P"  $ nf qrGivens (mkMatP @10 @10)+    , bench "50x50/P"  $ nf qrGivens (mkMatP @50 @50)+    ]+  ]
+ bench/Bench/Parallel.hs view
@@ -0,0 +1,46 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- | Benchmarks specifically for measuring parallelism scalability.+--+-- Runs matrix multiplication at various sizes with explicit thread counts+-- using massiv's ParN constructor.+module Bench.Parallel (parallelBenchmarks) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Comp(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMulComp)++mkMatP :: forall m n. (KnownNat m, KnownNat n) => Matrix m n M.P Double+mkMatP = makeMatrix @m @n @M.P $ \i j -> fromIntegral (i * 7 + j * 3 + 1) / 100.0++-- | Benchmarks measuring how performance scales with thread count.+parallelBenchmarks :: [Benchmark]+parallelBenchmarks =+  [ bgroup "scaling-gemm"+    [ bgroup "100x100"+        [ bench "Seq"     $ nf (uncurry (matMulComp @100 @100 @100 Seq))     (mkMatP @100 @100, mkMatP @100 @100)+        , bench "Par"     $ nf (uncurry (matMulComp @100 @100 @100 Par))     (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-1"  $ nf (uncurry (matMulComp @100 @100 @100 (ParN 1)))  (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-2"  $ nf (uncurry (matMulComp @100 @100 @100 (ParN 2)))  (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-4"  $ nf (uncurry (matMulComp @100 @100 @100 (ParN 4)))  (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-8"  $ nf (uncurry (matMulComp @100 @100 @100 (ParN 8)))  (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-16" $ nf (uncurry (matMulComp @100 @100 @100 (ParN 16))) (mkMatP @100 @100, mkMatP @100 @100)+        , bench "ParN-20" $ nf (uncurry (matMulComp @100 @100 @100 (ParN 20))) (mkMatP @100 @100, mkMatP @100 @100)+        ]+    , bgroup "200x200"+        [ bench "Seq"     $ nf (uncurry (matMulComp @200 @200 @200 Seq))     (mkMatP @200 @200, mkMatP @200 @200)+        , bench "Par"     $ nf (uncurry (matMulComp @200 @200 @200 Par))     (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-1"  $ nf (uncurry (matMulComp @200 @200 @200 (ParN 1)))  (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-2"  $ nf (uncurry (matMulComp @200 @200 @200 (ParN 2)))  (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-4"  $ nf (uncurry (matMulComp @200 @200 @200 (ParN 4)))  (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-8"  $ nf (uncurry (matMulComp @200 @200 @200 (ParN 8)))  (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-16" $ nf (uncurry (matMulComp @200 @200 @200 (ParN 16))) (mkMatP @200 @200, mkMatP @200 @200)+        , bench "ParN-20" $ nf (uncurry (matMulComp @200 @200 @200 (ParN 20))) (mkMatP @200 @200, mkMatP @200 @200)+        ]+    ]+  ]
+ bench/Bench/Solve.hs view
@@ -0,0 +1,52 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Bench.Solve (solveBenchmarks) where++import Criterion.Main+import qualified Data.Massiv.Array as M+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Solve.LU (lu, luSolve)+import Numeric.LinearAlgebra.Massiv.Solve.Cholesky (cholesky, choleskySolve)++-- Diagonally dominant matrix for LU+mkMatP :: forall n. KnownNat n => Matrix n n M.P Double+mkMatP = makeMatrix @n @n @M.P $ \i j ->+  fromIntegral (i * 7 + j * 3 + 1) / 100.0 + if i == j then fromIntegral (dimVal @n) else 0++-- SPD matrix: A = B^T B + nI+mkSPDP :: forall n. KnownNat n => Matrix n n M.P Double+mkSPDP =+  let nn = dimVal @n+      b = makeMatrix @n @n @M.P $ \i j -> fromIntegral (i * nn + j + 1) / fromIntegral (nn * nn)+  in makeMatrix @n @n @M.P $ \i j ->+    foldl' (\acc k -> acc + (b ! (i, k)) * (b ! (j, k))) (if i == j then 1 else 0) [0..nn-1]++mkVecP :: forall n. KnownNat n => Vector n M.P Double+mkVecP = makeVector @n @M.P $ \i -> fromIntegral (i + 1)++solveBenchmarks :: [Benchmark]+solveBenchmarks =+  [ bgroup "lu"+    [ bench "10x10/P"  $ nf lu (mkMatP @10)+    , bench "50x50/P"  $ nf lu (mkMatP @50)+    , bench "100x100/P" $ nf lu (mkMatP @100)+    ]+  , bgroup "luSolve"+    [ bench "10x10/P"  $ nf (uncurry luSolve) (mkMatP @10, mkVecP @10)+    , bench "50x50/P"  $ nf (uncurry luSolve) (mkMatP @50, mkVecP @50)+    , bench "100x100/P" $ nf (uncurry luSolve) (mkMatP @100, mkVecP @100)+    ]+  , bgroup "cholesky"+    [ bench "10x10/P"  $ nf cholesky (mkSPDP @10)+    , bench "50x50/P"  $ nf cholesky (mkSPDP @50)+    , bench "100x100/P" $ nf cholesky (mkSPDP @100)+    ]+  , bgroup "choleskySolve"+    [ bench "10x10/P"  $ nf (uncurry choleskySolve) (mkSPDP @10, mkVecP @10)+    , bench "50x50/P"  $ nf (uncurry choleskySolve) (mkSPDP @50, mkVecP @50)+    , bench "100x100/P" $ nf (uncurry choleskySolve) (mkSPDP @100, mkVecP @100)+    ]+  ]
+ bench/Main.hs view
@@ -0,0 +1,18 @@+module Main (main) where++import Criterion.Main++import Bench.BLAS (blasBenchmarks)+import Bench.Solve (solveBenchmarks)+import Bench.Orthogonal (orthogonalBenchmarks)+import Bench.Eigen (eigenBenchmarks)+import Bench.Parallel (parallelBenchmarks)++main :: IO ()+main = defaultMain+  [ bgroup "BLAS" blasBenchmarks+  , bgroup "Solve" solveBenchmarks+  , bgroup "Orthogonal" orthogonalBenchmarks+  , bgroup "Eigen" eigenBenchmarks+  , bgroup "Parallel" parallelBenchmarks+  ]
+ linear-massiv.cabal view
@@ -0,0 +1,159 @@+cabal-version: 3.0+name:          linear-massiv+version:       0.1.0.0+synopsis:      Type-safe numerical linear algebra backed by massiv arrays+description:+  Native Haskell implementations of algorithms from Golub & Van Loan's+  "Matrix Computations" (4th ed.) using massiv arrays as the backing store,+  with compile-time dimensional conformance via GHC type-level naturals.+  Extends the linear library's typeclasses for integration with existing code.+  .+  Co-authored by Claude Opus (Anthropic). This code should be considered a+  derived work of the various algorithmic examples and reference+  implementations drawn upon during development, including but not limited+  to LAPACK, OpenBLAS, and GVL4.+license:       BSD-3-Clause+build-type:    Simple++library+  hs-source-dirs: src+  exposed-modules:+    Numeric.LinearAlgebra.Massiv+    Numeric.LinearAlgebra.Massiv.Types+    Numeric.LinearAlgebra.Massiv.Internal+    Numeric.LinearAlgebra.Massiv.Norms+    Numeric.LinearAlgebra.Massiv.BLAS.Level1+    Numeric.LinearAlgebra.Massiv.BLAS.Level2+    Numeric.LinearAlgebra.Massiv.BLAS.Level3+    Numeric.LinearAlgebra.Massiv.Solve.Triangular+    Numeric.LinearAlgebra.Massiv.Solve.LU+    Numeric.LinearAlgebra.Massiv.Solve.Cholesky+    Numeric.LinearAlgebra.Massiv.Solve.Banded+    Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+    Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+    Numeric.LinearAlgebra.Massiv.Orthogonal.QR+    Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+    Numeric.LinearAlgebra.Massiv.Eigen.Power+    Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+    Numeric.LinearAlgebra.Massiv.Eigen.Schur+    Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+    Numeric.LinearAlgebra.Massiv.Eigen.SVD+    Numeric.LinearAlgebra.Massiv.Internal.Kernel+    Numeric.LinearAlgebra.Massiv.Linear+  build-depends:+    base >= 4.16 && < 5,+    ghc-prim,+    massiv >= 1.0 && < 2,+    linear >= 1.21 && < 2,+    vector >= 0.12 && < 1,+    primitive >= 0.7 && < 1,+    deepseq >= 1.4 && < 2+  default-language: GHC2021+  default-extensions:+    DataKinds+    TypeFamilies+    ScopedTypeVariables+    TypeApplications+    TypeOperators+    GADTs+    RankNTypes+    FlexibleContexts+    FlexibleInstances+    MultiParamTypeClasses+    StandaloneDeriving+    DerivingStrategies+  ghc-options: -Wall -O2++test-suite linear-massiv-test+  type: exitcode-stdio-1.0+  main-is: Spec.hs+  hs-source-dirs: test+  other-modules:+    Test.Types+    Test.Residuals+    Test.BLAS+    Test.Solve+    Test.Orthogonal+    Test.Eigen+    Test.Norms+  build-depends:+    base,+    linear-massiv,+    massiv,+    tasty >= 1.4,+    tasty-hunit >= 0.10,+    tasty-quickcheck >= 0.10,+    QuickCheck >= 2.14,+    linear,+    vector+  default-language: GHC2021+  default-extensions:+    DataKinds+    TypeFamilies+    ScopedTypeVariables+    TypeApplications+    TypeOperators+    GADTs+    RankNTypes+    FlexibleContexts+    FlexibleInstances+    MultiParamTypeClasses+  ghc-options: -Wall -threaded -rtsopts++benchmark linear-massiv-bench+  type: exitcode-stdio-1.0+  main-is: Main.hs+  hs-source-dirs: bench+  other-modules:+    Bench.BLAS+    Bench.Solve+    Bench.Orthogonal+    Bench.Eigen+    Bench.Parallel+  build-depends:+    base,+    linear-massiv,+    massiv,+    criterion >= 1.5,+    deepseq,+    linear+  default-language: GHC2021+  default-extensions:+    DataKinds+    TypeFamilies+    ScopedTypeVariables+    TypeApplications+    TypeOperators+    GADTs+    RankNTypes+    FlexibleContexts+    FlexibleInstances+    MultiParamTypeClasses+  ghc-options: -Wall -O2 -threaded -rtsopts "-with-rtsopts=-N"++benchmark linear-massiv-comparison+  type: exitcode-stdio-1.0+  main-is: Main.hs+  hs-source-dirs: bench-comparison+  build-depends:+    base,+    linear-massiv,+    massiv,+    hmatrix >= 0.20,+    linear,+    criterion >= 1.5,+    deepseq,+    vector+  default-language: GHC2021+  default-extensions:+    DataKinds+    TypeFamilies+    ScopedTypeVariables+    TypeApplications+    TypeOperators+    GADTs+    RankNTypes+    FlexibleContexts+    FlexibleInstances+    MultiParamTypeClasses+  ghc-options: -Wall -O2 -threaded -rtsopts "-with-rtsopts=-N"
+ src/Numeric/LinearAlgebra/Massiv.hs view
@@ -0,0 +1,236 @@+-- |+-- Module      : Numeric.LinearAlgebra.Massiv+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- @linear-massiv@: Type-safe numerical linear algebra backed by+-- <https://hackage.haskell.org/package/massiv massiv> arrays.+--+-- This library provides native Haskell implementations of algorithms from:+--+-- * Golub, G. H., & Van Loan, C. F. (2013). /Matrix Computations/ (4th ed.).+--   Johns Hopkins University Press. ISBN 978-1-4214-0794-4.+--+-- referred to throughout as __GVL4__.+--+-- = Derived Work Attribution+--+-- This code was co-authored by Claude Opus (Anthropic) and should be+-- considered a derived work of the various algorithmic examples and+-- reference implementations drawn upon during development, including but+-- not limited to:+--+-- * __LAPACK__ (Linear Algebra PACKage) — Anderson, E. et al. (1999).+--   /LAPACK Users' Guide/, 3rd ed., SIAM. The LAPACK testing methodology,+--   algorithm structures, and numerical stability techniques informed much+--   of the implementation.+--+-- * __OpenBLAS__ — Xianyi, Z., Qian, W., and Yunquan, Z. (2011--).+--   The tiled GEMM micro-kernel architecture, cache-blocking strategies,+--   and SIMD vectorisation patterns were inspired by OpenBLAS.+--+-- * __GVL4__ — The primary algorithmic reference, as noted above.+--+-- * __Higham__ — Higham, N. J. (2002). /Accuracy and Stability of Numerical+--   Algorithms/, 2nd ed., SIAM. Error analysis and numerical stability+--   frameworks.+--+-- = Design Principles+--+-- 1. __Type-level dimensional safety__: Matrix dimensions are tracked in the+--    type system via GHC @DataKinds@ and @KnownNat@ constraints. Dimensionally+--    incorrect operations (e.g., multiplying an \(m \times k\) matrix by an+--    \(n \times p\) matrix where \(k \neq n\)) are rejected at compile time.+--+-- 2. __Representation polymorphism__: All operations are parametric over+--    massiv's array representation @r@ (e.g., @P@ for primitive, @U@ for+--    unboxed, @S@ for storable, @B@ for boxed), constrained by+--    @'Data.Massiv.Array.Manifest' r e@. Users choose the representation at+--    the call site.+--+-- 3. __Parallelism via massiv__: Operations that construct arrays via+--    @makeArray@ inherit massiv's computation strategies. Use 'matMulComp'+--    with @Par@ or @ParN n@ for parallel matrix multiplication.+--+-- 4. __No FFI__: All algorithms are pure Haskell, enabling portability and+--    auditability. Benchmarks compare performance across massiv representations+--    and parallelism strategies.+--+-- = Internal Architecture: Two-Layer Design+--+-- @linear-massiv@ uses a two-layer architecture that separates the type-safe+-- public API from the performance-critical internal representation:+--+-- * __Public layer__: 'Matrix' and 'Vector' are @newtype@ wrappers around+--   massiv's @Array r Ix2 e@, providing compile-time dimension checking via+--   phantom @Nat@ parameters and representation polymorphism via @r@.+--+-- * __Internal layer__: Performance-critical inner loops (GEMM, QR,+--   tridiagonalisation, SVD, etc.) unwrap the massiv array to a raw+--   @ByteArray#@ \/ @MutableByteArray#@ and operate directly via GHC primops,+--   including @DoubleX4#@ AVX2 SIMD instructions compiled through the LLVM 17+--   backend.  Functions receive @(ByteArray, offset, stride)@ triples, enabling+--   zero-copy submatrix views for panel factorisations.+--+-- This separation is essential for performance.  Benchmarks (Round 3 of the+-- accompanying report) showed that massiv's per-element @M.readM@\/@M.write_@+-- abstraction layer imposed a 240–330× penalty on BLAS operations relative to+-- direct primop access, even though the underlying memory layout is identical.+-- The raw primop layer eliminates this overhead while the @newtype@ wrapper+-- preserves type safety at the API boundary.+--+-- == Why not @vector-sized@ or @linear@'s @V@?+--+-- The @<https://hackage.haskell.org/package/vector-sized vector-sized>@ package+-- provides an @Unbox (Vector n a)@ instance that stores+-- @Vector m (Vector n Double)@ as a contiguous flat @ByteArray@ of @m × n@+-- doubles.  While the __memory layout__ is correct, contiguous memory alone is+-- insufficient for high-performance numerical kernels:+--+-- * __Per-element typeclass dispatch__: Every access goes through+--   @basicUnsafeRead@ \/ @basicUnsafeWrite@ of the @Unbox@ data family.+--   Reading element @(i, j)@ requires indexing the outer vector to obtain a+--   @Vector n Double@ (constructing an intermediate slice), then indexing that.+--   @linear-massiv@ computes @off + i * stride + j@ and issues a single+--   @readDoubleArray#@ primop.+--+-- * __No SIMD access__: The 4×8 GEMM micro-kernel loads 4 consecutive doubles+--   via @indexDoubleArrayAsDoubleX4# ba# (off# +# i#)@—a direct 256-bit AVX2+--   load from a computed byte offset.  The @Unbox@ typeclass does not expose+--   the underlying @ByteArray#@, and GHC cannot optimise through the data+--   family indirection to produce equivalent code.+--+-- * __No mutable primop access__: In-place factorisations (LU, QR,+--   tridiagonalisation, bidiagonalisation) require @writeDoubleArray#@ on+--   @MutableByteArray#@ with computed offsets.  The @MVector@ abstraction+--   interposes allocation and function calls that prevent the tight unboxed+--   loops needed for competitive performance.+--+-- * __No zero-copy submatrix views__: Panel factorisations pass+--   @(ByteArray, offset, stride)@ triples to kernels, enabling zero-copy views+--   into submatrices.  @Vector n a@ does not naturally express "this row starts+--   at byte offset X in a larger backing array."+--+-- The @<https://hackage.haskell.org/package/linear linear>@ library's @V n a@+-- uses @Vector@ from the @vector@ package internally and is designed for small+-- fixed-size vectors (V2–V4) where GHC can fully unbox everything.  At+-- @n = 100–500@, @V n (V m Double)@ would be a vector-of-vectors with per-row+-- indirection—catastrophic for cache locality and SIMD vectorisation.+-- @linear-massiv@ provides conversion functions ('fromLinearV', 'fromV2', etc.)+-- in "Numeric.LinearAlgebra.Massiv.Linear" for interoperability.+--+-- = Quick Start+--+-- @+-- import Numeric.LinearAlgebra.Massiv+-- import qualified Data.Massiv.Array as M+--+-- -- Create a 3x3 matrix (Primitive representation, Double elements)+-- let a = 'makeMatrix' \@3 \@3 \@M.P $ \\i j ->+--           fromIntegral (i * 3 + j + 1) :: Double+--+-- -- QR factorization+-- let (q, r) = 'qr' a+--+-- -- Solve Ax = b via LU+-- let b = 'makeVector' \@3 \@M.P $ \\i -> fromIntegral (i + 1) :: Double+-- let x = 'luSolve' a b+-- @+--+-- = Module Organisation+--+-- == Core types and construction+--+-- * "Numeric.LinearAlgebra.Massiv.Types" — 'Matrix', 'Vector' newtypes with+--   phantom dimension parameters+-- * "Numeric.LinearAlgebra.Massiv.Internal" — Unsafe constructors, dimension+--   reification, array creation helpers+--+-- == BLAS-like operations (GVL4 Ch. 1)+--+-- * "Numeric.LinearAlgebra.Massiv.BLAS.Level1" — Vector–vector: 'dot', 'axpy', 'scal', 'nrm2'+-- * "Numeric.LinearAlgebra.Massiv.BLAS.Level2" — Matrix–vector: 'gemv', 'matvec', 'ger'+-- * "Numeric.LinearAlgebra.Massiv.BLAS.Level3" — Matrix–matrix: 'gemm', 'matMul', 'transpose'+--+-- == Direct solvers (GVL4 Chs. 3–4)+--+-- * "Numeric.LinearAlgebra.Massiv.Solve.Triangular" — Forward\/back substitution+-- * "Numeric.LinearAlgebra.Massiv.Solve.LU" — LU with partial pivoting+-- * "Numeric.LinearAlgebra.Massiv.Solve.Cholesky" — Cholesky for SPD matrices+-- * "Numeric.LinearAlgebra.Massiv.Solve.Banded" — Band LU, band Cholesky, tridiagonal solver+--+-- == Orthogonal factorizations (GVL4 Ch. 5)+--+-- * "Numeric.LinearAlgebra.Massiv.Orthogonal.Householder" — Householder reflections+-- * "Numeric.LinearAlgebra.Massiv.Orthogonal.Givens" — Givens rotations+-- * "Numeric.LinearAlgebra.Massiv.Orthogonal.QR" — QR factorization (Householder and Givens)+-- * "Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares" — Least squares via QR+--+-- == Eigenvalue problems and SVD (GVL4 Chs. 7–8)+--+-- * "Numeric.LinearAlgebra.Massiv.Eigen.Power" — Power, inverse, Rayleigh quotient iteration+-- * "Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg" — Hessenberg reduction+-- * "Numeric.LinearAlgebra.Massiv.Eigen.Schur" — Schur decomposition (QR algorithm)+-- * "Numeric.LinearAlgebra.Massiv.Eigen.Symmetric" — Symmetric eigenvalue (tridiagonal QR, Jacobi)+-- * "Numeric.LinearAlgebra.Massiv.Eigen.SVD" — Singular value decomposition+--+-- == Norms and condition numbers (GVL4 Ch. 2)+--+-- * "Numeric.LinearAlgebra.Massiv.Norms" — Frobenius, 1-, \(\infty\)-, and 2-norms+--+-- == Integration with the @linear@ library+--+-- * "Numeric.LinearAlgebra.Massiv.Linear" — Conversions to\/from @linear@'s @V@, @V2@, @V3@, @V4@+module Numeric.LinearAlgebra.Massiv+  ( -- * Core types+    module Numeric.LinearAlgebra.Massiv.Types+    -- * Construction helpers+  , module Numeric.LinearAlgebra.Massiv.Internal+    -- * BLAS operations+  , module Numeric.LinearAlgebra.Massiv.BLAS.Level1+  , module Numeric.LinearAlgebra.Massiv.BLAS.Level2+  , module Numeric.LinearAlgebra.Massiv.BLAS.Level3+    -- * Direct solvers+  , module Numeric.LinearAlgebra.Massiv.Solve.Triangular+  , module Numeric.LinearAlgebra.Massiv.Solve.LU+  , module Numeric.LinearAlgebra.Massiv.Solve.Cholesky+  , module Numeric.LinearAlgebra.Massiv.Solve.Banded+    -- * Orthogonal factorizations+  , module Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+  , module Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+  , module Numeric.LinearAlgebra.Massiv.Orthogonal.QR+  , module Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+    -- * Eigenvalue problems+  , module Numeric.LinearAlgebra.Massiv.Eigen.Power+  , module Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+  , module Numeric.LinearAlgebra.Massiv.Eigen.Schur+  , module Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+  , module Numeric.LinearAlgebra.Massiv.Eigen.SVD+    -- * Norms+  , module Numeric.LinearAlgebra.Massiv.Norms+    -- * Linear integration+  , module Numeric.LinearAlgebra.Massiv.Linear+  ) where++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1+import Numeric.LinearAlgebra.Massiv.BLAS.Level2+import Numeric.LinearAlgebra.Massiv.BLAS.Level3+import Numeric.LinearAlgebra.Massiv.Solve.Triangular+import Numeric.LinearAlgebra.Massiv.Solve.LU+import Numeric.LinearAlgebra.Massiv.Solve.Cholesky+import Numeric.LinearAlgebra.Massiv.Solve.Banded+import Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+import Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR+import Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+import Numeric.LinearAlgebra.Massiv.Eigen.Power+import Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+import Numeric.LinearAlgebra.Massiv.Eigen.Schur+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+import Numeric.LinearAlgebra.Massiv.Eigen.SVD+import Numeric.LinearAlgebra.Massiv.Norms+import Numeric.LinearAlgebra.Massiv.Linear
+ src/Numeric/LinearAlgebra/Massiv/BLAS/Level1.hs view
@@ -0,0 +1,203 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.BLAS.Level1+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = BLAS Level 1: Vector–Vector Operations+--+-- This module provides type-safe, dimension-indexed wrappers around the+-- standard BLAS Level 1 kernels for vector–vector operations.  Every+-- function carries the vector length @n@ as a phantom type-level natural,+-- so dimension mismatches are caught at compile time rather than at run+-- time.+--+-- The algorithms implemented here correspond to the elementary building+-- blocks described in:+--+-- * Golub, G. H. & Van Loan, C. F. (2013). /Matrix Computations/,+--   4th edition (GVL4). Johns Hopkins University Press.+--   __Chapter 1, Section 1.1__, pp. 4–8.+--+-- Specifically:+--+-- * __Algorithm 1.1.1__ (p. 4) — Inner product (dot product).+--   Given vectors \(x, y \in \mathbb{R}^{n}\), compute+--   \(c = x^{T} y = \sum_{i=1}^{n} x_i y_i\).+--+-- * __Algorithm 1.1.2 (Saxpy)__ (p. 4) — Scalar \(\alpha\) times+--   vector \(x\) plus vector \(y\):+--   \(y \leftarrow \alpha x + y\).+--   This is the fundamental vector-update operation upon which the+--   higher-level BLAS Level 2 and Level 3 routines are built.+--+-- Additionally the module exposes the common vector norms+-- (\(\lVert \cdot \rVert_1\) and \(\lVert \cdot \rVert_2\)) and+-- scalar–vector scaling, which together form the complete Level 1+-- BLAS surface.+--+-- == Complexity+--+-- All operations in this module are \(O(n)\) in the vector length.+module Numeric.LinearAlgebra.Massiv.BLAS.Level1+  ( -- * Dot product (Algorithm 1.1.1, GVL4 p. 4)+    dot+  , dotP+    -- * Scalar–vector operations (Algorithm 1.1.2, GVL4 p. 4)+  , scal+  , axpy+    -- * Vector norms (GVL4 Section 1.1, pp. 4–8)+  , nrm2+  , asum+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (unwrapByteArray, unwrapByteArrayOffset)+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Internal.Kernel (rawDot)++-- | Inner (dot) product of two vectors.+--+-- __GVL4 Reference:__ Algorithm 1.1.1, p. 4.+--+-- Given \(x, y \in \mathbb{R}^{n}\), computes the scalar+--+-- \[+--   c \;=\; x^{T} y \;=\; \sum_{i=1}^{n} x_i \, y_i+-- \]+--+-- ==== Type-safety guarantees+--+-- Both vectors carry the same compile-time dimension @n@, so a+-- length mismatch is a type error.+--+-- ==== Complexity+--+-- \(O(n)\) — exactly \(n\) multiplications and \(n\) additions+-- (or \(n - 1\) additions, depending on the fold seed).+dot :: (KnownNat n, M.Manifest r e, Num e)+    => Vector n r e -> Vector n r e -> e+dot (MkVector x) (MkVector y) =+  M.foldlS (+) 0 $ M.zipWith (*) x y+{-# NOINLINE [1] dot #-}++-- | Specialised raw-array dot product for P Double.+dotP :: forall n. KnownNat n => Vector n M.P Double -> Vector n M.P Double -> Double+dotP (MkVector x) (MkVector y) =+  rawDot (unwrapByteArray x) (unwrapByteArrayOffset x)+         (unwrapByteArray y) (unwrapByteArrayOffset y)+         (dimVal @n)+{-# NOINLINE dotP #-}++{-# RULES "dot/P/Double" forall (x :: Vector n M.P Double)+                                (y :: Vector n M.P Double).+    dot x y = dotP x y #-}++-- | Scale every element of a vector by a scalar.+--+-- __GVL4 Reference:__ Section 1.1, pp. 4–8 (scalar–vector operations).+--+-- Computes+--+-- \[+--   x \;\leftarrow\; \alpha \, x+-- \]+--+-- i.e., each component \(x_i\) is replaced by \(\alpha \, x_i\).+--+-- ==== Type-safety guarantees+--+-- The output vector retains the same compile-time dimension @n@ as the+-- input.+--+-- ==== Complexity+--+-- \(O(n)\) — one multiplication per element.+scal :: (KnownNat n, M.Manifest r e, Num e)+     => e -> Vector n r e -> Vector n r e+scal alpha (MkVector x) = MkVector $ M.compute $ M.map (* alpha) x++-- | Saxpy (Scalar Alpha X Plus Y) — the fundamental vector-update operation.+--+-- __GVL4 Reference:__ Algorithm 1.1.1 (Saxpy), p. 4.+--+-- Given a scalar \(\alpha\) and vectors \(x, y \in \mathbb{R}^{n}\),+-- computes+--+-- \[+--   y \;\leftarrow\; \alpha \, x + y+-- \]+--+-- The Saxpy kernel is the innermost building block of the BLAS hierarchy.+-- Every Gaxpy (Level 2) and matrix–matrix (Level 3) operation can be+-- expressed as a sequence of Saxpy calls (GVL4, Section 1.1, p. 4).+--+-- ==== Type-safety guarantees+--+-- Both input vectors and the result share the same compile-time+-- dimension @n@.+--+-- ==== Complexity+--+-- \(O(n)\) — one fused multiply-add per element.+axpy :: (KnownNat n, M.Manifest r e, Num e)+     => e -> Vector n r e -> Vector n r e -> Vector n r e+axpy alpha (MkVector x) (MkVector y) =+  MkVector $ M.compute $ M.zipWith (\xi yi -> alpha * xi + yi) x y++-- | Euclidean (2-) norm of a vector.+--+-- __GVL4 Reference:__ Section 1.1, pp. 4–8 (vector norms).+--+-- Computes+--+-- \[+--   \lVert x \rVert_2 \;=\; \sqrt{\sum_{i=1}^{n} x_i^{2}}+-- \]+--+-- ==== Type-safety guarantees+--+-- The input vector carries its length @n@ at the type level; the result+-- is a scalar of the same element type.+--+-- ==== Complexity+--+-- \(O(n)\) — one multiply-accumulate per element, plus a single square+-- root.+--+-- /Note:/ This implementation does not perform the scaling trick+-- described in GVL4 (p. 5) to avoid overflow\/underflow for+-- extreme element magnitudes.  For production use on+-- floating-point data with very large or very small entries,+-- consider a two-pass scaled variant.+nrm2 :: (KnownNat n, M.Manifest r e, Floating e)+     => Vector n r e -> e+nrm2 (MkVector x) = sqrt $ M.foldlS (\acc xi -> acc + xi * xi) 0 x++-- | Sum of absolute values — the 1-norm (Manhattan norm) of a vector.+--+-- __GVL4 Reference:__ Section 1.1, pp. 4–8 (vector norms).+--+-- Computes+--+-- \[+--   \lVert x \rVert_1 \;=\; \sum_{i=1}^{n} \lvert x_i \rvert+-- \]+--+-- ==== Type-safety guarantees+--+-- The input vector carries its length @n@ at the type level; the result+-- is a scalar of the same element type.+--+-- ==== Complexity+--+-- \(O(n)\) — one absolute value and one addition per element.+asum :: (KnownNat n, M.Manifest r e, Num e, Ord e)+     => Vector n r e -> e+asum (MkVector x) = M.foldlS (\acc xi -> acc + abs xi) 0 x
+ src/Numeric/LinearAlgebra/Massiv/BLAS/Level2.hs view
@@ -0,0 +1,177 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.BLAS.Level2+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = BLAS Level 2: Matrix–Vector Operations+--+-- This module provides type-safe, dimension-indexed wrappers around the+-- standard BLAS Level 2 kernels.  These are /matrix–vector/ operations+-- whose arithmetic cost is \(O(m \, n)\) for an \(m \times n\) matrix,+-- one level above the \(O(n)\) vector–vector operations of Level 1.+--+-- The central operation is the /Gaxpy/ — Generalized Saxpy — which+-- computes \(y \leftarrow \alpha A x + \beta y\).  It can be viewed+-- as a sequence of Saxpy updates (one per row or one per column),+-- giving rise to two natural loop orderings:+--+-- * Golub, G. H. & Van Loan, C. F. (2013). /Matrix Computations/,+--   4th edition (GVL4). Johns Hopkins University Press.+--   __Chapter 1, Sections 1.1.3–1.1.4__, pp. 8–12.+--+-- Specifically:+--+-- * __Algorithm 1.1.3__ (Row-Oriented Gaxpy, p. 8) — The @i@-th+--   component of the result is a dot product:+--   \(y_i \leftarrow a_i^{T} x + y_i\), where \(a_i^{T}\) is the+--   @i@-th row of \(A\).+--+-- * __Algorithm 1.1.4__ (Column-Oriented Gaxpy, p. 9) — The result+--   vector is updated one column at a time via Saxpy:+--   \(y \leftarrow A(:,\!j) \, x_j + y\), for \(j = 1, \ldots, n\).+--+-- The module also provides the rank-1 outer-product update+-- \(A \leftarrow A + \alpha x y^{T}\) (BLAS @GER@), which is the+-- matrix analogue of the Saxpy at Level 1 and plays a key role in LU+-- factorisation (GVL4, Section 3.2, p. 112).+--+-- == Complexity+--+-- All operations in this module are \(O(m \, n)\) for an+-- \(m \times n\) matrix.+module Numeric.LinearAlgebra.Massiv.BLAS.Level2+  ( -- * Matrix–vector multiply — Gaxpy (Algorithms 1.1.3–1.1.4, GVL4 pp. 8–9)+    gemv+  , matvec+  , matvecP+    -- * Rank-1 update (GVL4 Section 1.1.4, p. 10)+  , ger+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..), Comp(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Internal.Kernel (rawGemv)++-- | General matrix–vector multiply (BLAS @GEMV@).+--+-- __GVL4 Reference:__ Algorithms 1.1.3 (Row Gaxpy, p. 8) and 1.1.4+-- (Column Gaxpy, p. 9).+--+-- Given \(A \in \mathbb{R}^{m \times n}\),+-- \(x \in \mathbb{R}^{n}\), \(y \in \mathbb{R}^{m}\), and scalars+-- \(\alpha, \beta\), computes+--+-- \[+--   y \;\leftarrow\; \alpha \, A \, x \;+\; \beta \, y+-- \]+--+-- The implementation uses a /row-oriented/ traversal (Algorithm 1.1.3):+-- for each row \(i\) the dot product \(a_i^{T} x\) is formed, scaled by+-- \(\alpha\), and added to \(\beta \, y_i\).+--+-- ==== Type-safety guarantees+--+-- * \(A\) is @m x n@, \(x\) is @n@, \(y\) and the result are @m@.+-- * The shared inner dimension @n@ is enforced at compile time, so a+--   dimension mismatch is a type error.+--+-- ==== Complexity+--+-- \(O(m \, n)\) — two floating-point operations per matrix entry+-- (one multiply, one add in the inner product), plus \(O(m)\) work+-- for the \(\alpha / \beta\) scaling.+gemv :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+     => e -> Matrix m n r e -> Vector n r e -> e -> Vector m r e -> Vector m r e+gemv alpha mat x beta y =+  let r = dimVal @m+      c = dimVal @n+  in makeVector @m @r $ \i ->+    let axi = foldl (\acc j -> acc + (mat ! (i, j)) * (x !. j)) 0 [0..c-1]+    in alpha * axi + beta * (y !. i)++-- | Simple matrix–vector multiply (specialisation of 'gemv').+--+-- __GVL4 Reference:__ Algorithm 1.1.3 (Row Gaxpy, p. 8), with+-- \(\alpha = 1\) and \(\beta = 0\).+--+-- Given \(A \in \mathbb{R}^{m \times n}\) and+-- \(x \in \mathbb{R}^{n}\), computes+--+-- \[+--   y \;=\; A \, x+-- \]+--+-- This is a convenience wrapper equivalent to @'gemv' 1 a x 0 zero@+-- but avoids allocating or requiring an initial @y@ vector.+--+-- ==== Type-safety guarantees+--+-- * \(A\) is @m x n@, \(x\) is @n@, the result is @m@.+-- * The inner dimension @n@ is checked at compile time.+--+-- ==== Complexity+--+-- \(O(m \, n)\).+matvec :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+       => Matrix m n r e -> Vector n r e -> Vector m r e+matvec mat x =+  let c = dimVal @n+  in makeVector @m @r $ \i ->+    foldl (\acc j -> acc + (mat ! (i, j)) * (x !. j)) 0 [0..c-1]+{-# NOINLINE [1] matvec #-}++-- | Specialised raw-array matvec for P Double.+matvecP :: forall m n. (KnownNat m, KnownNat n)+              => Matrix m n M.P Double -> Vector n M.P Double -> Vector m M.P Double+matvecP (MkMatrix a) (MkVector x) =+  createVector @m @M.P $ \mc ->+    rawGemv (unwrapByteArray a) (unwrapByteArrayOffset a) (dimVal @n)+            (unwrapByteArray x) (unwrapByteArrayOffset x)+            (unwrapMutableByteArray mc) (unwrapMutableByteArrayOffset mc)+            (dimVal @m)+{-# NOINLINE matvecP #-}++{-# RULES "matvec/P/Double" forall (a :: Matrix m n M.P Double)+                                   (x :: Vector n M.P Double).+    matvec a x = matvecP a x #-}++-- | Rank-1 update — outer product (BLAS @GER@).+--+-- __GVL4 Reference:__ Section 1.1.4, p. 10.  The rank-1 update is+-- the matrix-level analogue of the Saxpy and appears as the inner+-- kernel in outer-product formulations of LU factorisation+-- (GVL4 Section 3.2, Algorithm 3.2.1, p. 112).+--+-- Given \(x \in \mathbb{R}^{m}\), \(y \in \mathbb{R}^{n}\),+-- \(A \in \mathbb{R}^{m \times n}\), and a scalar \(\alpha\),+-- computes+--+-- \[+--   A \;\leftarrow\; A \;+\; \alpha \, x \, y^{T}+-- \]+--+-- Equivalently, each entry is updated as+-- \(a_{ij} \leftarrow a_{ij} + \alpha \, x_i \, y_j\).+--+-- ==== Type-safety guarantees+--+-- * \(x\) is @m@, \(y\) is @n@, \(A\) and the result are @m x n@.+-- * All dimension constraints are enforced at compile time.+--+-- ==== Complexity+--+-- \(O(m \, n)\) — one fused multiply-add per matrix entry.+ger :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+    => e -> Vector m r e -> Vector n r e -> Matrix m n r e -> Matrix m n r e+ger alpha x y mat =+  makeMatrix @m @n @r $ \i j ->+    (mat ! (i, j)) + alpha * (x !. i) * (y !. j)
+ src/Numeric/LinearAlgebra/Massiv/BLAS/Level3.hs view
@@ -0,0 +1,405 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.BLAS.Level3+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = BLAS Level 3: Matrix–Matrix Operations+--+-- This module provides type-safe, dimension-indexed wrappers around the+-- standard BLAS Level 3 kernels.  These are /matrix–matrix/ operations+-- whose arithmetic cost is \(O(m \, n \, k)\) for an+-- \(m \times k\) by \(k \times n\) multiply, one level above the+-- \(O(m \, n)\) matrix–vector operations of Level 2.+--+-- The algorithms implemented here correspond to the six loop orderings+-- of the triple-loop matrix multiplication described in:+--+-- * Golub, G. H. & Van Loan, C. F. (2013). /Matrix Computations/,+--   4th edition (GVL4). Johns Hopkins University Press.+--   __Chapter 1, Sections 1.1.5–1.1.8__, pp. 12–18.+--+-- Specifically:+--+-- * __Algorithm 1.1.5__ (ijk variant, p. 12) — The "row-oriented+--   inner-product" form.  For each entry \(c_{ij}\) the inner product+--   of row \(i\) of \(A\) with column \(j\) of \(B\) is computed.+--   This is the variant implemented by 'gemm', 'matMul', and+--   'matMulComp'.+--+-- * __Algorithms 1.1.6–1.1.8__ (pp. 13–15) — The jki (Gaxpy),+--   kji (outer-product), and other orderings.  These alternatives+--   differ in data-access pattern but compute the same result.  The+--   present implementation uses the ijk ordering; cache-oblivious or+--   blocked variants can be added in the future.+--+-- The module also provides elementary matrix arithmetic (addition,+-- subtraction, scaling, transpose) and a triangular matrix–matrix+-- multiply ('trmmLeft') that exploits the triangular structure to+-- halve the work.+--+-- == Complexity+--+-- * 'gemm', 'matMul', 'matMulComp': \(O(m \, k \, n)\).+-- * 'transpose', 'mAdd', 'mSub', 'mScale': \(O(m \, n)\).+-- * 'trmmLeft': \(O(n^{3}/2)\) (triangular, in-place structure).+module Numeric.LinearAlgebra.Massiv.BLAS.Level3+  ( -- * Matrix–matrix multiply (Algorithm 1.1.5, GVL4 p. 12)+    gemm+  , matMul+  , matMulP+  , matMulPPar+  , matMulComp+    -- * Elementary matrix operations (GVL4 Section 1.1, pp. 4–18)+  , transpose+  , transposeP+  , matMulAtAP+  , mAdd+  , mSub+  , mScale+    -- * Triangular matrix multiply (GVL4 Section 1.1.8, p. 15)+  , trmmLeft+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..), Comp(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)+import GHC.Exts (Int(..), isTrue#, (>=#), (*#), (+#))+import GHC.Prim+import GHC.ST (ST(..))+import Data.Array.Byte (MutableByteArray(..))+import Data.Primitive.ByteArray (ByteArray(..), newByteArray, unsafeFreezeByteArray)+import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Control.Concurrent (forkOn, newEmptyMVar, putMVar, takeMVar, getNumCapabilities)+import System.IO.Unsafe (unsafePerformIO)+import GHC.IO (stToIO)+import Numeric.LinearAlgebra.Massiv.Internal.Kernel (rawGemmKernel, rawGemmBISlice, rawGemmBIBJSlice, rawSyrkLowerKernel)++-- | Block size for cache-tiled GEMM (generic fallback path).+gemmBlockSize :: Int+gemmBlockSize = 32+{-# INLINE gemmBlockSize #-}++-- | General matrix multiply (BLAS @GEMM@).+--+-- __GVL4 Reference:__ Algorithm 1.1.5 (ijk matrix multiply), p. 12.+--+-- Given \(A \in \mathbb{R}^{m \times k}\),+-- \(B \in \mathbb{R}^{k \times n}\),+-- \(C \in \mathbb{R}^{m \times n}\), and scalars+-- \(\alpha, \beta\), computes+--+-- \[+--   C \;\leftarrow\; \alpha \, A \, B \;+\; \beta \, C+-- \]+gemm :: forall m k n r e. (KnownNat m, KnownNat k, KnownNat n, M.Manifest r e, Num e)+     => e -> Matrix m k r e -> Matrix k n r e -> e -> Matrix m n r e -> Matrix m n r e+gemm alpha a b beta c =+  let mm = dimVal @m+      kk = dimVal @k+      nn = dimVal @n+      bs = gemmBlockSize+  in createMatrix @m @n @r $ \mc -> do+    -- Initialize C with β·C₀+    mapM_ (\i -> mapM_ (\j -> do+      M.write_ mc (i :. j) (beta * (c ! (i, j)))+      ) [0..nn-1]) [0..mm-1]+    -- Tiled ikj loop: for each block triple, accumulate α·A·B+    let go_bi bi = do+          let iEnd = min (bi + bs) mm+          let go_bk bk = do+                let kEnd = min (bk + bs) kk+                let go_bj bj = do+                      let jEnd = min (bj + bs) nn+                      -- Inner micro-kernel: ikj within the block+                      mapM_ (\i -> mapM_ (\l -> do+                        let aik = alpha * (a ! (i, l))+                        mapM_ (\j -> do+                          cij <- M.readM mc (i :. j)+                          M.write_ mc (i :. j) (cij + aik * (b ! (l, j)))+                          ) [bj..jEnd-1]+                        ) [bk..kEnd-1]) [bi..iEnd-1]+                mapM_ go_bj [0, bs .. nn-1]+          mapM_ go_bk [0, bs .. kk-1]+    mapM_ go_bi [0, bs .. mm-1]++-- | Simple matrix multiply (specialisation of 'gemm').+--+-- __GVL4 Reference:__ Algorithm 1.1.5 (ijk matrix multiply), p. 12,+-- with \(\alpha = 1\) and \(\beta = 0\).+--+-- Given \(A \in \mathbb{R}^{m \times k}\) and+-- \(B \in \mathbb{R}^{k \times n}\), computes+--+-- \[+--   C \;=\; A \, B+-- \]+matMul :: forall m k n r e. (KnownNat m, KnownNat k, KnownNat n, M.Manifest r e, Num e)+       => Matrix m k r e -> Matrix k n r e -> Matrix m n r e+matMul a b = matMulGeneric a b+{-# NOINLINE [1] matMul #-}++-- | Generic fallback for non-P or non-Double representations.+matMulGeneric :: forall m k n r e. (KnownNat m, KnownNat k, KnownNat n, M.Manifest r e, Num e)+              => Matrix m k r e -> Matrix k n r e -> Matrix m n r e+matMulGeneric a b =+  let mm = dimVal @m+      kk = dimVal @k+      nn = dimVal @n+      bs = gemmBlockSize+  in createMatrix @m @n @r $ \mc -> do+    -- Initialize C to zero+    mapM_ (\i -> mapM_ (\j ->+      M.write_ mc (i :. j) 0+      ) [0..nn-1]) [0..mm-1]+    -- Tiled ikj loop+    let go_bi bi = do+          let iEnd = min (bi + bs) mm+          let go_bk bk = do+                let kEnd = min (bk + bs) kk+                let go_bj bj = do+                      let jEnd = min (bj + bs) nn+                      mapM_ (\i -> mapM_ (\l -> do+                        let aik = a ! (i, l)+                        mapM_ (\j -> do+                          cij <- M.readM mc (i :. j)+                          M.write_ mc (i :. j) (cij + aik * (b ! (l, j)))+                          ) [bj..jEnd-1]+                        ) [bk..kEnd-1]) [bi..iEnd-1]+                mapM_ go_bj [0, bs .. nn-1]+          mapM_ go_bk [0, bs .. kk-1]+    mapM_ go_bi [0, bs .. mm-1]++-- | Specialised raw-array GEMM for @P Double@.+-- Bypasses massiv's per-element abstraction and uses raw ByteArray# primops+-- with AVX2 DoubleX4# SIMD for the inner kernel.+matMulP :: forall m k n. (KnownNat m, KnownNat k, KnownNat n)+        => Matrix m k M.P Double -> Matrix k n M.P Double -> Matrix m n M.P Double+matMulP (MkMatrix arrA) (MkMatrix arrB) =+  let mm = dimVal @m+      kk = dimVal @k+      nn = dimVal @n+      baA = unwrapByteArray arrA+      offA = unwrapByteArrayOffset arrA+      baB = unwrapByteArray arrB+      offB = unwrapByteArrayOffset arrB+  in createMatrix @m @n @M.P $ \mc -> do+    -- Zero-initialise C+    let mbaC = unwrapMutableByteArray mc+        offC = unwrapMutableByteArrayOffset mc+        !(I# mm#) = mm+        !(I# nn#) = nn+    ST $ \s0 ->+      let go i s+            | isTrue# (i >=# (mm# *# nn#)) = s+            | otherwise = case writeDoubleArray# (unMBA# mbaC) (unI offC +# i) 0.0## s of+                            s' -> go (i +# 1#) s'+      in (# go 0# s0, () #)+    -- Run the raw SIMD kernel+    rawGemmKernel baA offA baB offB mbaC offC mm kk nn+{-# NOINLINE matMulP #-}++-- | Parallel specialised GEMM for @P Double@.+-- Uses 2D grid partitioning when numThreads >= 4 and both dimensions are large+-- enough (min(m,n) >= 128), otherwise falls back to 1D row partitioning.+-- 2D partitioning reduces per-thread B cache traffic by a factor of sqrt(p).+-- Falls back to single-threaded 'matMulP' when @getNumCapabilities == 1@.+matMulPPar :: forall m k n. (KnownNat m, KnownNat k, KnownNat n)+           => Matrix m k M.P Double -> Matrix k n M.P Double -> Matrix m n M.P Double+matMulPPar a b = unsafePerformIO $ do+  let !mm = dimVal @m+      !kk = dimVal @k+      !nn = dimVal @n+      !baA = unwrapByteArray (unMatrix a)+      !offA = unwrapByteArrayOffset (unMatrix a)+      !baB = unwrapByteArray (unMatrix b)+      !offB = unwrapByteArrayOffset (unMatrix b)+  caps <- getNumCapabilities+  -- Adaptive thread count: ensure each thread gets enough rows to+  -- amortize fork/join overhead (minimum 16 rows per thread).+  let !minRowsPerThread = 16+      !maxThreads = max 1 (mm `div` minRowsPerThread)+      !numThreads = min caps (min mm maxThreads)+  if numThreads <= 1+    then pure (matMulP a b)+    else do+      -- Allocate mutable C, zero-initialise+      mc <- stToIO $ M.newMArray (Sz (mm :. nn)) (0 :: Double)+      let !mbaC = unwrapMutableByteArray mc+          !offC = unwrapMutableByteArrayOffset mc+      -- Choose 1D or 2D decomposition+      let !use2D = numThreads >= 4 && mm >= 128 && nn >= 128+      if use2D+        then do+          -- 2D grid: pr rows × pc columns, pr * pc = numThreads+          -- Choose pr, pc to balance aspect ratio: pr/pc ≈ mm/nn+          let (pr, pc) = gridDims numThreads mm nn+              !rChunk = (mm + pr - 1) `div` pr+              !cChunk = (nn + pc - 1) `div` pc+          mvars <- sequence+            [ do let !biStart = tr * rChunk+                     !biEnd = min (biStart + rChunk) mm+                     !bjStart = tc * cChunk+                     !bjEnd = min (bjStart + cChunk) nn+                 mv <- newEmptyMVar+                 _ <- forkOn (tr * pc + tc) $ do+                   stToIO $ rawGemmBIBJSlice baA offA baB offB mbaC offC+                              biStart biEnd bjStart bjEnd mm kk nn+                   putMVar mv ()+                 pure mv+            | tr <- [0..pr-1], tc <- [0..pc-1]+            ]+          mapM_ takeMVar mvars+        else do+          -- 1D row partitioning (original path)+          let !chunkSize = (mm + numThreads - 1) `div` numThreads+          mvars <- mapM (\t -> do+            let !biStart = t * chunkSize+                !biEnd = min (biStart + chunkSize) mm+            mv <- newEmptyMVar+            _ <- forkOn t $ do+              stToIO $ rawGemmBISlice baA offA baB offB mbaC offC biStart biEnd mm kk nn+              putMVar mv ()+            pure mv+            ) [0..numThreads-1]+          mapM_ takeMVar mvars+      -- Freeze and wrap+      arr <- stToIO $ M.freezeS mc+      pure (MkMatrix arr)+{-# NOINLINE matMulPPar #-}++-- | Compute 2D grid dimensions (pr × pc) for p threads such that+-- pr * pc = p and the aspect ratio pr/pc approximates m/n.+gridDims :: Int -> Int -> Int -> (Int, Int)+gridDims p m n =+  let sqrtP = floor (sqrt (fromIntegral p :: Double)) :: Int+      -- Try all factorizations of p and pick the one with best aspect match+      factors = [(i, p `div` i) | i <- [1..sqrtP], p `mod` i == 0]+      targetRatio = fromIntegral m / fromIntegral (max 1 n) :: Double+      score (pr, pc) = abs (fromIntegral pr / fromIntegral (max 1 pc) - targetRatio)+      best = minimumBy (\a' b' -> compare (score a') (score b')) factors+      -- Also consider the transpose (pc, pr)+      bestT = let (pr, pc) = best in if score (pc, pr) < score best then (pc, pr) else best+  in bestT+  where+    minimumBy _ [x] = x+    minimumBy f (x:xs) = foldl' (\a' b' -> if f a' b' == GT then b' else a') x xs+    minimumBy _ [] = (1, p)  -- fallback++{-# RULES+"matMul/P/Double" forall (a :: Matrix m k M.P Double)+                         (b :: Matrix k n M.P Double).+    matMul a b = matMulP a b+  #-}++-- | Matrix multiply with explicit computation strategy.+matMulComp :: forall m k n r e. (KnownNat m, KnownNat k, KnownNat n, M.Manifest r e, Num e)+           => Comp -> Matrix m k r e -> Matrix k n r e -> Matrix m n r e+matMulComp comp a b =+  case comp of+    Seq -> matMul a b+    _   -> -- For parallel: use delayed array with ikj-reordered inner product+           let kk = dimVal @k+           in makeMatrixComp @m @n @r comp $ \i j ->+             foldl' (\acc l -> acc + (a ! (i, l)) * (b ! (l, j))) 0 [0..kk-1]++-- | Matrix transpose.+transpose :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+          => Matrix m n r e -> Matrix n m r e+transpose (MkMatrix arr) = MkMatrix $ M.compute $ M.transposeInner arr++-- | P-specialised raw-primop matrix transpose.+-- Avoids per-element overhead of massiv's delayed transpose.+transposeP :: forall m n. (KnownNat m, KnownNat n)+           => Matrix m n M.P Double -> Matrix n m M.P Double+transposeP (MkMatrix a) =+  let !mm = dimVal @m+      !nn = dimVal @n+      !(I# mm#) = mm+      !(I# nn#) = nn+      !(ByteArray ba#) = unwrapByteArray a+      !(I# off#) = unwrapByteArrayOffset a+  in createMatrix @n @m @M.P $ \mu ->+    let !(MutableByteArray mba#) = unwrapMutableByteArray mu+        !(I# offR#) = unwrapMutableByteArrayOffset mu+    in ST $ \s0 ->+      -- Iterate source rows (sequential read, strided write)+      let goRow j s+            | isTrue# (j >=# mm#) = s+            | otherwise =+                let goCol i s1+                      | isTrue# (i >=# nn#) = s1+                      | otherwise =+                          let !src = off# +# j *# nn# +# i+                              !dst = offR# +# i *# mm# +# j+                              !v = indexDoubleArray# ba# src+                          in case writeDoubleArray# mba# dst v s1 of+                               s2 -> goCol (i +# 1#) s2+                in goRow (j +# 1#) (goCol 0# s)+      in (# goRow 0# s0, () #)+{-# NOINLINE transposeP #-}++-- | P-specialised A^T * A without materialising A^T.+-- Computes C = A^T * A using a fused DSYRK kernel that processes only the+-- lower triangle (halving flops) and mirrors to the upper triangle.+-- Avoids materialising A^T entirely — one allocation, no transpose pass.+matMulAtAP :: forall m n. (KnownNat m, KnownNat n)+           => Matrix m n M.P Double -> Matrix n n M.P Double+matMulAtAP (MkMatrix arrA) =+  let mm = dimVal @m+      nn = dimVal @n+      baA = unwrapByteArray arrA+      offA = unwrapByteArrayOffset arrA+  in createMatrix @n @n @M.P $ \mc -> do+    -- Zero-initialise C (n×n)+    let mbaC = unwrapMutableByteArray mc+        offC = unwrapMutableByteArrayOffset mc+        !(I# nn#) = nn+    ST $ \s0 ->+      let go i s+            | isTrue# (i >=# (nn# *# nn#)) = s+            | otherwise = case writeDoubleArray# (unMBA# mbaC) (unI offC +# i) 0.0## s of+                            s' -> go (i +# 1#) s'+      in (# go 0# s0, () #)+    -- Run the fused SYRK kernel: C = A^T * A (lower triangle + mirror)+    rawSyrkLowerKernel baA offA mbaC offC mm nn+{-# NOINLINE matMulAtAP #-}++-- | Element-wise matrix addition.+mAdd :: (KnownNat m, KnownNat n, M.Manifest r e, Num e)+     => Matrix m n r e -> Matrix m n r e -> Matrix m n r e+mAdd (MkMatrix a) (MkMatrix b) = MkMatrix $ M.compute $ M.zipWith (+) a b++-- | Element-wise matrix subtraction.+mSub :: (KnownNat m, KnownNat n, M.Manifest r e, Num e)+     => Matrix m n r e -> Matrix m n r e -> Matrix m n r e+mSub (MkMatrix a) (MkMatrix b) = MkMatrix $ M.compute $ M.zipWith (-) a b++-- | Scalar–matrix multiply.+mScale :: (KnownNat m, KnownNat n, M.Manifest r e, Num e)+       => e -> Matrix m n r e -> Matrix m n r e+mScale alpha (MkMatrix a) = MkMatrix $ M.compute $ M.map (* alpha) a++-- | Left-multiply by a lower-triangular matrix (BLAS @TRMM@, left side).+trmmLeft :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+         => Matrix n n r e -> Matrix n n r e -> Matrix n n r e+trmmLeft l b =+  let nn = dimVal @n+  in makeMatrix @n @n @r $ \i j ->+    foldl' (\acc k -> acc + (l ! (i, k)) * (b ! (k, j))) 0 [0..min i (nn-1)]++-- Helpers to unwrap newtypes for raw primop access+unMBA# :: MutableByteArray s -> MutableByteArray# s+unMBA# (MutableByteArray mba) = mba+{-# INLINE unMBA# #-}++unI :: Int -> Int#+unI (I# i) = i+{-# INLINE unI #-}
+ src/Numeric/LinearAlgebra/Massiv/Eigen/Hessenberg.hs view
@@ -0,0 +1,127 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Reduction of a general square matrix to upper Hessenberg form via+-- Householder similarity transformations, following Golub & Van Loan,+-- /Matrix Computations/, 4th edition (GVL4), Section 7.4, pp. 383--393.+--+-- An upper Hessenberg matrix \(H\) satisfies \(h_{ij} = 0\) for+-- \(i > j + 1\); that is, all entries below the first subdiagonal are zero.+-- Given \(A \in \mathbb{R}^{n \times n}\), the algorithm computes an+-- orthogonal matrix \(Q\) (accumulated as a product of \(n - 2\) Householder+-- reflectors) such that+--+-- \[+--   A = Q H Q^T+-- \]+--+-- This is the standard first phase in eigenvalue algorithms (e.g. the+-- implicit QR algorithm in "Numeric.LinearAlgebra.Massiv.Eigen.Schur"),+-- because QR steps preserve Hessenberg structure and are much cheaper on a+-- Hessenberg matrix than on a full matrix.+--+-- __Algorithm:__ Householder reduction to Hessenberg form (GVL4+-- Algorithm 7.4.2, p. 387).+--+-- __Complexity:__ \(\frac{10}{3} n^3\) floating-point operations.+module Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+  ( -- * Hessenberg reduction (Algorithm 7.4.2)+    hessenberg+  , hessenbergInPlace+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Reduce a general matrix to upper Hessenberg form (GVL4 Algorithm 7.4.2,+-- p. 387).+--+-- Computes the factorisation+--+-- \[+--   A = Q \, H \, Q^T+-- \]+--+-- where \(Q\) is orthogonal (a product of \(n - 2\) Householder reflectors)+-- and \(H\) is upper Hessenberg, i.e. \(h_{ij} = 0\) for \(i > j + 1\).+--+-- At step \(k\) the algorithm determines a Householder reflector+-- \(P_k = I - \beta_k v_k v_k^T\) that zeroes entries \(k+2, \ldots, n\) in+-- column \(k\) of the current matrix, and applies it as a similarity+-- transformation \(H \leftarrow P_k H P_k\).+--+-- __Complexity:__ \(\frac{10}{3} n^3\) flops (GVL4, p. 389).+--+-- Returns @(Q, H)@.+hessenberg :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+           => Matrix n n r e -> (Matrix n n r e, Matrix n n r e)+hessenberg a =+  let nn = dimVal @n+      go :: Int -> Matrix n n r e -> Matrix n n r e -> (Matrix n n r e, Matrix n n r e)+      go k q_ h_+        | k >= nn - 2 = (q_, h_)+        | otherwise =+          -- Compute Householder to zero out h(k+2:n, k)+          let x0 = h_ ! (k+1, k)+              sigma = foldl' (\acc i -> acc + (h_ ! (i, k)) * (h_ ! (i, k))) 0 [k+2..nn-1]+          in if sigma == 0 && x0 >= 0+             then go (k + 1) q_ h_+             else+              let mu = sqrt (x0 * x0 + sigma)+                  v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                  beta = 2 * v0 * v0 / (sigma + v0 * v0)+                  -- v is zero for indices 0..k, v(k+1)=1, v(i) = h(i,k)/v0 for i > k+1+                  v = makeVector @n @r $ \i ->+                    if i <= k then 0+                    else if i == k + 1 then 1+                    else (h_ ! (i, k)) / v0+                  -- H ← (I - β·v·vᵀ)·H·(I - β·v·vᵀ) = similarity transform+                  -- First: H ← (I - β·v·vᵀ)·H+                  h1 = applyFromLeft v beta h_+                  -- Then: H ← H·(I - β·v·vᵀ)+                  h2 = applyFromRight h1 v beta+                  -- Q ← Q·(I - β·v·vᵀ)+                  q_new = applyFromRight q_ v beta+              in go (k + 1) q_new h2++      q0 = identityMatrix @n @r+  in go 0 q0 a++-- Helper: apply Householder from left+applyFromLeft :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+              => Vector n r e -> e -> Matrix n n r e -> Matrix n n r e+applyFromLeft v beta h =+  let nn = dimVal @n+  in makeMatrix @n @n @r $ \i j ->+    let wj = beta * foldl' (\acc k -> acc + (v !. k) * (h ! (k, j))) 0 [0..nn-1]+    in (h ! (i, j)) - (v !. i) * wj++-- Helper: apply Householder from right+applyFromRight :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+               => Matrix n n r e -> Vector n r e -> e -> Matrix n n r e+applyFromRight h v beta =+  let nn = dimVal @n+  in makeMatrix @n @n @r $ \i j ->+    let wi = beta * foldl' (\acc k -> acc + (h ! (i, k)) * (v !. k)) 0 [0..nn-1]+    in (h ! (i, j)) - wi * (v !. j)++-- | Compute only the upper Hessenberg matrix \(H\), discarding the+-- orthogonal factor \(Q\).+--+-- This is a convenience wrapper around 'hessenberg' that returns only the+-- second component of the pair.  Use this when only the Hessenberg form is+-- needed and the transformation matrix is not required (e.g. when computing+-- eigenvalues but not eigenvectors).+hessenbergInPlace :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+                  => Matrix n n r e -> Matrix n n r e+hessenbergInPlace a = snd (hessenberg a)
+ src/Numeric/LinearAlgebra/Massiv/Eigen/Power.hs view
@@ -0,0 +1,182 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Eigen.Power+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Power iteration methods for computing individual eigenvalue\/eigenvector+-- pairs of a general square matrix.+--+-- This module implements three iterative projection techniques drawn from+-- Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4), Section 7.3,+-- pp. 372--382:+--+-- * __Power method__ (Algorithm 7.3.3, p. 375) — converges to the dominant+--   eigenpair at a rate governed by the ratio \(|\lambda_2 / \lambda_1|\) per+--   iteration, where \(\lambda_1\) is the eigenvalue of largest modulus.+--+-- * __Inverse iteration__ (Section 7.3.1, p. 377) — given a shift \(\mu\),+--   converges to the eigenvalue closest to \(\mu\) by applying the power+--   method to \((A - \mu I)^{-1}\).+--+-- * __Rayleigh quotient iteration__ (Section 7.3.2, p. 379) — an adaptive+--   variant of inverse iteration in which the shift is updated at every step+--   to equal the current Rayleigh quotient.  For symmetric matrices this+--   achieves /cubic/ convergence; for general matrices the convergence is+--   /quadratic/.+--+-- All three routines return an approximate eigenvalue \(\lambda\) and its+-- associated eigenvector \(q\) satisfying \(Aq \approx \lambda q\).+module Numeric.LinearAlgebra.Massiv.Eigen.Power+  ( -- * Power method (Algorithm 7.3.3)+    powerMethod+    -- * Inverse iteration (Section 7.3.1)+  , inverseIteration+    -- * Rayleigh quotient iteration (Section 7.3.2)+  , rayleighQuotient+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1 (dot, scal, nrm2)+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.Solve.LU (luSolve)++-- | Power method for computing the dominant eigenpair (GVL4 Algorithm 7.3.3,+-- p. 375).+--+-- Given a square matrix \(A \in \mathbb{R}^{n \times n}\) with eigenvalues+-- ordered \(|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|\), the+-- power method generates a sequence of vectors+--+-- \[+--   z_k = A q_{k-1}, \qquad q_k = z_k / \|z_k\|_2+-- \]+--+-- that converges to the eigenvector associated with \(\lambda_1\).  The+-- corresponding eigenvalue is estimated via the Rayleigh quotient+-- \(\lambda \approx q_k^T A q_k\).+--+-- __Convergence rate:__ the error contracts by a factor of+-- \(|\lambda_2 / \lambda_1|\) per iteration (GVL4, p. 375).  The method+-- therefore requires a dominant eigenvalue that is well-separated from the+-- rest of the spectrum.+--+-- Returns @(eigenvalue, eigenvector)@ once the eigenvalue estimate changes by+-- less than the given tolerance, or after the specified number of iterations.+powerMethod :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+            => Matrix n n r e+            -> Vector n r e    -- ^ Initial guess \(q_0\) (should be unit norm)+            -> Int             -- ^ Maximum iterations+            -> e               -- ^ Convergence tolerance+            -> (e, Vector n r e)+powerMethod a q0 maxIter tol = go 0 q0 0+  where+    go :: Int -> Vector n r e -> e -> (e, Vector n r e)+    go iter q prevLambda+      | iter >= maxIter = (prevLambda, q)+      | otherwise =+        let z = matvec a q                        -- z = A·q+            znorm = nrm2 z                        -- ‖z‖₂+            qNew = scal (1 / znorm) z             -- q = z / ‖z‖₂+            lambda = dot qNew (matvec a qNew)     -- λ = qᵀAq (Rayleigh quotient)+        in if abs (lambda - prevLambda) < tol+           then (lambda, qNew)+           else go (iter + 1) qNew lambda++-- | Inverse iteration for computing the eigenpair closest to a given shift+-- (GVL4 Section 7.3.1, p. 377).+--+-- Given a shift \(\mu\) that approximates some eigenvalue of \(A\), inverse+-- iteration applies the power method to the matrix \((A - \mu I)^{-1}\).+-- Each step solves the linear system+--+-- \[+--   (A - \mu I)\, z_k = q_{k-1}, \qquad q_k = z_k / \|z_k\|_2+-- \]+--+-- and converges to the eigenvalue \(\lambda_j\) that minimises+-- \(|\lambda_j - \mu|\).  The convergence rate is+-- \(|\lambda_j - \mu| / |\lambda_i - \mu|\) per iteration, where+-- \(\lambda_i\) is the second-closest eigenvalue to \(\mu\).+--+-- The eigenvalue estimate is refined at each step via the Rayleigh quotient+-- \(\lambda \approx q_k^T A q_k\) (using the /original/ matrix \(A\)).+--+-- Returns @(eigenvalue, eigenvector)@.+inverseIteration :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+                 => Matrix n n r e+                 -> e               -- ^ Shift \(\mu\)+                 -> Vector n r e    -- ^ Initial guess+                 -> Int             -- ^ Maximum iterations+                 -> e               -- ^ Convergence tolerance+                 -> (e, Vector n r e)+inverseIteration a mu q0 maxIter tol = go 0 q0 0+  where+    aShifted = makeMatrix @n @n @r $ \i j ->+      if i == j then (a ! (i, j)) - mu else a ! (i, j)++    go :: Int -> Vector n r e -> e -> (e, Vector n r e)+    go iter q prevLambda+      | iter >= maxIter = (prevLambda, q)+      | otherwise =+        let z = luSolve aShifted q                -- Solve (A - μI)z = q+            znorm = nrm2 z+            qNew = scal (1 / znorm) z+            lambda = dot qNew (matvec a qNew)     -- Rayleigh quotient with original A+        in if abs (lambda - prevLambda) < tol+           then (lambda, qNew)+           else go (iter + 1) qNew lambda++-- | Rayleigh quotient iteration (GVL4 Section 7.3.2, p. 379).+--+-- An adaptive variant of inverse iteration in which the shift \(\mu_k\) is+-- set equal to the current Rayleigh quotient at every step:+--+-- \[+--   \mu_k = q_k^T A q_k, \qquad (A - \mu_k I)\, z_{k+1} = q_k, \qquad+--   q_{k+1} = z_{k+1} / \|z_{k+1}\|_2+-- \]+--+-- __Convergence:__+--+-- * For /symmetric/ matrices \(A = A^T\), the iteration converges+--   /cubically/ — the residual \(\|Aq - \lambda q\|\) is cubed at each step+--   (GVL4, p. 379).+-- * For general (non-symmetric) matrices, convergence is /quadratic/.+--+-- Because the shift changes at each iteration, a fresh LU factorisation of+-- \(A - \mu_k I\) is computed every step.  Despite this extra cost the rapid+-- convergence usually makes Rayleigh quotient iteration the method of choice+-- when a good initial vector is available.+--+-- Returns @(eigenvalue, eigenvector)@.+rayleighQuotient :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+                 => Matrix n n r e+                 -> Vector n r e    -- ^ Initial guess+                 -> Int             -- ^ Maximum iterations+                 -> e               -- ^ Convergence tolerance+                 -> (e, Vector n r e)+rayleighQuotient a q0 maxIter tol = go 0 q0 (dot q0 (matvec a q0))+  where+    go :: Int -> Vector n r e -> e -> (e, Vector n r e)+    go iter q lambda+      | iter >= maxIter = (lambda, q)+      | otherwise =+        let nn = dimVal @n+            aShifted = makeMatrix @n @n @r $ \i j ->+              if i == j then (a ! (i, j)) - lambda else a ! (i, j)+            z = luSolve aShifted q+            znorm = nrm2 z+            qNew = scal (1 / znorm) z+            lambdaNew = dot qNew (matvec a qNew)+        in if abs (lambdaNew - lambda) < tol+           then (lambdaNew, qNew)+           else go (iter + 1) qNew lambdaNew
+ src/Numeric/LinearAlgebra/Massiv/Eigen/SVD.hs view
@@ -0,0 +1,1992 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE BangPatterns #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE UnboxedTuples #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Eigen.SVD+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Singular Value Decomposition (SVD) of a general real matrix, following+-- Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4), Section 8.6,+-- pp. 498--512.+--+-- __Theorem 8.6.1 (SVD Existence, p. 499):__ For any+-- \(A \in \mathbb{R}^{m \times n}\) with \(m \geq n\) there exist orthogonal+-- matrices \(U \in \mathbb{R}^{m \times m}\) and+-- \(V \in \mathbb{R}^{n \times n}\) such that+--+-- \[+--   A = U \, \Sigma \, V^T, \qquad+--   \Sigma = \mathrm{diag}(\sigma_1, \ldots, \sigma_n),+--   \qquad \sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_n \geq 0+-- \]+--+-- The \(\sigma_i\) are the /singular values/ of \(A\) and equal the+-- non-negative square roots of the eigenvalues of \(A^T A\).+module Numeric.LinearAlgebra.Massiv.Eigen.SVD+  ( -- * Full SVD+    svd+  , svdP+  , svdAtAP+  , svdGKP+    -- * Singular values only+  , singularValues+  , singularValuesP+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..), newByteArray,+                                 unsafeFreezeByteArray)+import GHC.TypeNats (KnownNat)+import Control.Monad (forM_, when)+import Control.Monad.ST (runST)+import Data.List (sortBy)+import Data.Ord ()+import GHC.Exts+import GHC.ST (ST(..))++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose, matMulAtAP)+-- matvecP no longer needed: U-matrix now computed via single GEMM+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+  ( symmetricEigen, symmetricEigenP, symmetricEigenPDC+  -- D&C secular equation infrastructure (reused for bidiagonal SVD)+  , secularSolve, deflatePartition, dcEigenvectors+  , sumZSq+  , readRawI+  )+import Numeric.LinearAlgebra.Massiv.Internal.Kernel+  ( rawMutSumSqColumn, rawMutSumSqRow+  , rawMutHouseholderApply, rawMutHouseholderApplyRow+  , rawMutQAccum+  , rawMutApplyGivensColumns+  , rawMutApplyGivensColumnsCM+  , rawTransposeToColMajor, rawTransposeFromColMajor+  , rawGemmKernel, rawZeroDoubles, rawNegateDoubles+  , rawCopyColumn )++-- | Compute the full Singular Value Decomposition (GVL4 Theorem 8.6.1,+-- p. 499).+--+-- For an \(m \times n\) matrix \(A\) with \(m \geq n\), computes+--+-- \[+--   A = U \, \Sigma \, V^T+-- \]+--+-- where+--+--   * \(U \in \mathbb{R}^{m \times m}\) is orthogonal (columns are the+--     /left singular vectors/),+--   * \(\Sigma = \mathrm{diag}(\sigma_1, \ldots, \sigma_n)\) with+--     \(\sigma_1 \geq \cdots \geq \sigma_n \geq 0\) (the /singular values/),+--   * \(V \in \mathbb{R}^{n \times n}\) is orthogonal (columns are the+--     /right singular vectors/).+--+-- __Method:__ Forms \(A^T A\) and calls+-- 'Numeric.LinearAlgebra.Massiv.Eigen.Symmetric.symmetricEigen' to obtain+-- the eigendecomposition \(A^T A = V \Lambda V^T\).  Singular values are+-- recovered as \(\sigma_i = \sqrt{\max(0, \lambda_i)}\) and left singular+-- vectors as \(u_i = A v_i / \sigma_i\).  For zero singular values the+-- corresponding column of \(U\) is set to the appropriate standard basis+-- vector.+--+-- Returns @(U, sigma, V)@.+svd :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+    => Matrix m n r e+    -> (Matrix m m r e, Vector n r e, Matrix n n r e)+svd a =+  let nn = dimVal @n+      at = transpose a+      ata = matMul at a  -- n×n symmetric positive semidefinite+      -- Eigendecomposition of AᵀA+      (eigvalsRaw_, vRaw_) = symmetricEigen ata (30 * nn) 1e-12+      -- Sort eigenvalues descending; build O(1) permutation array+      permBA_ = buildPermArray+                  (map snd $ sortBy (\(a_,_) (b_,_) -> compare (Down a_) (Down b_))+                             [(eigvalsRaw_ !. i, i) | i <- [0..nn-1]])+                  nn+      v = makeMatrix @n @n @r $ \i j -> vRaw_ ! (i, indexPermArray permBA_ j)+      -- Singular values = sqrt of sorted eigenvalues (clamp negatives to 0)+      sigma = makeVector @n @r $ \j ->+        let ev = eigvalsRaw_ !. indexPermArray permBA_ j+        in if ev > 0 then sqrt ev else 0+      -- Compute U: u_i = A·v_i / σ_i+      -- First, build U by computing A·V column by column+      u = makeMatrix @m @m @r $ \i j ->+        if j < nn then+          let sj = sigma !. j+          in if sj > 1e-14+             then -- u_j = (1/σ_j) · Σ_k A(i,k) · V(k,j)+               let av_ = foldl' (\acc k -> acc + (a ! (i, k)) * (v ! (k, j))) 0 [0..nn-1]+               in av_ / sj+             else -- Zero singular value; use arbitrary orthogonal vector+               if i == j then 1 else 0+        else+          -- Extra columns for m > n: extend to full orthogonal basis+          if i == j then 1 else 0+  in (u, sigma, v)++-- | P-specialised full SVD using raw ByteArray# SIMD kernels throughout.+--+-- Wires 'matMulP' (SIMD GEMM), 'symmetricEigenP' (raw primop QR iteration),+-- and 'matvecP' (SIMD matrix–vector product) into the SVD pipeline.+-- | P-specialised full SVD.  Uses the A^T A eigendecomposition path by default+-- as it is currently faster than the Golub-Kahan bidiagonalisation path+-- (svdGKP) at all sizes.  The GK path will become the default once blocked+-- bidiagonalisation is implemented.+svdP :: forall m n. (KnownNat m, KnownNat n)+     => Matrix m n M.P Double+     -> (Matrix m m M.P Double, Vector n M.P Double, Matrix n n M.P Double)+svdP = svdAtAP+{-# NOINLINE svdP #-}++-- | SVD via A^T A eigendecomposition.+-- Forms A^T A, eigendecomposes via 'symmetricEigenP', recovers singular+-- values as square roots and left singular vectors via matrix-vector products.+svdAtAP :: forall m n. (KnownNat m, KnownNat n)+        => Matrix m n M.P Double+        -> (Matrix m m M.P Double, Vector n M.P Double, Matrix n n M.P Double)+svdAtAP a =+  let !mm = dimVal @m+      !nn = dimVal @n+      !ata = matMulAtAP a  -- n×n symmetric positive semidefinite, fast transpose + SIMD GEMM+      -- Eigendecomposition of AᵀA: D&C for large, QR iteration for small+      (!eigvalsRaw, !vRaw) = if nn >= 50+        then symmetricEigenPDC ata 1e-12+        else symmetricEigenP ata (max 30 (6 * nn)) 1e-12+      -- Sort eigenvalues descending; build O(1) permutation array+      !permList = map snd $ sortBy (\(a_,_) (b_,_) -> compare (Down a_) (Down b_))+                        [(eigvalsRaw !. i, i) | i <- [0..nn-1]]+      !permBA = buildPermArray permList nn+      -- Rearrange V columns via O(1) indexed permutation using raw ByteArray copy+      !v = createMatrix @n @n @M.P $ \mv -> do+        let !baV   = unwrapByteArray (unMatrix vRaw)+            !offV  = unwrapByteArrayOffset (unMatrix vRaw)+            !mbaVP = unwrapMutableByteArray mv+            !offVP = unwrapMutableByteArrayOffset mv+            !(ByteArray baV#) = baV+            !(I# offV#) = offV+            !(MutableByteArray mbaVP#) = mbaVP+            !(I# offVP#) = offVP+            !(I# nnV#) = nn+        -- Copy columns: V_new[i,j] = V_raw[i, perm[j]]+        ST $ \s0 ->+          let goRow i s+                | isTrue# (i >=# nnV#) = s+                | otherwise =+                    let goCol j s1+                          | isTrue# (j >=# nnV#) = s1+                          | otherwise =+                              let !(I# pj) = indexPermArray permBA (I# j)+                                  !val = indexDoubleArray# baV# (offV# +# i *# nnV# +# pj)+                              in case writeDoubleArray# mbaVP# (offVP# +# i *# nnV# +# j) val s1 of+                                   s2 -> goCol (j +# 1#) s2+                    in goRow (i +# 1#) (goCol 0# s)+          in (# goRow 0# s0, () #)+      -- Singular values = sqrt of sorted eigenvalues (clamp negatives to 0)+      sigma = makeVector @n @M.P $ \j ->+        let !(ByteArray baEV#) = unwrapByteArray (unVector eigvalsRaw)+            !(I# offEV#) = unwrapByteArrayOffset (unVector eigvalsRaw)+            !(I# pj#) = indexPermArray permBA j+            ev = case indexDoubleArray# baEV# (offEV# +# pj#) of v_ -> D# v_+        in if ev > 0 then sqrt ev else 0+      -- Compute U = A · V · diag(1/σ) via pre-scaled V and single GEMM.+      -- This avoids the intermediate m×n AV matrix and a separate scaling pass.+      -- V_scaled[i,j] = V[i,j] / σ_j (zero for σ_j ≤ ε).+      !vScaled = createMatrix @n @n @M.P @Double $ \mvs -> do+        let !baV   = unwrapByteArray (unMatrix v)+            !offVs = unwrapByteArrayOffset (unMatrix v)+            !mbaVS = unwrapMutableByteArray mvs+            !offVS = unwrapMutableByteArrayOffset mvs+        -- Pre-compute invSigma+        mbaInvS <- newByteArray (nn * 8)+        forM_ [0..nn-1] $ \j -> do+          let sj = sigma !. j+          writeRawD mbaInvS 0 j (if sj > 1e-14 then 1.0 / sj else 0.0)+        !(ByteArray baInvS#) <- unsafeFreezeByteArray mbaInvS+        -- Scale each column: V_scaled[i,j] = V[i,j] * invSigma[j]+        let !(ByteArray baV#) = baV+            !(I# offVs#) = offVs+            !(MutableByteArray mbaVS#) = mbaVS+            !(I# offVS#) = offVS+            !(I# nnV#) = nn+            !nn4 = nn - (nn `rem` 4)+            !(I# nn4#) = nn4+        ST $ \s0 ->+          let goRow i s+                | isTrue# (i >=# nnV#) = s+                | otherwise =+                    let !srcOff = offVs# +# i *# nnV#+                        !dstOff = offVS# +# i *# nnV#+                        goSimd j s1+                          | isTrue# (j >=# nn4#) = s1+                          | otherwise =+                              let vv = indexDoubleArrayAsDoubleX4# baV# (srcOff +# j)+                                  sv = indexDoubleArrayAsDoubleX4# baInvS# j+                                  !p  = timesDoubleX4# vv sv+                              in case writeDoubleArrayAsDoubleX4# mbaVS# (dstOff +# j) p s1 of+                                   s2 -> goSimd (j +# 4#) s2+                        goScalar j s1+                          | isTrue# (j >=# nnV#) = s1+                          | otherwise =+                              let vVal = indexDoubleArray# baV# (srcOff +# j)+                                  sVal = indexDoubleArray# baInvS# j+                              in case writeDoubleArray# mbaVS# (dstOff +# j) (vVal *## sVal) s1 of+                                   s2 -> goScalar (j +# 1#) s2+                    in goRow (i +# 1#) (goScalar nn4# (goSimd 0# s))+          in (# goRow 0# s0, () #)+      -- U = A · V_scaled: GEMM writes m×n result directly.+      -- For mm == nn (square), GEMM writes directly into U.+      -- For mm > nn (rectangular), GEMM writes into temp then copy columns.+      u = createMatrix @m @m @M.P $ \mu -> do+        let !mbaU  = unwrapMutableByteArray mu+            !offU  = unwrapMutableByteArrayOffset mu+            !(I# mm#) = mm+        -- Zero all of U+        rawZeroDoubles mbaU offU (mm * mm)+        let !baA  = unwrapByteArray (unMatrix a)+            !offA = unwrapByteArrayOffset (unMatrix a)+            !baVS = unwrapByteArray (unMatrix vScaled)+            !offVS = unwrapByteArrayOffset (unMatrix vScaled)+        if mm == nn+          then+            -- Direct GEMM into U (stride mm == nn, so layout matches)+            rawGemmKernel baA offA baVS offVS mbaU offU mm nn nn+          else do+            -- GEMM into temp (m×n), then copy columns into U (m×m)+            mbaTemp <- newByteArray (mm * nn * 8)+            rawZeroDoubles mbaTemp 0 (mm * nn)+            rawGemmKernel baA offA baVS offVS mbaTemp 0 mm nn nn+            baTemp <- unsafeFreezeByteArray mbaTemp+            -- Copy: U[i, 0..nn-1] = temp[i, 0..nn-1]+            let !(ByteArray baT#) = baTemp+                !(MutableByteArray mbaU#) = mbaU+                !(I# offU#) = offU+                !(I# nn#) = nn+            ST $ \s0 ->+              let goCopy i s+                    | isTrue# (i >=# mm#) = s+                    | otherwise =+                        let goCol j s1+                              | isTrue# (j >=# nn#) = s1+                              | otherwise =+                                  let !val = indexDoubleArray# baT# (i *# nn# +# j)+                                  in case writeDoubleArray# mbaU# (offU# +# i *# mm# +# j) val s1 of+                                       s2 -> goCol (j +# 1#) s2+                        in goCopy (i +# 1#) (goCol 0# s)+              in (# goCopy 0# s0, () #)+        -- Fix zero singular values: set diagonal U[j,j] = 1.0+        forM_ [0..nn-1] $ \j -> do+          let sj = sigma !. j+          when (sj <= 1e-14) $+            writeRawD mbaU offU (j * mm + j) 1.0+        -- Extra columns for m > n: extend to full orthogonal basis+        forM_ [nn..mm-1] $ \j ->+          writeRawD mbaU offU (j * mm + j) 1.0+  in (u, sigma, v)+{-# NOINLINE svdAtAP #-}++-- | Read a Double from an immutable ByteArray at element index.+readBA :: ByteArray -> Int -> Int -> Double+readBA (ByteArray ba) (I# off) (I# i) =+  case indexDoubleArray# ba (off +# i) of v -> D# v+{-# INLINE readBA #-}++-- | Build an unboxed Int permutation array from a list for O(1) indexed access.+buildPermArray :: [Int] -> Int -> ByteArray+buildPermArray xs n = runST $ do+  mba <- newByteArray (n * 8)  -- 8 bytes per Int on 64-bit+  let go _ []     = pure ()+      go i (x:rest) = do+        let !(MutableByteArray mba#) = mba+            !(I# i#) = i+            !(I# x#) = x+        ST $ \s -> case writeIntArray# mba# i# x# s of s' -> (# s', () #)+        go (i + 1) rest+  go 0 xs+  unsafeFreezeByteArray mba+{-# INLINE buildPermArray #-}++-- | O(1) index into a permutation ByteArray.+indexPermArray :: ByteArray -> Int -> Int+indexPermArray (ByteArray ba#) (I# i#) =+  case indexIntArray# ba# i# of x# -> I# x#+{-# INLINE indexPermArray #-}++-- | Compute only the singular values of \(A\), sorted in descending order.+singularValues :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+               => Matrix m n r e -> Vector n r e+singularValues a =+  let nn = dimVal @n+      at = transpose a+      ata = matMul at a+      (eigvals, _) = symmetricEigen ata (30 * nn) 1e-12+      -- Sort eigenvalues descending, take sqrt+      evList = map (\i -> eigvals !. i) [0..nn-1]+      sorted = sortBy (\x y -> compare (Down x) (Down y)) evList+  in makeVector @n @r $ \i ->+    let ev = sorted !! i+    in if ev > 0 then sqrt ev else 0++-- | P-specialised singular values using raw SIMD GEMM and raw primop eigenvalue solver.+singularValuesP :: forall m n. (KnownNat m, KnownNat n)+                => Matrix m n M.P Double -> Vector n M.P Double+singularValuesP a =+  let nn = dimVal @n+      !ata = matMulAtAP a+      (!eigvals, _) = symmetricEigenP ata (10 * nn) 1e-12+      evList = map (\i -> eigvals !. i) [0..nn-1]+      sorted = sortBy (\x y -> compare (Down x) (Down y)) evList+  in makeVector @n @M.P $ \i ->+    let ev = sorted !! i+    in if ev > 0 then sqrt ev else 0++-- ============================================================================+-- Golub-Kahan bidiagonalisation SVD (GVL4 Algorithm 5.4.2 + 8.6.2)+-- ============================================================================++-- | Full Golub-Kahan SVD pipeline.+-- Phase 1: Bidiagonalise A → U₀ B V₀^T+-- Phase 2: Implicit-shift QR on bidiagonal B, accumulating rotations into U, V+-- Phase 3: Assemble final U, sigma, V; ensure σᵢ ≥ 0; sort descending+svdGKP :: forall m n. (KnownNat m, KnownNat n)+       => Matrix m n M.P Double+       -> (Matrix m m M.P Double, Vector n M.P Double, Matrix n n M.P Double)+svdGKP (MkMatrix a_) = runST $ do+  let !mm = dimVal @m+      !nn = dimVal @n++  -- Copy input into mutable working storage+  mA <- M.thawS a_+  let mbaA = unwrapMutableByteArray mA+      offA = unwrapMutableByteArrayOffset mA++  -- Allocate arrays for Householder betas+  mbaBetaL <- newByteArray (nn * 8)  -- left Householder betas+  mbaBetaR <- newByteArray (nn * 8)  -- right Householder betas++  -- Phase 1: Bidiagonalise A in-place (BLAS-3 panel for large, Level-2 for small)+  bidiagonalizePPanel mbaA offA mm nn mbaBetaL mbaBetaR++  -- Extract diagonal d and superdiagonal e from bidiagonalised A+  mbaD <- newByteArray (nn * 8)+  mbaE <- newByteArray (nn * 8)+  forM_ [0..nn-1] $ \k -> do+    dk <- readRawD mbaA offA (k * nn + k)+    writeRawD mbaD 0 k dk+  forM_ [0..nn-2] $ \k -> do+    ek <- readRawD mbaA offA (k * nn + (k+1))+    writeRawD mbaE 0 k ek++  -- Freeze A for Householder vector extraction+  frozenA <- M.freezeS mA++  -- Phase 2: Accumulate U₀ and V₀ from stored Householder vectors+  -- U₀ = H₀ H₁ ... H_{n-1} (left reflectors, stored in columns of A)+  mU <- M.newMArray @M.P (Sz (mm :. mm)) (0 :: Double)+  let mbaU = unwrapMutableByteArray mU+      offU = unwrapMutableByteArrayOffset mU+  -- Initialise U = I+  forM_ [0..mm-1] $ \i ->+    writeRawD mbaU offU (i * mm + i) 1.0++  let baA = unwrapByteArray frozenA+      offFA = unwrapByteArrayOffset frozenA++  -- Accumulate left Householder reflectors into U (forward: U = H₀ H₁ ⋯ H_{n-1})+  -- Left reflector k: v stored in column k of A, rows k+1..m-1, with v[k]=1 implicit+  if nn <= 16+    then+      -- Small matrix: per-row accumulation (Level-2)+      forM_ [0..nn-1] $ \k -> do+        betaK <- readRawD mbaBetaL 0 k+        when (betaK /= 0) $+          forM_ [0..mm-1] $ \row ->+            rawMutQAccum mbaU offU mm baA offFA nn betaK k mm row+    else do+      -- Blocked WY: batch nb Householder vectors at a time+      let !nbU = min 48 nn+      mbaYU  <- newByteArray (mm * nbU * 8)+      mbaTfU <- newByteArray (nbU * nbU * 8)+      mbaW1U <- newByteArray (mm * nbU * 8)+      mbaW2U <- newByteArray (mm * nbU * 8)+      mbaYTU <- newByteArray (nbU * mm * 8)+      mbaGU  <- newByteArray (nbU * nbU * 8)++      let goBlockU !k0+            | k0 >= nn = pure ()+            | otherwise = do+                let !bsz = min nbU (nn - k0)+                -- Pack Y (mm × bsz): Y[:,j] = left Householder vector k0+j+                rawZeroDoubles mbaYU 0 (mm * bsz)+                forM_ [0..bsz-1] $ \j -> do+                  let !k = k0 + j+                  writeRawD mbaYU 0 (k * bsz + j) 1.0+                  forM_ [k+1..mm-1] $ \l ->+                    writeRawD mbaYU 0 (l * bsz + j) (readBA baA offFA (l * nn + k))++                -- Transpose Y → Y^T (bsz × mm) for GEMM reuse+                rawZeroDoubles mbaYTU 0 (bsz * mm)+                forM_ [0..bsz-1] $ \j -> do+                  let !k = k0 + j+                  writeRawD mbaYTU 0 (j * mm + k) 1.0+                  forM_ [k+1..mm-1] $ \l ->+                    writeRawD mbaYTU 0 (j * mm + l) (readBA baA offFA (l * nn + k))++                baYU  <- unsafeFreezeByteArray mbaYU+                baYTU <- unsafeFreezeByteArray mbaYTU++                -- G = Y^T × Y (bsz × bsz)+                rawZeroDoubles mbaGU 0 (bsz * bsz)+                rawGemmKernel baYTU 0 baYU 0 mbaGU 0 bsz mm bsz++                -- Build T-factor (bsz × bsz upper-triangular)+                rawZeroDoubles mbaTfU 0 (bsz * bsz)+                forM_ [0..bsz-1] $ \j -> do+                  betaj <- readRawD mbaBetaL 0 (k0 + j)+                  writeRawD mbaTfU 0 (j * bsz + j) betaj+                  when (j > 0 && betaj /= 0) $ do+                    -- T[0..j-1, j] = -betaj * T[0..j-1, 0..j-1] * G[0..j-1, j]+                    forM_ [0..j-1] $ \i -> do+                      g_ij <- readRawD mbaGU 0 (i * bsz + j)+                      writeRawD mbaW1U 0 i g_ij+                    forM_ [0..j-1] $ \i -> do+                      let triLoop !l !acc+                            | l >= j = pure acc+                            | otherwise = do+                                til <- readRawD mbaTfU 0 (i * bsz + l)+                                dl  <- readRawD mbaW1U 0 l+                                triLoop (l+1) (acc + til * dl)+                      z <- triLoop 0 0+                      writeRawD mbaTfU 0 (i * bsz + j) (negate betaj * z)++                -- W1 = Q · Y (mm×mm * mm×bsz → mm×bsz)+                baQU <- unsafeFreezeByteArray mbaU+                rawZeroDoubles mbaW1U 0 (mm * bsz)+                rawGemmKernel baQU offU baYU 0 mbaW1U 0 mm mm bsz++                -- W2 = W1 · T (mm×bsz * bsz×bsz → mm×bsz)+                baW1U <- unsafeFreezeByteArray mbaW1U+                baTfU <- unsafeFreezeByteArray mbaTfU+                rawZeroDoubles mbaW2U 0 (mm * bsz)+                rawGemmKernel baW1U 0 baTfU 0 mbaW2U 0 mm bsz bsz++                -- Negate W2+                rawNegateDoubles mbaW2U 0 (mm * bsz)++                -- Q += (-W2) · Y^T (mm×bsz * bsz×mm → mm×mm)+                baNW2U <- unsafeFreezeByteArray mbaW2U+                rawGemmKernel baNW2U 0 baYTU 0 mbaU offU mm bsz mm++                goBlockU (k0 + bsz)+      goBlockU 0++  -- V₀ = G₁ G₂ ... G_{n-3} (right reflectors)+  mV <- M.newMArray @M.P (Sz (nn :. nn)) (0 :: Double)+  let mbaV = unwrapMutableByteArray mV+      offV = unwrapMutableByteArrayOffset mV+  -- Initialise V = I+  forM_ [0..nn-1] $ \i ->+    writeRawD mbaV offV (i * nn + i) 1.0++  -- Accumulate right Householder reflectors into V (forward: V = G₀ G₁ ⋯ G_{n-3})+  -- Right reflector k: v stored in row k of A, cols k+2..n-1, with v[k+1]=1 implicit+  if nn < 19+    then+      -- Small: per-row Level-2+      when (nn >= 3) $+        forM_ [0..nn-3] $ \k -> do+          betaK <- readRawD mbaBetaR 0 k+          when (betaK /= 0) $+            forM_ [0..nn-1] $ \row ->+              rightQAccum mbaV offV nn baA offFA nn betaK k nn row+    else when (nn >= 3) $ do+      -- Blocked WY for right reflectors+      let !nRefl = nn - 2  -- right reflectors 0..nn-3+          !nbV = min 48 nRefl+      mbaYV  <- newByteArray (nn * nbV * 8)+      mbaTfV <- newByteArray (nbV * nbV * 8)+      mbaW1V <- newByteArray (nn * nbV * 8)+      mbaW2V <- newByteArray (nn * nbV * 8)+      mbaYTV <- newByteArray (nbV * nn * 8)+      mbaGV  <- newByteArray (nbV * nbV * 8)++      let goBlockV !k0+            | k0 >= nRefl = pure ()+            | otherwise = do+                let !bsz = min nbV (nRefl - k0)+                -- Pack Y (nn × bsz): Y[:,j] = right Householder vector k0+j+                -- Right vector k has implicit 1 at position k+1, stored values at k+2..nn-1+                rawZeroDoubles mbaYV 0 (nn * bsz)+                forM_ [0..bsz-1] $ \j -> do+                  let !k = k0 + j+                  writeRawD mbaYV 0 ((k+1) * bsz + j) 1.0+                  forM_ [k+2..nn-1] $ \l ->+                    writeRawD mbaYV 0 (l * bsz + j) (readBA baA offFA (k * nn + l))++                -- Transpose Y → Y^T (bsz × nn)+                rawZeroDoubles mbaYTV 0 (bsz * nn)+                forM_ [0..bsz-1] $ \j -> do+                  let !k = k0 + j+                  writeRawD mbaYTV 0 (j * nn + (k+1)) 1.0+                  forM_ [k+2..nn-1] $ \l ->+                    writeRawD mbaYTV 0 (j * nn + l) (readBA baA offFA (k * nn + l))++                baYV  <- unsafeFreezeByteArray mbaYV+                baYTV <- unsafeFreezeByteArray mbaYTV++                -- G = Y^T × Y (bsz × bsz)+                rawZeroDoubles mbaGV 0 (bsz * bsz)+                rawGemmKernel baYTV 0 baYV 0 mbaGV 0 bsz nn bsz++                -- Build T-factor (bsz × bsz upper-triangular)+                rawZeroDoubles mbaTfV 0 (bsz * bsz)+                forM_ [0..bsz-1] $ \j -> do+                  betaj <- readRawD mbaBetaR 0 (k0 + j)+                  writeRawD mbaTfV 0 (j * bsz + j) betaj+                  when (j > 0 && betaj /= 0) $ do+                    -- T[0..j-1, j] = -betaj * T[0..j-1, 0..j-1] * G[0..j-1, j]+                    forM_ [0..j-1] $ \i -> do+                      g_ij <- readRawD mbaGV 0 (i * bsz + j)+                      writeRawD mbaW1V 0 i g_ij+                    forM_ [0..j-1] $ \i -> do+                      let triLoop !l !acc+                            | l >= j = pure acc+                            | otherwise = do+                                til <- readRawD mbaTfV 0 (i * bsz + l)+                                dl  <- readRawD mbaW1V 0 l+                                triLoop (l+1) (acc + til * dl)+                      z <- triLoop 0 0+                      writeRawD mbaTfV 0 (i * bsz + j) (negate betaj * z)++                -- W1 = V · Y (nn×nn * nn×bsz → nn×bsz)+                baQV <- unsafeFreezeByteArray mbaV+                rawZeroDoubles mbaW1V 0 (nn * bsz)+                rawGemmKernel baQV offV baYV 0 mbaW1V 0 nn nn bsz++                -- W2 = W1 · T (nn×bsz * bsz×bsz → nn×bsz)+                baW1V <- unsafeFreezeByteArray mbaW1V+                baTfV <- unsafeFreezeByteArray mbaTfV+                rawZeroDoubles mbaW2V 0 (nn * bsz)+                rawGemmKernel baW1V 0 baTfV 0 mbaW2V 0 nn bsz bsz++                -- Negate W2+                rawNegateDoubles mbaW2V 0 (nn * bsz)++                -- V += (-W2) · Y^T (nn×bsz * bsz×nn → nn×nn)+                baNW2V <- unsafeFreezeByteArray mbaW2V+                rawGemmKernel baNW2V 0 baYTV 0 mbaV offV nn bsz nn++                goBlockV (k0 + bsz)+      goBlockV 0++  -- Phase 3: Bidiagonal SVD (D&C for large, QR iteration for small)+  if nn >= dcBidiagThreshold+    then dcBidiagSVD mbaD 0 mbaE 0 mbaU offU mm mbaV offV nn nn 1e-14+    else bidiagQRIterPCM mbaD 0 mbaE 0 mbaU offU mm mbaV offV nn nn (30 * nn)++  -- Phase 4: Ensure σᵢ ≥ 0 (flip sign of U column if needed)+  forM_ [0..nn-1] $ \k -> do+    dk <- readRawD mbaD 0 k+    when (dk < 0) $ do+      writeRawD mbaD 0 k (negate dk)+      -- Flip column k of U+      forM_ [0..mm-1] $ \i -> do+        uik <- readRawD mbaU offU (i * mm + k)+        writeRawD mbaU offU (i * mm + k) (negate uik)++  -- Phase 5: Sort singular values descending and permute U, V columns+  pairs <- mapM (\k -> do dk <- readRawD mbaD 0 k; return (dk, k)) [0..nn-1]+  let !sorted = sortBy (\(a1,_) (b1,_) -> compare (Down a1) (Down b1)) pairs++  frozenU <- M.freezeS mU+  frozenV <- M.freezeS mV+  let baU = unwrapByteArray frozenU+      offFU = unwrapByteArrayOffset frozenU+      baV = unwrapByteArray frozenV+      offFV = unwrapByteArrayOffset frozenV++  let !sigmaVec = makeVector @n @M.P $ \i -> fst (sorted !! i)+      !uMat = makeMatrix @m @m @M.P $ \i j ->+        if j < nn+          then let origCol = snd (sorted !! j)+               in readBA baU offFU (i * mm + origCol)+          else if i == j then 1 else 0+      !vMat = makeMatrix @n @n @M.P $ \i j ->+        let origCol = snd (sorted !! j)+        in readBA baV offFV (i * nn + origCol)++  return (uMat, sigmaVec, vMat)+{-# NOINLINE svdGKP #-}++-- | In-place bidiagonalisation of an m×n matrix stored in a MutableByteArray.+-- GVL4 Algorithm 5.4.2, p. 284.+--+-- After this, the matrix has:+-- - Diagonal d[k] = A[k,k]+-- - Superdiagonal e[k] = A[k,k+1]+-- - Left Householder vectors stored in column k below diagonal (rows k+1..m-1)+-- - Right Householder vectors stored in row k right of superdiag (cols k+2..n-1)+-- - Householder betas stored in mbaBetaL and mbaBetaR+bidiagonalizeP :: MutableByteArray s -> Int -> Int -> Int+               -> MutableByteArray s -> MutableByteArray s -> ST s ()+bidiagonalizeP mbaA offA mm nn mbaBetaL mbaBetaR = do+  forM_ [0..nn-1] $ \k -> do+    -- Left Householder: zero out A[k+1:m, k]+    -- Compute Householder vector for column k, rows k..m-1+    if k < mm - 1+      then do+        -- sigma = Σ A[i,k]² for i in k+1..m-1+        sigma <- rawMutSumSqColumn mbaA offA nn (k+1) mm k+        x0 <- readRawD mbaA offA (k * nn + k)+        if sigma < 1e-300+          then writeRawD mbaBetaL 0 k 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                beta = 2 * v0 * v0 / (sigma + v0 * v0)+            -- Store v: normalise by v0+            -- v[k] will become 1 (implicit), v[k+1..m-1] = A[i,k]/v0+            forM_ [k+1..mm-1] $ \i -> do+              aik <- readRawD mbaA offA (i * nn + k)+              writeRawD mbaA offA (i * nn + k) (aik / v0)+            -- Set A[k,k] = mu (the diagonal value after reflection)+            writeRawD mbaA offA (k * nn + k) mu+            writeRawD mbaBetaL 0 k beta+            -- Apply left Householder to columns k+1..n-1+            -- Using rawMutHouseholderApply which reads v from column k, rows k+1..m-1+            forM_ [k+1..nn-1] $ \col ->+              rawMutHouseholderApply mbaA offA nn beta k mm col+      else+        writeRawD mbaBetaL 0 k 0++    -- Right Householder: zero out A[k, k+2:n]+    if k < nn - 2+      then do+        -- sigma = Σ A[k,j]² for j in k+2..n-1+        sigma <- rawMutSumSqRow mbaA offA nn k (k+2) nn+        x0 <- readRawD mbaA offA (k * nn + (k+1))+        if sigma < 1e-300+          then writeRawD mbaBetaR 0 k 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                beta = 2 * v0 * v0 / (sigma + v0 * v0)+            -- Store v: normalise by v0+            -- v[k+1] will become 1 (implicit), v[k+2..n-1] = A[k,j]/v0+            forM_ [k+2..nn-1] $ \j -> do+              akj <- readRawD mbaA offA (k * nn + j)+              writeRawD mbaA offA (k * nn + j) (akj / v0)+            -- Set A[k,k+1] = mu (the superdiagonal value)+            writeRawD mbaA offA (k * nn + (k+1)) mu+            writeRawD mbaBetaR 0 k beta+            -- Apply right Householder to rows k+1..m-1+            -- v is stored in row k, cols k+2..n-1, with implicit v[k+1]=1+            forM_ [k+1..mm-1] $ \row ->+              rawMutHouseholderApplyRow mbaA offA nn beta k (k+1) nn row+      else+        when (k < nn - 1) $ writeRawD mbaBetaR 0 k 0+{-# NOINLINE bidiagonalizeP #-}++-- | BLAS-3 panel bidiagonalisation (DLABRD-style, GVL4 §5.4.3).+-- Processes nb columns at a time, deferring trailing updates via X, Y+-- accumulators and applying them as rank-nb GEMMs.+-- Falls back to Level-2 bidiagonalizeP for n < panelBidiagCrossover.+bidiagonalizePPanel :: MutableByteArray s -> Int -> Int -> Int+                    -> MutableByteArray s -> MutableByteArray s -> ST s ()+bidiagonalizePPanel mbaA offA mm nn mbaBetaL mbaBetaR+  | nn < panelBidiagCrossover = bidiagonalizeP mbaA offA mm nn mbaBetaL mbaBetaR+  | otherwise = do+      let !nb = min 32 (max 8 (nn `div` 6))+      -- Allocate accumulators: X (mm × nb), Y (nn × nb), row-major+      mbaX <- newByteArray (mm * nb * 8)+      mbaY <- newByteArray (nn * nb * 8)+      -- Temp vectors for dot products+      mbaZL <- newByteArray (nb * 8)  -- V_L^T * v or Y^T * u+      mbaZX <- newByteArray (nb * 8)  -- X^T * v or V_R^T * u+      -- Buffers for trailing GEMM+      mbaVLbuf <- newByteArray (mm * nb * 8)+      mbaYTbuf <- newByteArray (nb * nn * 8)+      mbaXbuf  <- newByteArray (mm * nb * 8)+      mbaVRTbuf <- newByteArray (nb * nn * 8)+      mbaTrail <- newByteArray (mm * nn * 8)++      let goPanel !k0+            | k0 >= nn - 1 = pure ()+            | otherwise = do+                let !bs = min nb (nn - 1 - k0)+                if bs < 2  -- last column: use Level-2+                  then bidiagLastCols mbaA offA mm nn mbaBetaL mbaBetaR k0+                  else do+                    rawZeroDoubles mbaX 0 (mm * bs)+                    rawZeroDoubles mbaY 0 (nn * bs)+                    -- Panel phase+                    panelBidiagStep mbaA offA mm nn mbaBetaL mbaBetaR+                                    mbaX mbaY mbaZL mbaZX k0 bs+                    -- Trailing update+                    let !remR = mm - k0 - bs+                        !remC = nn - k0 - bs+                    when (remR > 0 && remC > 0) $+                      applyTrailingUpdate mbaA offA mm nn mbaX mbaY+                                          mbaVLbuf mbaYTbuf mbaXbuf mbaVRTbuf mbaTrail+                                          k0 bs remR remC+                    goPanel (k0 + bs)+      goPanel 0+  where+    panelBidiagCrossover = 64+{-# NOINLINE bidiagonalizePPanel #-}++-- | Finish remaining columns with Level-2 bidiagonalisation.+bidiagLastCols :: MutableByteArray s -> Int -> Int -> Int+               -> MutableByteArray s -> MutableByteArray s -> Int -> ST s ()+bidiagLastCols mbaA offA mm nn mbaBetaL mbaBetaR k0 = do+  forM_ [k0..nn-1] $ \k -> do+    -- Left Householder+    if k < mm - 1+      then do+        sigma <- rawMutSumSqColumn mbaA offA nn (k+1) mm k+        x0 <- readRawD mbaA offA (k * nn + k)+        if sigma < 1e-300+          then writeRawD mbaBetaL 0 k 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                beta = 2 * v0 * v0 / (sigma + v0 * v0)+            forM_ [k+1..mm-1] $ \i -> do+              aik <- readRawD mbaA offA (i * nn + k)+              writeRawD mbaA offA (i * nn + k) (aik / v0)+            writeRawD mbaA offA (k * nn + k) mu+            writeRawD mbaBetaL 0 k beta+            forM_ [k+1..nn-1] $ \col ->+              rawMutHouseholderApply mbaA offA nn beta k mm col+      else writeRawD mbaBetaL 0 k 0+    -- Right Householder+    if k < nn - 2+      then do+        sigma <- rawMutSumSqRow mbaA offA nn k (k+2) nn+        x0 <- readRawD mbaA offA (k * nn + (k+1))+        if sigma < 1e-300+          then writeRawD mbaBetaR 0 k 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                beta = 2 * v0 * v0 / (sigma + v0 * v0)+            forM_ [k+2..nn-1] $ \j -> do+              akj <- readRawD mbaA offA (k * nn + j)+              writeRawD mbaA offA (k * nn + j) (akj / v0)+            writeRawD mbaA offA (k * nn + (k+1)) mu+            writeRawD mbaBetaR 0 k beta+            forM_ [k+1..mm-1] $ \row ->+              rawMutHouseholderApplyRow mbaA offA nn beta k (k+1) nn row+      else when (k < nn - 1) $ writeRawD mbaBetaR 0 k 0++-- | DLABRD-style panel step: compute bs left/right Householder reflectors+-- starting at column k0, maintaining X and Y accumulators.+-- After this, A_eff[i,c] = A[i,c] - V_L[i,:]*Y[c,:]^T - X[i,:]*V_R[c,:]^T+-- for all i >= k0+bs, c >= k0+bs.+panelBidiagStep :: MutableByteArray s -> Int -> Int -> Int+                -> MutableByteArray s -> MutableByteArray s+                -> MutableByteArray s -> MutableByteArray s+                -> MutableByteArray s -> MutableByteArray s+                -> Int -> Int -> ST s ()+panelBidiagStep mbaA offA mm nn mbaBetaL mbaBetaR mbaX mbaY mbaZL mbaZX k0 bs = do+  forM_ [0..bs-1] $ \j -> do+    let !k = k0 + j++    -- ================================================================+    -- PART A: Left Householder on column k+    -- ================================================================++    -- Step A1: Read corrected column k into A (in-place correction for rows k..m-1).+    -- A_corr[i,k] = A[i,k] - sum_{l<j} V_L[i,l]*Y[k,l] - sum_{l<j} X[i,l]*V_R[k,l]+    when (j > 0) $+      forM_ [k..mm-1] $ \i -> do+        aik <- readRawD mbaA offA (i * nn + k)+        -- V_L[i,l] * Y[k,l] sum+        cVLY <- panelDot_VLY mbaA offA nn mbaY bs k0 i k j+        -- X[i,l] * V_R[k,l] sum+        cXVR <- panelDot_XVR mbaA offA nn mbaX bs k0 i k j+        writeRawD mbaA offA (i * nn + k) (aik - cVLY - cXVR)++    -- Step A2: Left Householder from corrected column k, rows k..m-1+    if k < mm - 1+      then do+        sigma <- rawMutSumSqColumn mbaA offA nn (k+1) mm k+        x0 <- readRawD mbaA offA (k * nn + k)+        if sigma < 1e-300+          then do+            writeRawD mbaBetaL 0 k 0+            -- Zero Y column j+            forM_ [0..nn-1] $ \c -> writeRawD mbaY 0 (c * bs + j) 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                beta = 2 * v0 * v0 / (sigma + v0 * v0)+            -- Normalise and store HH vector in column k+            forM_ [k+1..mm-1] $ \i -> do+              aik <- readRawD mbaA offA (i * nn + k)+              writeRawD mbaA offA (i * nn + k) (aik / v0)+            writeRawD mbaA offA (k * nn + k) mu+            writeRawD mbaBetaL 0 k beta++            -- Step A3: Compute Y column j+            -- Precompute zL[l] = V_L[:,l]^T * v for l = 0..j-1+            forM_ [0..j-1] $ \l -> do+              d <- dotVL_v mbaA offA nn k0 k mm l+              writeRawD mbaZL 0 l d+            -- Precompute zX[l] = X[:,l]^T * v for l = 0..j-1+            forM_ [0..j-1] $ \l -> do+              d <- dotX_v mbaA offA nn mbaX bs k mm l+              writeRawD mbaZX 0 l d+            -- Y[c, j] = beta * (A^T*v[c] - sum_l Y[c,l]*zL[l] - sum_l V_R[c,l]*zX[l])+            forM_ [0..k] $ \c -> writeRawD mbaY 0 (c * bs + j) 0+            forM_ [k+1..nn-1] $ \c -> do+              atv <- dotAT_v mbaA offA nn k mm c+              ycorr <- dotAccum mbaY bs c mbaZL j+              vrcorr <- dotVR_zX mbaA offA nn k0 mbaZX c j+              writeRawD mbaY 0 (c * bs + j) (beta * (atv - ycorr - vrcorr))+      else do+        writeRawD mbaBetaL 0 k 0+        forM_ [0..nn-1] $ \c -> writeRawD mbaY 0 (c * bs + j) 0++    -- ================================================================+    -- PART B: Correct row k, then right Householder (if applicable)+    -- ================================================================++    -- Step B1: ALWAYS correct row k for columns k+1..n-1.+    -- This is needed both for the right HH (if k < nn-2) and for the+    -- superdiagonal entry e[k] = A[k, k+1] and trailing column values.+    -- A_eff[k,c] = A[k,c] - sum_{l<=j} V_L[k,l]*Y[c,l] - sum_{l<j} X[k,l]*V_R[c,l]+    when (k < nn - 1) $+      forM_ [k+1..nn-1] $ \c -> do+        akc <- readRawD mbaA offA (k * nn + c)+        cVLY <- panelDot_VLY mbaA offA nn mbaY bs k0 k c (j+1)+        cXVR <- panelDot_XVR mbaA offA nn mbaX bs k0 k c j+        writeRawD mbaA offA (k * nn + c) (akc - cVLY - cXVR)++    -- Step B2: Right Householder from corrected row k (only if k < nn-2)+    if k < nn - 2+      then do+        sigma <- rawMutSumSqRow mbaA offA nn k (k+2) nn+        x0 <- readRawD mbaA offA (k * nn + (k+1))+        if sigma < 1e-300+          then do+            writeRawD mbaBetaR 0 k 0+            forM_ [0..mm-1] $ \i -> writeRawD mbaX 0 (i * bs + j) 0+          else do+            let mu = sqrt (x0 * x0 + sigma)+                v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                gamma = 2 * v0 * v0 / (sigma + v0 * v0)+            forM_ [k+2..nn-1] $ \c -> do+              akc <- readRawD mbaA offA (k * nn + c)+              writeRawD mbaA offA (k * nn + c) (akc / v0)+            writeRawD mbaA offA (k * nn + (k+1)) mu+            writeRawD mbaBetaR 0 k gamma++            -- Step B3: Compute X column j+            -- Precompute zL'[l] = Y[:,l]^T * u for l = 0..j+            forM_ [0..j] $ \l -> do+              d <- dotY_u mbaA offA nn mbaY bs k l+              writeRawD mbaZL 0 l d+            -- Precompute zX'[l] = V_R[:,l]^T * u for l = 0..j-1+            forM_ [0..j-1] $ \l -> do+              d <- dotVR_u mbaA offA nn k0 k l+              writeRawD mbaZX 0 l d+            -- X[i, j] = gamma * (A*u[i] - sum_l V_L[i,l]*zL'[l] - sum_l X[i,l]*zX'[l])+            forM_ [0..k] $ \i -> writeRawD mbaX 0 (i * bs + j) 0+            forM_ [k+1..mm-1] $ \i -> do+              au <- dotA_u mbaA offA nn k i+              vlcorr <- dotVL_zL mbaA offA nn k0 mbaZL i (j+1)+              xcorr <- dotX_zX mbaX bs mbaZX i j+              writeRawD mbaX 0 (i * bs + j) (gamma * (au - vlcorr - xcorr))+      else do+        when (k < nn - 1) $ writeRawD mbaBetaR 0 k 0+        forM_ [0..mm-1] $ \i -> writeRawD mbaX 0 (i * bs + j) 0++-- Helper: sum_l V_L[i,l]*Y[c,l] for l = 0..nL-1+panelDot_VLY :: MutableByteArray s -> Int -> Int -> MutableByteArray s -> Int+             -> Int -> Int -> Int -> Int -> ST s Double+panelDot_VLY mbaA offA nn mbaY bs k0 i c nL = go 0 0+  where+    go !l !acc+      | l >= nL = pure acc+      | otherwise = do+          let !kl = k0 + l+          vl <- if i == kl then pure 1.0+                else if i > kl then readRawD mbaA offA (i * nn + kl)+                else pure 0.0+          ycl <- readRawD mbaY 0 (c * bs + l)+          go (l+1) (acc + vl * ycl)++-- Helper: sum_l X[i,l]*V_R[c,l] for l = 0..nR-1+panelDot_XVR :: MutableByteArray s -> Int -> Int -> MutableByteArray s -> Int+             -> Int -> Int -> Int -> Int -> ST s Double+panelDot_XVR mbaA offA nn mbaX bs k0 i c nR = go 0 0+  where+    go !l !acc+      | l >= nR = pure acc+      | otherwise = do+          xil <- readRawD mbaX 0 (i * bs + l)+          let !kl = k0 + l+          vr <- if c == kl + 1 then pure 1.0+                else if c > kl + 1 then readRawD mbaA offA (kl * nn + c)+                else pure 0.0+          go (l+1) (acc + xil * vr)++-- Helper: V_L[:,l]^T * v where v = [1, A[k+1:m-1, k]]+dotVL_v :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> Int -> ST s Double+dotVL_v mbaA offA nn k0 k mm l = do+  -- v[i-k]: v[0]=1, v[i-k]=A[i,k] for i>k+  -- V_L[i,l]: 1 if i==kl, A[i,kl] if i>kl, 0 if i<kl+  -- Since k >= k0+j and l < j, kl < k, so V_L[k,l] = A[k,kl]+  vlk <- readRawD mbaA offA (k * nn + kl)+  go (k+1) vlk+  where+    !kl = k0 + l+    go !i !acc+      | i >= mm = pure acc+      | otherwise = do+          vli <- readRawD mbaA offA (i * nn + kl)+          vi  <- readRawD mbaA offA (i * nn + k)+          go (i+1) (acc + vli * vi)++-- Helper: X[:,l]^T * v where v = [1, A[k+1:m-1, k]]+dotX_v :: MutableByteArray s -> Int -> Int -> MutableByteArray s -> Int+       -> Int -> Int -> Int -> ST s Double+dotX_v mbaA offA nn mbaX bs k mm l = do+  xkl <- readRawD mbaX 0 (k * bs + l)+  go (k+1) xkl+  where+    go !i !acc+      | i >= mm = pure acc+      | otherwise = do+          xil <- readRawD mbaX 0 (i * bs + l)+          vi  <- readRawD mbaA offA (i * nn + k)+          go (i+1) (acc + xil * vi)++-- Helper: A^T * v at column c, where v = [1, A[k+1:m-1, k]]+dotAT_v :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> ST s Double+dotAT_v mbaA offA nn k mm c = do+  akc <- readRawD mbaA offA (k * nn + c)+  go (k+1) akc+  where+    go !i !acc+      | i >= mm = pure acc+      | otherwise = do+          aic <- readRawD mbaA offA (i * nn + c)+          vi  <- readRawD mbaA offA (i * nn + k)+          go (i+1) (acc + aic * vi)++-- Helper: sum_l Y[c,l]*zL[l] for l = 0..nL-1+dotAccum :: MutableByteArray s -> Int -> Int -> MutableByteArray s -> Int -> ST s Double+dotAccum mbaY bs c mbaZL nL = go 0 0+  where+    go !l !acc+      | l >= nL = pure acc+      | otherwise = do+          ycl <- readRawD mbaY 0 (c * bs + l)+          zl  <- readRawD mbaZL 0 l+          go (l+1) (acc + ycl * zl)++-- Helper: sum_l V_R[c,l]*zX[l] for l = 0..nR-1+dotVR_zX :: MutableByteArray s -> Int -> Int -> Int -> MutableByteArray s+         -> Int -> Int -> ST s Double+dotVR_zX mbaA offA nn k0 mbaZX c nR = go 0 0+  where+    go !l !acc+      | l >= nR = pure acc+      | otherwise = do+          let !kl = k0 + l+          vr <- if c == kl + 1 then pure 1.0+                else if c > kl + 1 then readRawD mbaA offA (kl * nn + c)+                else pure 0.0+          zx <- readRawD mbaZX 0 l+          go (l+1) (acc + vr * zx)++-- Helper: Y[:,l]^T * u where u = [1, A[k, k+2:n-1]]+dotY_u :: MutableByteArray s -> Int -> Int -> MutableByteArray s -> Int+       -> Int -> Int -> ST s Double+dotY_u mbaA offA nn mbaY bs k l = do+  yk1l <- readRawD mbaY 0 ((k+1) * bs + l)+  go (k+2) yk1l+  where+    go !c !acc+      | c >= nn = pure acc+      | otherwise = do+          ycl <- readRawD mbaY 0 (c * bs + l)+          uc  <- readRawD mbaA offA (k * nn + c)+          go (c+1) (acc + ycl * uc)++-- Helper: V_R[:,l]^T * u where u = [1, A[k, k+2:n-1]]+dotVR_u :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> ST s Double+dotVR_u mbaA offA nn k0 k l = do+  -- u[c-k-1]: u[0]=1 at c=k+1, u[c-k-1]=A[k,c] for c>k+1+  -- V_R[c,l]: 1 if c==kl+1, A[kl,c] if c>kl+1, 0 if c<=kl+  -- We need sum_{c=k+1}^{n-1} V_R[c,l] * u[c-k-1]+  -- Since k > kl (k=k0+j, l<j), V_R[k+1,l] = A[kl, k+1] (if k+1 > kl+1, i.e., k > kl)+  vrkp1 <- if k + 1 == kl + 1 then pure 1.0+            else readRawD mbaA offA (kl * nn + (k+1))+  go (k+2) vrkp1+  where+    !kl = k0 + l+    go !c !acc+      | c >= nn = pure acc+      | otherwise = do+          vrc <- if c == kl + 1 then pure 1.0+                 else readRawD mbaA offA (kl * nn + c)+          uc  <- readRawD mbaA offA (k * nn + c)+          go (c+1) (acc + vrc * uc)++-- Helper: A * u at row i, where u = [1, A[k, k+2:n-1]]+dotA_u :: MutableByteArray s -> Int -> Int -> Int -> Int -> ST s Double+dotA_u mbaA offA nn k i = do+  aikp1 <- readRawD mbaA offA (i * nn + (k+1))+  go (k+2) aikp1+  where+    go !c !acc+      | c >= nn = pure acc+      | otherwise = do+          aic <- readRawD mbaA offA (i * nn + c)+          uc  <- readRawD mbaA offA (k * nn + c)+          go (c+1) (acc + aic * uc)++-- Helper: sum_l V_L[i,l]*zL[l] for l = 0..nL-1+dotVL_zL :: MutableByteArray s -> Int -> Int -> Int -> MutableByteArray s+         -> Int -> Int -> ST s Double+dotVL_zL mbaA offA nn k0 mbaZL i nL = go 0 0+  where+    go !l !acc+      | l >= nL = pure acc+      | otherwise = do+          let !kl = k0 + l+          vl <- if i == kl then pure 1.0+                else if i > kl then readRawD mbaA offA (i * nn + kl)+                else pure 0.0+          zl <- readRawD mbaZL 0 l+          go (l+1) (acc + vl * zl)++-- Helper: sum_l X[i,l]*zX[l] for l = 0..nR-1+dotX_zX :: MutableByteArray s -> Int -> MutableByteArray s -> Int -> Int -> ST s Double+dotX_zX mbaX bs mbaZX i nR = go 0 0+  where+    go !l !acc+      | l >= nR = pure acc+      | otherwise = do+          xil <- readRawD mbaX 0 (i * bs + l)+          zx  <- readRawD mbaZX 0 l+          go (l+1) (acc + xil * zx)++-- | Apply trailing GEMM update after a panel step.+-- A[k0+bs:m, k0+bs:n] -= V_L_trail * Y_trail^T + X_trail * V_R_trail^T+applyTrailingUpdate :: MutableByteArray s -> Int -> Int -> Int+                    -> MutableByteArray s -> MutableByteArray s+                    -> MutableByteArray s -> MutableByteArray s+                    -> MutableByteArray s -> MutableByteArray s+                    -> MutableByteArray s+                    -> Int -> Int -> Int -> Int -> ST s ()+applyTrailingUpdate mbaA offA _mm nn mbaX mbaY+                    mbaVLbuf mbaYTbuf mbaXbuf mbaVRTbuf mbaTrail+                    k0 bs remR remC = do+  let !trailRowStart = k0 + bs+      !trailColStart = k0 + bs++  -- Copy A_trail to contiguous buffer (remR × remC)+  forM_ [0..remR-1] $ \i ->+    forM_ [0..remC-1] $ \c -> do+      val <- readRawD mbaA offA ((trailRowStart + i) * nn + trailColStart + c)+      writeRawD mbaTrail 0 (i * remC + c) val++  -- Build V_L_trail (remR × bs): V_L[trailRowStart+i, l] for i=0..remR-1, l=0..bs-1+  -- For all trail rows, i >= trailRowStart > k0+l, so V_L[i,l] = A[i, k0+l]+  forM_ [0..remR-1] $ \i ->+    forM_ [0..bs-1] $ \l ->+      readRawD mbaA offA ((trailRowStart + i) * nn + (k0 + l)) >>=+        writeRawD mbaVLbuf 0 (i * bs + l)++  -- Build Y_trail^T (bs × remC): Y_trail^T[l, c] = Y[trailColStart+c, l]+  forM_ [0..bs-1] $ \l ->+    forM_ [0..remC-1] $ \c ->+      readRawD mbaY 0 ((trailColStart + c) * bs + l) >>=+        writeRawD mbaYTbuf 0 (l * remC + c)++  -- GEMM 1: trail -= V_L_trail * Y_trail^T+  -- Negate V_L_trail: nVL = -V_L_trail+  rawNegateDoubles mbaVLbuf 0 (remR * bs)+  baVL <- unsafeFreezeByteArray mbaVLbuf+  baYT <- unsafeFreezeByteArray mbaYTbuf+  rawGemmKernel baVL 0 baYT 0 mbaTrail 0 remR bs remC++  -- Build X_trail (remR × bs): X[trailRowStart+i, l]+  forM_ [0..remR-1] $ \i ->+    forM_ [0..bs-1] $ \l ->+      readRawD mbaX 0 ((trailRowStart + i) * bs + l) >>=+        writeRawD mbaXbuf 0 (i * bs + l)++  -- Build V_R_trail^T (bs × remC): V_R_trail^T[l, c] = V_R[trailColStart+c, l]+  -- V_R[c, l] = 1 if c==k0+l+1, A[k0+l, c] if c>k0+l+1, 0 if c<=k0+l+  -- Must handle implicit 1: when trailColStart+c == k0+l+1 (i.e., l=bs-1, c=0)+  forM_ [0..bs-1] $ \l ->+    forM_ [0..remC-1] $ \c -> do+      let !globalC = trailColStart + c+          !kl = k0 + l+      val <- if globalC == kl + 1 then pure 1.0+             else if globalC > kl + 1 then readRawD mbaA offA (kl * nn + globalC)+             else pure 0.0+      writeRawD mbaVRTbuf 0 (l * remC + c) val++  -- GEMM 2: trail -= X_trail * V_R_trail^T+  rawNegateDoubles mbaXbuf 0 (remR * bs)+  baX  <- unsafeFreezeByteArray mbaXbuf+  baVRT <- unsafeFreezeByteArray mbaVRTbuf+  rawGemmKernel baX 0 baVRT 0 mbaTrail 0 remR bs remC++  -- Copy trail back to A+  forM_ [0..remR-1] $ \i ->+    forM_ [0..remC-1] $ \c -> do+      val <- readRawD mbaTrail 0 (i * remC + c)+      writeRawD mbaA offA ((trailRowStart + i) * nn + trailColStart + c) val++-- | Implicit-shift bidiagonal QR iteration (GVL4 Algorithm 8.6.2).+-- Operates on diagonal d and superdiagonal e of an upper bidiagonal matrix.+-- Accumulates left rotations into U (m×n columns) and right rotations into V (n×n).+--+-- Each iteration: (1) find the active unreduced block [p..q] by scanning from+-- the bottom for deflation, then scanning up for split; (2) apply one QR step+-- to [p..q]; (3) repeat until fully deflated or maxIter reached.+bidiagQRIterP :: MutableByteArray s -> Int  -- d + offset+              -> MutableByteArray s -> Int  -- e + offset+              -> MutableByteArray s -> Int -> Int  -- U + offset + ucols+              -> MutableByteArray s -> Int -> Int  -- V + offset + vcols+              -> Int -> Int  -- n, maxIter+              -> ST s ()+bidiagQRIterP mbaD offD mbaE offE mbaU offU ucols mbaV offV vcols nn maxIter = go 0+  where+    go !iter+      | iter >= maxIter = return ()+      | otherwise = do+          -- Step 1: Find q — the bottom of the unreduced block.+          -- Scan from nn-1 downward, deflating negligible e[q-1].+          q <- deflateHi (nn - 1)+          if q <= 0+            then return ()  -- fully deflated+            else do+              -- Step 2: Find p — the top of the unreduced block.+              -- Scan from q-1 downward, looking for a split.+              p <- findLo (q - 1)+              -- Step 3: Apply one QR step to [p..q]+              bidiagQRStep mbaD offD mbaE offE mbaU offU ucols mbaV offV vcols p q+              go (iter + 1)++    -- Scan from hi down: deflate any trailing negligible superdiagonals.+    -- Returns the index of the bottom row of the active block (0 if fully deflated).+    deflateHi !hi+      | hi <= 0 = return 0+      | otherwise = do+          ehi <- readRawD mbaE offE (hi - 1)+          dhi <- readRawD mbaD offD hi+          dhi1 <- readRawD mbaD offD (hi - 1)+          let tol = 1e-14 * (abs dhi1 + abs dhi)+          if abs ehi <= tol+            then do+              writeRawD mbaE offE (hi - 1) 0+              deflateHi (hi - 1)+            else return hi++    -- Scan from idx downward to find the top of the unreduced block.+    -- Returns the smallest p such that B[p..q] is unreduced.+    findLo !idx+      | idx <= 0 = return 0+      | otherwise = do+          eidx <- readRawD mbaE offE (idx - 1)+          didx <- readRawD mbaD offD idx+          didx1 <- readRawD mbaD offD (idx - 1)+          let tol = 1e-14 * (abs didx1 + abs didx)+          if abs eidx <= tol+            then do+              writeRawD mbaE offE (idx - 1) 0+              return idx+            else findLo (idx - 1)+{-# NOINLINE bidiagQRIterP #-}++-- | Column-major bidiagonal QR iteration with AED and stall detection.+-- Transposes U (mm×mm) and V (nn×nn) to column-major layout for SIMD Givens,+-- runs bidiag QR with aggressive early deflation, then transposes back.+-- Falls back to row-major path for nn < 10 (transpose overhead dominates).+bidiagQRIterPCM :: MutableByteArray s -> Int  -- d + offset+                -> MutableByteArray s -> Int  -- e + offset+                -> MutableByteArray s -> Int -> Int  -- U + offset + ucols (= mm)+                -> MutableByteArray s -> Int -> Int  -- V + offset + vcols (= nn)+                -> Int -> Int  -- nn, maxIter+                -> ST s ()+bidiagQRIterPCM mbaD offD mbaE offE mbaU offU mm mbaV offV nn n maxIter+  | n < 10 = bidiagQRIterP mbaD offD mbaE offE mbaU offU mm mbaV offV nn n maxIter+  | otherwise = do+      -- Transpose U (mm×mm) and V (nn×nn) to column-major+      tmpU <- newByteArray (mm * mm * 8)+      tmpV <- newByteArray (nn * nn * 8)+      rawTransposeToColMajor mbaU offU tmpU 0 mm+      rawTransposeToColMajor mbaV offV tmpV 0 nn+      -- Run CM iteration+      goCM 0 (n - 1) 0+        mbaD offD mbaE offE tmpU 0 mm tmpV 0 nn n maxIter+      -- Transpose back to row-major+      rawTransposeFromColMajor tmpU 0 mbaU offU mm+      rawTransposeFromColMajor tmpV 0 mbaV offV nn+{-# NOINLINE bidiagQRIterPCM #-}++-- | CM iteration core with AED and stall detection.+-- Parameters: iter, lastQ, stallCount, then the usual d/e/U/V arrays + n + maxIter.+goCM :: Int -> Int -> Int+     -> MutableByteArray s -> Int -> MutableByteArray s -> Int+     -> MutableByteArray s -> Int -> Int+     -> MutableByteArray s -> Int -> Int+     -> Int -> Int -> ST s ()+goCM !iter !lastQ !stall mbaD offD mbaE offE mbaU offU mm mbaV offV nn n maxIter+  | iter >= maxIter = return ()+  | stall >= 20     = return ()  -- stall detection: bail after 20 steps without deflation+  | otherwise = do+      -- AED: scan bottom w superdiagonal entries for aggressive early deflation+      let w = min 6 ((n + 2) `div` 3)+      aedScan (n - 1) w+      -- Find q — bottom of unreduced block+      q <- defHiCM (n - 1)+      if q <= 0+        then return ()  -- fully deflated+        else do+          -- Find p — top of unreduced block+          p <- findLoCM (q - 1)+          -- Apply one CM QR step to [p..q]+          bidiagQRStepCM mbaD offD mbaE offE mbaU offU mm mbaV offV nn p q+          let !newStall = if q == lastQ then stall + 1 else 0+          goCM (iter + 1) q newStall mbaD offD mbaE offE mbaU offU mm mbaV offV nn n maxIter+  where+    -- AED: scan bottom w entries, deflating negligible superdiagonals+    aedScan _ 0 = return ()+    aedScan k remaining+      | k <= 0 = return ()+      | otherwise = do+          ek <- readRawD mbaE offE (k - 1)+          dk <- readRawD mbaD offD k+          dk1 <- readRawD mbaD offD (k - 1)+          let tol = 1e-14 * (abs dk1 + abs dk)+          if abs ek <= tol+            then do+              writeRawD mbaE offE (k - 1) 0+              aedScan (k - 1) (remaining - 1)+            else return ()  -- stop at first non-negligible entry++    defHiCM !hi+      | hi <= 0 = return 0+      | otherwise = do+          ehi <- readRawD mbaE offE (hi - 1)+          dhi <- readRawD mbaD offD hi+          dhi1 <- readRawD mbaD offD (hi - 1)+          let tol = 1e-14 * (abs dhi1 + abs dhi)+          if abs ehi <= tol+            then do+              writeRawD mbaE offE (hi - 1) 0+              defHiCM (hi - 1)+            else return hi++    findLoCM !idx+      | idx <= 0 = return 0+      | otherwise = do+          eidx <- readRawD mbaE offE (idx - 1)+          didx <- readRawD mbaD offD idx+          didx1 <- readRawD mbaD offD (idx - 1)+          let tol = 1e-14 * (abs didx1 + abs didx)+          if abs eidx <= tol+            then do+              writeRawD mbaE offE (idx - 1) 0+              return idx+            else findLoCM (idx - 1)+{-# NOINLINE goCM #-}++-- | One implicit-shift QR step on bidiagonal [lo..hi] using column-major U,V.+-- Same as bidiagQRStep but calls rawMutApplyGivensColumnsCM for SIMD.+bidiagQRStepCM :: MutableByteArray s -> Int  -- d + offset+               -> MutableByteArray s -> Int  -- e + offset+               -> MutableByteArray s -> Int -> Int  -- U_CM + offset + mm+               -> MutableByteArray s -> Int -> Int  -- V_CM + offset + nn+               -> Int -> Int  -- lo, hi+               -> ST s ()+bidiagQRStepCM mbaD offD mbaE offE mbaU offU mm mbaV offV nn lo hi = do+  -- Compute Wilkinson shift from trailing 2×2 of T = B^T B+  dhi1 <- readRawD mbaD offD (hi - 1)+  dhi  <- readRawD mbaD offD hi+  ehi1 <- readRawD mbaE offE (hi - 1)+  ehi2 <- if hi >= 2 then readRawD mbaE offE (hi - 2) else return 0++  let t11 = dhi1 * dhi1 + (if hi - 1 > lo then ehi2 * ehi2 else 0)+      t12 = dhi1 * ehi1+      t22 = dhi * dhi + ehi1 * ehi1+      delta = (t11 - t22) / 2+      signD = if delta >= 0 then 1 else -1+      mu = t22 - t12 * t12 / (delta + signD * sqrt (delta * delta + t12 * t12))++  dlo <- readRawD mbaD offD lo+  elo <- readRawD mbaE offE lo+  let y = dlo * dlo - mu+      z = dlo * elo++  goChase lo y z+  where+    goChase k y_ z_ = do+      let (cosR, sinR) = givens y_ z_+      dk  <- readRawD mbaD offD k+      ek  <- readRawD mbaE offE k+      dk1 <- readRawD mbaD offD (k + 1)++      let dk'  = cosR * dk + sinR * ek+          ek'  = -sinR * dk + cosR * ek+          bulgeL = sinR * dk1+          dk1'   = cosR * dk1++      writeRawD mbaD offD k dk'+      writeRawD mbaE offE k ek'+      writeRawD mbaD offD (k + 1) dk1'++      -- Update e[k-1]: right Givens rotates entry from row above+      when (k > lo) $+        writeRawD mbaE offE (k - 1) (cosR * y_ + sinR * z_)++      -- Accumulate right rotation into V (column-major, SIMD)+      rawMutApplyGivensColumnsCM mbaV offV nn cosR sinR k (k+1) nn++      let (cosL, sinL) = givens dk' bulgeL++      let dk''  = cosL * dk' + sinL * bulgeL+          ek''  = cosL * ek' + sinL * dk1'+          dk1'' = -sinL * ek' + cosL * dk1'++      writeRawD mbaD offD k dk''+      writeRawD mbaE offE k ek''+      writeRawD mbaD offD (k + 1) dk1''++      when (k + 1 < hi) $ do+        ek1 <- readRawD mbaE offE (k + 1)+        let bulgeR = sinL * ek1+            ek1'   = cosL * ek1+        writeRawD mbaE offE (k + 1) ek1'++        -- Accumulate left rotation into U (column-major, SIMD)+        rawMutApplyGivensColumnsCM mbaU offU mm cosL sinL k (k+1) mm++        goChase (k + 1) ek'' bulgeR++      when (k + 1 >= hi) $+        rawMutApplyGivensColumnsCM mbaU offU mm cosL sinL k (k+1) mm++-- | One implicit-shift QR step on bidiagonal [lo..hi].+-- Computes Wilkinson shift from bottom 2×2 of B^T B,+-- then chases bulge via Givens rotations.+bidiagQRStep :: MutableByteArray s -> Int  -- d + offset+             -> MutableByteArray s -> Int  -- e + offset+             -> MutableByteArray s -> Int -> Int  -- U + offset + ucols+             -> MutableByteArray s -> Int -> Int  -- V + offset + vcols+             -> Int -> Int  -- lo, hi+             -> ST s ()+bidiagQRStep mbaD offD mbaE offE mbaU offU ucols mbaV offV vcols lo hi = do+  -- Compute Wilkinson shift from trailing 2×2 of T = B^T B+  dhi1 <- readRawD mbaD offD (hi - 1)+  dhi  <- readRawD mbaD offD hi+  ehi1 <- readRawD mbaE offE (hi - 1)+  ehi2 <- if hi >= 2 then readRawD mbaE offE (hi - 2) else return 0++  -- T = B^T B trailing 2×2:+  -- t11 = d[hi-1]^2 + e[hi-2]^2  (e[hi-2] = 0 if hi-1 == lo)+  -- t12 = d[hi-1] * e[hi-1]+  -- t22 = d[hi]^2 + e[hi-1]^2+  let t11 = dhi1 * dhi1 + (if hi - 1 > lo then ehi2 * ehi2 else 0)+      t12 = dhi1 * ehi1+      t22 = dhi * dhi + ehi1 * ehi1++  -- Wilkinson shift: eigenvalue of [[t11,t12],[t12,t22]] closer to t22+  let delta = (t11 - t22) / 2+      signD = if delta >= 0 then 1 else -1+      mu = t22 - t12 * t12 / (delta + signD * sqrt (delta * delta + t12 * t12))++  -- Initial values for bulge chase+  dlo <- readRawD mbaD offD lo+  elo <- readRawD mbaE offE lo+  let y = dlo * dlo - mu+      z = dlo * elo++  -- Chase bulge from lo to hi+  go lo y z+  where+    go k y_ z_ = do+      -- Right Givens rotation G(k,k+1,θ) to zero z in [y; z]+      let (cosR, sinR) = givens y_ z_+      -- Apply to columns k, k+1 of B (affects d[k], e[k], d[k+1], and possibly e[k-1])+      dk  <- readRawD mbaD offD k+      ek  <- readRawD mbaE offE k+      dk1 <- readRawD mbaD offD (k + 1)++      -- B * G^T: columns k and k+1 get mixed+      let dk'  = cosR * dk + sinR * ek+          ek'  = -sinR * dk + cosR * ek+          -- This creates a bulge at B[k+1,k]+          bulgeL = sinR * dk1+          dk1'   = cosR * dk1++      writeRawD mbaD offD k dk'+      writeRawD mbaE offE k ek'+      writeRawD mbaD offD (k + 1) dk1'++      -- Update e[k-1]: the right Givens also rotates the entry from the row above.+      -- For k > lo: B[k-1,k] was y_, B[k-1,k+1] was z_ (the bulge).+      -- After rotation: B[k-1,k] = cosR*y_ + sinR*z_ (= r), B[k-1,k+1] = 0.+      when (k > lo) $+        writeRawD mbaE offE (k - 1) (cosR * y_ + sinR * z_)++      -- Accumulate right rotation into V (columns k, k+1)+      rawMutApplyGivensColumns mbaV offV vcols cosR sinR k (k+1) vcols++      -- Left Givens rotation G(k,k+1,θ) to zero the bulge at (k+1, k)+      let (cosL, sinL) = givens dk' bulgeL++      -- G * B: rows k and k+1 get mixed+      let dk''  = cosL * dk' + sinL * bulgeL+          ek''  = cosL * ek' + sinL * dk1'+          dk1'' = -sinL * ek' + cosL * dk1'++      writeRawD mbaD offD k dk''+      writeRawD mbaE offE k ek''+      writeRawD mbaD offD (k + 1) dk1''++      -- This may create a new bulge at position (k, k+2) if k+1 < hi+      when (k + 1 < hi) $ do+        ek1 <- readRawD mbaE offE (k + 1)+        let bulgeR = sinL * ek1+            ek1'   = cosL * ek1+        writeRawD mbaE offE (k + 1) ek1'++        -- Accumulate left rotation into U (columns k, k+1)+        rawMutApplyGivensColumns mbaU offU ucols cosL sinL k (k+1) ucols++        -- Continue chase+        go (k + 1) ek'' bulgeR++      when (k + 1 >= hi) $+        -- Accumulate final left rotation+        rawMutApplyGivensColumns mbaU offU ucols cosL sinL k (k+1) ucols++-- | Compute Givens rotation coefficients (c, s) such that+-- @[c, s; -s, c] [a; b] = [r; 0]@, i.e. @-s*a + c*b = 0@ and @r = c*a + s*b > 0@.+--+-- This convention is chosen so that the bidiag QR bulge-chase formulas+-- @dk' = c*dk + s*ek@, @ek' = -s*dk + c*ek@ etc. are directly correct+-- for both left and right Givens rotations (GVL4 Algorithm 8.6.2).+givens :: Double -> Double -> (Double, Double)+givens a b+  | b == 0    = (1, 0)+  | abs b > abs a =+      let tau = a / b+          s   = 1 / sqrt (1 + tau * tau)+          c   = s * tau+      in (c, s)+  | otherwise =+      let tau = b / a+          c   = 1 / sqrt (1 + tau * tau)+          s   = c * tau+      in (c, s)+{-# INLINE givens #-}++-- | Accumulate a right Householder reflector into V.+-- Right reflector k: v stored in row k of frozen A, cols k+2..n-1, with v[k+1]=1 (implicit).+-- V = V * (I - beta * v * v^T)+-- For each row of V: V[row, k+1..n-1] -= (beta * Σ V[row,l] * v[l]) * v+rightQAccum :: MutableByteArray s -> Int -> Int  -- V + offset + vcols+            -> ByteArray -> Int -> Int           -- frozen A + offset + acols+            -> Double -> Int -> Int -> Int       -- beta, k, n, row+            -> ST s ()+rightQAccum mbaV offV vcols (ByteArray baA) offFA acols beta k nn row = do+  -- Phase 1: wi = beta * (V[row, k+1] + Σ_{l=k+2}^{n-1} V[row, l] * A[k, l])+  qrk1 <- readRawD mbaV offV (row * vcols + (k+1))+  acc <- goSum (k+2) 0+  let wi = beta * (qrk1 + acc)+  -- Phase 2: V[row, k+1] -= wi (implicit v[k+1]=1)+  writeRawD mbaV offV (row * vcols + (k+1)) (qrk1 - wi)+  -- V[row, l] -= wi * A[k, l] for l in k+2..n-1+  goUpdate (k+2) wi+  where+    goSum l acc_+      | l >= nn = return acc_+      | otherwise = do+          let vl = readBAI baA offFA (k * acols + l)+          qrl <- readRawD mbaV offV (row * vcols + l)+          goSum (l + 1) (acc_ + qrl * vl)++    goUpdate l wi+      | l >= nn = return ()+      | otherwise = do+          let vl = readBAI baA offFA (k * acols + l)+          qrl <- readRawD mbaV offV (row * vcols + l)+          writeRawD mbaV offV (row * vcols + l) (qrl - wi * vl)+          goUpdate (l + 1) wi++    readBAI ba_ off_ i_ =+      case indexDoubleArray# ba_ (case off_ of I# o -> o +# case i_ of I# ii -> ii) of+        v -> D# v+{-# INLINE rightQAccum #-}++-- | Read a Double from a MutableByteArray at element index.+readRawD :: MutableByteArray s -> Int -> Int -> ST s Double+readRawD (MutableByteArray mba) (I# off) (I# i) = ST $ \s ->+  case readDoubleArray# mba (off +# i) s of (# s', v #) -> (# s', D# v #)+{-# INLINE readRawD #-}++-- | Write a Double to a MutableByteArray at element index.+writeRawD :: MutableByteArray s -> Int -> Int -> Double -> ST s ()+writeRawD (MutableByteArray mba) (I# off) (I# i) (D# v) = ST $ \s ->+  case writeDoubleArray# mba (off +# i) v s of s' -> (# s', () #)+{-# INLINE writeRawD #-}++-- ============================================================================+-- Divide-and-conquer bidiagonal SVD (Gu-Eisenstat 1995, cf. LAPACK DBDSDC)+-- ============================================================================++-- | Small-subproblem threshold for D&C bidiagonal SVD.+-- Subproblems at or below this size use bidiag QR iteration.+dcBidiagThreshold :: Int+dcBidiagThreshold = 32++-- | Direct 2×2 upper bidiagonal SVD.+-- Given [[d0, e0], [0, d1]], compute singular values and rotation angles.+-- Returns (sigma_large, sigma_small, c_left, s_left, c_right, s_right) where+-- the left and right Givens rotations diagonalise B.+bidiag2x2SVD :: Double -> Double -> Double -> (Double, Double, Double, Double, Double, Double)+bidiag2x2SVD d0 e0 d1 =+  -- B^T B = [[d0², d0*e0], [d0*e0, e0²+d1²]]+  -- Eigenvalues of this 2×2 symmetric matrix give σ²+  let !a11 = d0 * d0+      !a12 = d0 * e0+      !a22 = e0 * e0 + d1 * d1+      !tr  = a11 + a22+      !det = a11 * a22 - a12 * a12+      !disc = max 0 (tr * tr - 4 * det)+      !sqrtDisc = sqrt disc+      !lam1 = (tr + sqrtDisc) / 2+      !lam2 = (tr - sqrtDisc) / 2+      !sig1 = sqrt (max 0 lam1)+      !sig2 = sqrt (max 0 lam2)+      -- Right rotation angle: diagonalise B^T B+      -- (a11 - lam2) * cv + a12 * sv = 0 => cv/sv = -a12/(a11-lam2)+      -- Or equivalently: atan2(a12, lam1 - a22)+      !cv = if abs a12 < 1e-300+            then 1+            else let d = a11 - lam2+                     r = sqrt (d * d + a12 * a12)+                 in d / r+      !sv = if abs a12 < 1e-300+            then 0+            else let d = a11 - lam2+                     r = sqrt (d * d + a12 * a12)+                 in a12 / r+      -- Left rotation: B * V = U * Sigma+      -- (d0*cv + e0*sv, -d0*sv + e0*cv)   = (sig1*cu, sig2*su_neg)+      -- (d1*sv,         d1*cv)             = (sig1*(-su), sig2*cu)+      -- From first row: sig1*cu = d0*cv + e0*sv+      !bv00 = d0 * cv + e0 * sv+      !bv10 = d1 * sv+      !r_l = sqrt (bv00 * bv00 + bv10 * bv10)+      !cu = if r_l < 1e-300 then 1 else bv00 / r_l+      !su = if r_l < 1e-300 then 0 else bv10 / r_l+  in (sig1, sig2, cu, su, cv, sv)+{-# INLINE bidiag2x2SVD #-}++-- | Divide-and-conquer bidiagonal SVD.+-- Replaces bidiagQRIterPCM for computing the SVD of a bidiagonal matrix.+--+-- Input: d[0..nn-1] (diagonal), e[0..nn-2] (superdiagonal).+-- Output: d[] overwritten with singular values,+--         U (mm×mm) and V (nn×nn) updated with accumulated rotations.+dcBidiagSVD :: forall s. MutableByteArray s -> Int    -- d + offset+            -> MutableByteArray s -> Int    -- e + offset+            -> MutableByteArray s -> Int -> Int  -- U + offset + mm+            -> MutableByteArray s -> Int -> Int  -- V + offset + nn+            -> Int                          -- nn (bidiag dimension)+            -> Double                       -- tolerance+            -> ST s ()+dcBidiagSVD mbaD offD mbaE offE mbaU offU mm mbaV offV nn0 nn tol = do+  -- Pre-allocate all workspace once at maximum size+  let !maxN = nn+  wsLam    <- newByteArray (maxN * 8)       -- new eigenvalues (squared)+  wsZ      <- newByteArray (maxN * 8)       -- z-vector+  wsDSort  <- newByteArray (maxN * 8)       -- sorted d² values+  wsZSort  <- newByteArray (maxN * 8)       -- sorted z values+  wsDOrig  <- newByteArray (maxN * 8)       -- original d values (unsquared)+  wsIdx    <- newByteArray (maxN * 8)       -- sort permutation (stored as Double)+  wsPerm   <- newByteArray (maxN * 8)       -- deflation permutation (Int)+  wsW      <- newByteArray (maxN * maxN * 8)  -- V-eigenvector matrix W_V+  wsWU     <- newByteArray (maxN * maxN * 8)  -- U-eigenvector matrix W_U++  -- Local accumulators for V (nn×nn) and U (nn×nn)+  -- U-local is nn×nn because we track rotations in singular-value index space+  wsVlocal <- newByteArray (maxN * maxN * 8)+  wsUlocal <- newByteArray (maxN * maxN * 8)++  -- GEMM workspace+  wsVsub   <- newByteArray (maxN * maxN * 8)  -- V column extraction buffer+  wsVres   <- newByteArray (maxN * maxN * 8)  -- V GEMM result+  wsUsub   <- newByteArray (maxN * maxN * 8)  -- U column extraction buffer+  wsUres   <- newByteArray (maxN * maxN * 8)  -- U GEMM result+  wsQtemp  <- newByteArray (maxN * maxN * 8)  -- QR base case scratch++  -- Initialise local accumulators as identity+  rawZeroDoubles wsVlocal 0 (maxN * maxN)+  rawZeroDoubles wsUlocal 0 (maxN * maxN)+  forM_ [0..maxN-1] $ \i -> do+    writeRawD wsVlocal 0 (i * maxN + i) 1+    writeRawD wsUlocal 0 (i * maxN + i) 1++  let -- Convert global index to local+      toLocal g = g++      -- Apply a k×k rotation matrix to wsVlocal columns [colOff..colOff+k-1]+      applyRotToVlocal !colOff !k rotMat = do+        forM_ [0..k-1] $ \j ->+          rawCopyColumn wsVlocal 0 maxN (colOff + j) wsVsub 0 k j maxN+        baVsub <- unsafeFreezeByteArray wsVsub+        baRot  <- unsafeFreezeByteArray rotMat+        rawZeroDoubles wsVres 0 (maxN * k)+        rawGemmKernel baVsub 0 baRot 0 wsVres 0 maxN k k+        forM_ [0..k-1] $ \j ->+          rawCopyColumn wsVres 0 k j wsVlocal 0 maxN (colOff + j) maxN++      -- Apply a k×k rotation matrix to wsUlocal columns [colOff..colOff+k-1]+      applyRotToUlocal !colOff !k rotMat = do+        forM_ [0..k-1] $ \j ->+          rawCopyColumn wsUlocal 0 maxN (colOff + j) wsUsub 0 k j maxN+        baUsub <- unsafeFreezeByteArray wsUsub+        baRot  <- unsafeFreezeByteArray rotMat+        rawZeroDoubles wsUres 0 (maxN * k)+        rawGemmKernel baUsub 0 baRot 0 wsUres 0 maxN k k+        forM_ [0..k-1] $ \j ->+          rawCopyColumn wsUres 0 k j wsUlocal 0 maxN (colOff + j) maxN++      -- The recursive D&C function+      dcGo :: Int -> Int -> ST s ()  -- s from ScopedTypeVariables+      dcGo lo hi+        -- Trivial: single element+        | lo >= hi = return ()++        -- Base case: 2×2 direct SVD+        | hi == lo + 1 = do+            d0_ <- readRawD mbaD offD lo+            e0_ <- readRawD mbaE offE lo+            d1_ <- readRawD mbaD offD hi+            let (!sig1, !sig2, !cu, !su, !cv, !sv) = bidiag2x2SVD d0_ e0_ d1_+            writeRawD mbaD offD lo sig1+            writeRawD mbaD offD hi sig2+            writeRawD mbaE offE lo 0+            -- Apply left Givens to Ulocal columns+            let !loL = toLocal lo+                !hiL = toLocal hi+            rawMutApplyGivensColumns wsUlocal 0 maxN cu su loL hiL maxN+            -- Apply right Givens to Vlocal columns+            rawMutApplyGivensColumns wsVlocal 0 maxN cv sv loL hiL maxN++        -- Small subproblem: use bidiag QR + GEMM to local accumulators+        | hi - lo + 1 <= dcBidiagThreshold = do+            let !k = hi - lo + 1+                !loL = toLocal lo+            -- Initialise k×k identities for U and V rotations+            rawZeroDoubles wsQtemp 0 (k * k)+            rawZeroDoubles wsW 0 (k * k)+            forM_ [0..k-1] $ \i -> do+              writeRawD wsQtemp 0 (i * k + i) 1+              writeRawD wsW     0 (i * k + i) 1+            -- Run bidiag QR iteration: wsQtemp accumulates left, wsW accumulates right+            bidiagQRIterP mbaD (offD + lo) mbaE (offE + lo) wsQtemp 0 k wsW 0 k k (30 * k)+            -- Apply rotations to local accumulators via GEMM+            applyRotToUlocal loL k wsQtemp+            applyRotToVlocal loL k wsW++        -- D&C merge+        | otherwise = do+            let !k   = (lo + hi) `div` 2+                !n1  = k - lo + 1+                !n2  = hi - k+                !nn_ = hi - lo + 1+                !kL  = toLocal k+                !loL = toLocal lo++            -- Read and modify the coupling element+            beta <- readRawD mbaE offE k+            dk   <- readRawD mbaD offD k+            dk1  <- readRawD mbaD offD (k + 1)+            let !absBeta = abs beta+                !rho = absBeta+            writeRawD mbaD offD k     (dk - absBeta)+            writeRawD mbaD offD (k+1) (dk1 - absBeta)+            writeRawD mbaE offE k 0++            -- Recurse on left [lo..k] and right [k+1..hi] subproblems+            dcGo lo k+            dcGo (k + 1) hi++            -- === Merge phase ===++            -- Extract z-vector from Vlocal accumulator rows+            -- z[0..n1-1] = last row of V₁ = row kL, columns loL..loL+n1-1+            forM_ [0..n1-1] $ \i -> do+              qv <- readRawD wsVlocal 0 (kL * maxN + (loL + i))+              writeRawD wsZ 0 i qv+            -- z[n1..nn_-1] = first row of V₂ = row (kL+1), columns loL+n1..loL+nn_-1+            forM_ [0..n2-1] $ \i -> do+              qv <- readRawD wsVlocal 0 ((kL + 1) * maxN + (loL + n1 + i))+              let !zv = if beta < 0 then negate qv else qv+              writeRawD wsZ 0 (n1 + i) zv++            -- Save original d-values (unsquared) for U-eigenvector computation+            forM_ [0..nn_-1] $ \i -> do+              di <- readRawD mbaD offD (lo + i)+              writeRawD wsDOrig 0 i di++            -- Square d-values for secular equation and copy into sort buffers+            forM_ [0..nn_-1] $ \i -> do+              di <- readRawD mbaD offD (lo + i)+              writeRawD wsDSort 0 i (di * di)+              writeRawD wsZSort 0 i =<< readRawD wsZ 0 i+              writeRawD wsIdx 0 i (fromIntegral i)++            -- Sort by d² values (insertion sort)+            forM_ [1..nn_-1] $ \i -> do+              di   <- readRawD wsDSort 0 i+              zi   <- readRawD wsZSort 0 i+              idxi <- readRawD wsIdx 0 i+              dOi  <- readRawD wsDOrig 0 i+              let insertAt !j+                    | j < 0 = do+                        writeRawD wsDSort 0 0 di+                        writeRawD wsZSort 0 0 zi+                        writeRawD wsIdx   0 0 idxi+                        writeRawD wsDOrig 0 0 dOi+                    | otherwise = do+                        dj <- readRawD wsDSort 0 j+                        if dj > di+                          then do+                            writeRawD wsDSort 0 (j+1) dj+                            writeRawD wsZSort 0 (j+1) =<< readRawD wsZSort 0 j+                            writeRawD wsIdx   0 (j+1) =<< readRawD wsIdx 0 j+                            writeRawD wsDOrig 0 (j+1) =<< readRawD wsDOrig 0 j+                            insertAt (j - 1)+                          else do+                            writeRawD wsDSort 0 (j+1) di+                            writeRawD wsZSort 0 (j+1) zi+                            writeRawD wsIdx   0 (j+1) idxi+                            writeRawD wsDOrig 0 (j+1) dOi+              insertAt (i - 1)++            -- Close-d deflation on squared values (cf. LAPACK dlaed2)+            dMaxSq <- readRawD wsDSort 0 (nn_ - 1)+            dMinSq <- readRawD wsDSort 0 0+            let !closeDTol = 8 * 2.220446049250313e-16+                          * max (abs dMaxSq) (abs dMinSq + rho)+            forM_ [0..nn_-2] $ \i -> do+              di  <- readRawD wsDSort 0 i+              di1 <- readRawD wsDSort 0 (i + 1)+              when (abs (di1 - di) <= closeDTol) $ do+                zi  <- readRawD wsZSort 0 i+                zi1 <- readRawD wsZSort 0 (i + 1)+                let !r = sqrt (zi * zi + zi1 * zi1)+                when (r > 1e-300) $ do+                  let !c = zi1 / r+                      !s = zi / r+                  writeRawD wsZSort 0 i 0+                  writeRawD wsZSort 0 (i + 1) r+                  -- Apply Givens to Vlocal columns (same as tridiagonal)+                  origI  <- readRawD wsIdx 0 i+                  origI1 <- readRawD wsIdx 0 (i + 1)+                  let !colI  = loL + (round origI  :: Int)+                      !colI1 = loL + (round origI1 :: Int)+                  rawMutApplyGivensColumns wsVlocal 0 maxN c s colI colI1 maxN+                  -- Also apply to Ulocal+                  rawMutApplyGivensColumns wsUlocal 0 maxN c s colI colI1 maxN++            -- Perturbation-based deflation+            zn2 <- sumZSq wsZSort 0 nn_+            let !eps_ = 2.220446049250313e-16+                !matNorm = max (abs dMaxSq) (abs dMinSq) + rho * zn2+                !basicDeflTol = max (tol * sqrt zn2) (8 * eps_ * matNorm)+                !pertDeflTol = sqrt (eps_ * (1 + matNorm) / max rho 1e-300)+                !deflTol = max basicDeflTol pertDeflTol+            kND <- deflatePartition wsZSort 0 wsPerm 0 nn_ deflTol++            -- Extract Vlocal columns permuted by sort order into wsVsub (maxN × nn_)+            forM_ [0..nn_-1] $ \sortedJ -> do+              origIdx <- readRawD wsIdx 0 sortedJ+              let !origJ = round origIdx :: Int+                  !srcCol = loL + origJ+              rawCopyColumn wsVlocal 0 maxN srcCol wsVsub 0 nn_ sortedJ maxN++            -- Also extract Ulocal columns+            forM_ [0..nn_-1] $ \sortedJ -> do+              origIdx <- readRawD wsIdx 0 sortedJ+              let !origJ = round origIdx :: Int+                  !srcCol = loL + origJ+              rawCopyColumn wsUlocal 0 maxN srcCol wsUsub 0 nn_ sortedJ maxN++            if kND == 0+              then do+                -- All deflated: singular values from sorted d², vectors from sorted cols+                forM_ [0..nn_-1] $ \i -> do+                  rawCopyColumn wsVsub 0 nn_ i wsVlocal 0 maxN (loL + i) maxN+                  rawCopyColumn wsUsub 0 nn_ i wsUlocal 0 maxN (loL + i) maxN+                forM_ [0..nn_-1] $ \i -> do+                  dsq <- readRawD wsDSort 0 i+                  writeRawD mbaD offD (lo + i) (sqrt (max 0 dsq))++              else if kND == nn_+                then do+                  -- No deflation: full secular solve + eigenvectors + dual GEMM+                  secularSolve wsLam 0 wsDSort 0 wsZSort 0 rho nn_ deflTol++                  -- V-eigenvectors via Gu-Eisenstat on (d², z, μ)+                  dcEigenvectors wsW 0 wsDSort 0 wsZSort 0 wsLam 0 rho nn_++                  -- U-eigenvectors: W_U[j,i] = dOrig[j] * W_V[j,i], then normalize+                  dcEigenvectorsBidiagU wsWU 0 wsDOrig 0 wsW 0 nn_++                  -- V-GEMM: wsVres = Vsub * W_V+                  do baVs <- unsafeFreezeByteArray wsVsub+                     baWv <- unsafeFreezeByteArray wsW+                     rawZeroDoubles wsVres 0 (maxN * nn_)+                     rawGemmKernel baVs 0 baWv 0 wsVres 0 maxN nn_ nn_+                  forM_ [0..nn_-1] $ \i ->+                    rawCopyColumn wsVres 0 nn_ i wsVlocal 0 maxN (loL + i) maxN++                  -- U-GEMM: wsUres = Usub * W_U+                  do baUs <- unsafeFreezeByteArray wsUsub+                     baWu <- unsafeFreezeByteArray wsWU+                     rawZeroDoubles wsUres 0 (maxN * nn_)+                     rawGemmKernel baUs 0 baWu 0 wsUres 0 maxN nn_ nn_+                  forM_ [0..nn_-1] $ \i ->+                    rawCopyColumn wsUres 0 nn_ i wsUlocal 0 maxN (loL + i) maxN++                  -- Write singular values = sqrt(|mu|)+                  forM_ [0..nn_-1] $ \i -> do+                    mu <- readRawD wsLam 0 i+                    writeRawD mbaD offD (lo + i) (sqrt (max 0 (abs mu)))++                else do+                  -- Partial deflation: reduced secular solve + reduced GEMM+                  -- Build compressed d²_nd and z_nd in wsQtemp+                  forM_ [0..kND-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    dpi <- readRawD wsDSort 0 pi_+                    zpi <- readRawD wsZSort 0 pi_+                    writeRawD wsQtemp 0 j dpi+                    writeRawD wsQtemp 0 (kND + j) zpi++                  -- Also build compressed dOrig_nd for U-eigenvectors+                  forM_ [0..kND-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    dO  <- readRawD wsDOrig 0 pi_+                    writeRawD wsQtemp 0 (2 * kND + j) dO++                  secularSolve wsLam 0 wsQtemp 0 wsQtemp kND rho kND deflTol+                  dcEigenvectors wsW 0 wsQtemp 0 wsQtemp kND wsLam 0 rho kND+                  dcEigenvectorsBidiagU wsWU 0 wsQtemp (2 * kND) wsW 0 kND++                  -- Copy deflated columns to local accumulators+                  forM_ [kND..nn_-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    rawCopyColumn wsVsub 0 nn_ pi_ wsVlocal 0 maxN (loL + j) maxN+                    rawCopyColumn wsUsub 0 nn_ pi_ wsUlocal 0 maxN (loL + j) maxN++                  -- Extract V_nd (maxN × kND) from non-deflated columns+                  forM_ [0..kND-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    rawCopyColumn wsVsub 0 nn_ pi_ wsVres 0 kND j maxN++                  -- V-GEMM: wsVsub(maxN×kND) = V_nd(maxN×kND) * W_V(kND×kND)+                  do baVnd <- unsafeFreezeByteArray wsVres+                     baWv  <- unsafeFreezeByteArray wsW+                     rawZeroDoubles wsVsub 0 (maxN * kND)+                     rawGemmKernel baVnd 0 baWv 0 wsVsub 0 maxN kND kND+                  forM_ [0..kND-1] $ \j ->+                    rawCopyColumn wsVsub 0 kND j wsVlocal 0 maxN (loL + j) maxN++                  -- Extract U_nd (maxN × kND) from non-deflated columns+                  forM_ [0..kND-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    rawCopyColumn wsUsub 0 nn_ pi_ wsUres 0 kND j maxN++                  -- U-GEMM: wsUsub(maxN×kND) = U_nd(maxN×kND) * W_U(kND×kND)+                  do baUnd <- unsafeFreezeByteArray wsUres+                     baWu  <- unsafeFreezeByteArray wsWU+                     rawZeroDoubles wsUsub 0 (maxN * kND)+                     rawGemmKernel baUnd 0 baWu 0 wsUsub 0 maxN kND kND+                  forM_ [0..kND-1] $ \j ->+                    rawCopyColumn wsUsub 0 kND j wsUlocal 0 maxN (loL + j) maxN++                  -- Write eigenvalues: non-deflated from secular, deflated from sorted d²+                  forM_ [0..kND-1] $ \i -> do+                    mu <- readRawD wsLam 0 i+                    writeRawD mbaD offD (lo + i) (sqrt (max 0 (abs mu)))+                  forM_ [kND..nn_-1] $ \j -> do+                    pi_ <- readRawI wsPerm 0 j+                    dsq <- readRawD wsDSort 0 pi_+                    writeRawD mbaD offD (lo + j) (sqrt (max 0 dsq))++  -- Run the D&C recursion+  dcGo 0 (nn - 1)++  -- Final step: apply local accumulators to global U and V via GEMM+  -- V[:, 0..nn-1] = V[:, 0..nn-1] * Vlocal+  forM_ [0..nn-1] $ \j ->+    rawCopyColumn mbaV offV nn0 j wsVsub 0 nn j nn0+  do baVs <- unsafeFreezeByteArray wsVsub+     baVl <- unsafeFreezeByteArray wsVlocal+     rawZeroDoubles wsVres 0 (nn0 * nn)+     rawGemmKernel baVs 0 baVl 0 wsVres 0 nn0 nn nn+  forM_ [0..nn-1] $ \j ->+    rawCopyColumn wsVres 0 nn j mbaV offV nn0 j nn0++  -- U[:, 0..nn-1] = U[:, 0..nn-1] * Ulocal+  forM_ [0..nn-1] $ \j ->+    rawCopyColumn mbaU offU mm j wsUsub 0 nn j mm+  do baUs <- unsafeFreezeByteArray wsUsub+     baUl <- unsafeFreezeByteArray wsUlocal+     rawZeroDoubles wsUres 0 (mm * nn)+     rawGemmKernel baUs 0 baUl 0 wsUres 0 mm nn nn+  forM_ [0..nn-1] $ \j ->+    rawCopyColumn wsUres 0 nn j mbaU offU mm j mm++-- | Compute U-eigenvectors from V-eigenvectors and original (unsquared) d-values.+-- W_U[j,i] = dOrig[j] * W_V[j,i], then normalize each column.+dcEigenvectorsBidiagU :: MutableByteArray s -> Int  -- W_U output (nn × nn)+                      -> MutableByteArray s -> Int  -- dOrig (unsquared singular values)+                      -> MutableByteArray s -> Int  -- W_V (V-eigenvectors, already computed)+                      -> Int                        -- nn+                      -> ST s ()+dcEigenvectorsBidiagU mbaWU offWU mbaDOrig offDO mbaWV offWV nn = do+  forM_ [0..nn-1] $ \i -> do+    -- W_U[:,i] = diag(dOrig) * W_V[:,i], then normalize+    norm2 <- goCol i 0 0+    let !invNorm = if norm2 > 0 then 1 / sqrt norm2 else 1+    forM_ [0..nn-1] $ \j -> do+      wuji <- readRawD mbaWU offWU (j * nn + i)+      writeRawD mbaWU offWU (j * nn + i) (wuji * invNorm)+  where+    goCol !i !j !acc+      | j >= nn = pure acc+      | otherwise = do+          dj   <- readRawD mbaDOrig offDO j+          wvji <- readRawD mbaWV offWV (j * nn + i)+          let !wuji = dj * wvji+          writeRawD mbaWU offWU (j * nn + i) wuji+          goCol i (j + 1) (acc + wuji * wuji)+{-# NOINLINE dcBidiagSVD #-}
+ src/Numeric/LinearAlgebra/Massiv/Eigen/Schur.hs view
@@ -0,0 +1,259 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Eigen.Schur+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Real Schur decomposition via the practical QR algorithm, following+-- Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4), Section 7.5,+-- pp. 393--417.+--+-- __Theorem 7.5.1 (Real Schur Decomposition, p. 393):__ For every+-- \(A \in \mathbb{R}^{n \times n}\) there exists an orthogonal matrix \(Q\)+-- such that+--+-- \[+--   A = Q \, T \, Q^T+-- \]+--+-- where \(T\) is upper /quasi/-triangular: its diagonal consists of \(1+-- \times 1\) blocks (real eigenvalues) and \(2 \times 2\) blocks whose+-- eigenvalues are complex conjugate pairs \(\alpha \pm \beta i\).+--+-- __Algorithm:__ The implementation follows GVL4 Algorithm 7.5.1 (Practical+-- QR Algorithm, p. 395):+--+--   1. Reduce \(A\) to upper Hessenberg form \(H\) via+--      "Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg".+--   2. Apply implicit single-shift QR iterations with Givens rotations on+--      \(H\), using the /Wilkinson shift/ (eigenvalue of the trailing \(2+--      \times 2\) block closest to \(h_{nn}\), p. 397) to accelerate+--      convergence.+--   3. Deflate converged eigenvalues from the bottom of the active+--      Hessenberg window.+--+-- The Wilkinson shift ensures global convergence; in practice, most+-- eigenvalues converge in only one or two iterations (GVL4, p. 397).+module Numeric.LinearAlgebra.Massiv.Eigen.Schur+  ( -- * Schur decomposition (Algorithm 7.5.1)+    schur+    -- * Eigenvalues from Schur form+  , eigenvalues+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg (hessenberg)+import Numeric.LinearAlgebra.Massiv.Orthogonal.Givens (givensRotation)++-- | Real Schur decomposition (GVL4 Theorem 7.5.1, p. 393; Algorithm 7.5.1,+-- p. 395).+--+-- Computes orthogonal \(Q\) and upper quasi-triangular \(T\) satisfying+--+-- \[+--   A = Q \, T \, Q^T+-- \]+--+-- The matrix \(T\) has the same eigenvalues as \(A\).  Its diagonal blocks+-- are either:+--+--   * \(1 \times 1\) — corresponding to a real eigenvalue, or+--   * \(2 \times 2\) — corresponding to a pair of complex conjugate+--     eigenvalues \(\alpha \pm \beta i\).+--+-- Internally the algorithm first reduces \(A\) to upper Hessenberg form via+-- 'Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg.hessenberg', then applies+-- implicit single-shift QR iterations using the /Wilkinson shift/ (GVL4,+-- p. 397) and Givens rotations.+--+-- Returns @(Q, T)@.+schur :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+      => Matrix n n r e+      -> Int   -- ^ Maximum iterations+      -> e     -- ^ Convergence tolerance+      -> (Matrix n n r e, Matrix n n r e)+schur a maxIter tol =+  let nn = dimVal @n+      -- Step 1: Reduce to Hessenberg form+      (q0, h0) = hessenberg a+      -- Step 2: QR iteration on Hessenberg matrix+      (qFinal, tFinal) = qrIteration nn q0 h0 maxIter tol+  in (qFinal, tFinal)++-- | Implicit QR iteration on an upper Hessenberg matrix.+qrIteration :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+            => Int -> Matrix n n r e -> Matrix n n r e -> Int -> e+            -> (Matrix n n r e, Matrix n n r e)+qrIteration nn q h maxIter tol = go 0 q h (nn - 1)+  where+    go :: Int -> Matrix n n r e -> Matrix n n r e -> Int -> (Matrix n n r e, Matrix n n r e)+    go iter q_ h_ p+      | iter >= maxIter = (q_, h_)+      | p <= 0 = (q_, h_)+      | otherwise =+        -- Check for convergence of h(p, p-1)+        let subdiag = abs (h_ ! (p, p - 1))+            diagSum = abs (h_ ! (p - 1, p - 1)) + abs (h_ ! (p, p))+        in if subdiag <= tol * diagSum+           then+             -- Deflate: set subdiagonal to zero, reduce problem size+             let h_new = makeMatrix @n @n @r $ \i j ->+                   if i == p && j == p - 1 then 0 else h_ ! (i, j)+             in go iter q_ h_new (p - 1)+           else+             -- Apply one QR step with Wilkinson shift+             let shift = wilkinsonShift (h_ ! (p-1, p-1)) (h_ ! (p-1, p))+                                         (h_ ! (p, p-1))   (h_ ! (p, p))+                 -- Shifted QR step: H - σI = QR, H_new = RQ + σI+                 -- Implemented via Givens rotations on Hessenberg matrix+                 (q_new, h_new) = qrStepGivens q_ h_ shift p+             in go (iter + 1) q_new h_new p++-- | Wilkinson shift (GVL4, p. 397).+--+-- Given the trailing \(2 \times 2\) block+--+-- \[+--   \begin{bmatrix} a & b \\ c & d \end{bmatrix}+-- \]+--+-- the Wilkinson shift is the eigenvalue of this block that is closest to+-- \(d\) (the bottom-right entry).  When the eigenvalues of the block are+-- complex the shift defaults to \(d\).+wilkinsonShift :: (Floating e, Ord e) => e -> e -> e -> e -> e+wilkinsonShift a b c d =+  let trace_ = a + d+      det_ = a * d - b * c+      disc = trace_ * trace_ / 4 - det_+  in if disc < 0+     then d  -- Complex eigenvalues; use d as shift+     else+       let sqrtDisc = sqrt disc+           mu1 = trace_ / 2 + sqrtDisc+           mu2 = trace_ / 2 - sqrtDisc+       in if abs (mu1 - d) < abs (mu2 - d) then mu1 else mu2++-- | One QR step on Hessenberg matrix using Givens rotations.+-- H ← shift, QR factorize, then H = RQ + shift.+qrStepGivens :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+             => Matrix n n r e -> Matrix n n r e -> e -> Int+             -> (Matrix n n r e, Matrix n n r e)+qrStepGivens q h shift p =+  let nn = dimVal @n+      -- Apply shift: H ← H - σI+      h_shifted = makeMatrix @n @n @r $ \i j ->+        if i == j then (h ! (i, j)) - shift else h ! (i, j)+      -- QR factorization via Givens rotations (only on the active part)+      (rotations, r) = applyGivensQR h_shifted p+      -- Form RQ + σI+      h_new = formRQ r rotations shift p+      -- Update Q+      q_new = updateQ q rotations p+  in (q_new, h_new)++-- | Apply Givens rotations to zero out subdiagonal of Hessenberg matrix.+-- Returns list of (c, s, row_index) and the resulting R.+applyGivensQR :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+              => Matrix n n r e -> Int -> ([(e, e, Int)], Matrix n n r e)+applyGivensQR h p = foldl step ([], h) [0..p-1]+  where+    nn = dimVal @n+    step (rots, hh) k =+      let (c, s) = givensRotation (hh ! (k, k)) (hh ! (k+1, k))+          hh' = applyGivensLeftSq c s k (k+1) hh+      in (rots ++ [(c, s, k)], hh')++-- | Apply Givens from left to a square matrix.+applyGivensLeftSq :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+                  => e -> e -> Int -> Int -> Matrix n n r e -> Matrix n n r e+applyGivensLeftSq c s ri rk h =+  makeMatrix @n @n @r $ \i j ->+    if i == ri then+      c * (h ! (ri, j)) - s * (h ! (rk, j))+    else if i == rk then+      s * (h ! (ri, j)) + c * (h ! (rk, j))+    else+      h ! (i, j)++-- | Form RQ + σI from R and the Givens rotations.+formRQ :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+       => Matrix n n r e -> [(e, e, Int)] -> e -> Int -> Matrix n n r e+formRQ r rots shift _ =+  let -- Apply rotations from the right: R·G₁ᵀ·G₂ᵀ·...+      rq = foldl (\mat (c, s, k) ->+        applyGivensRightSq c s k (k+1) mat+        ) r rots+  in -- Add back shift+    makeMatrix @(MatDim n) @(MatDim n) $ \i j ->+      if i == j then (rq ! (i, j)) + shift else rq ! (i, j)++type MatDim n = n  -- type alias to avoid ambiguity++-- | Apply Givens from right to a square matrix.+applyGivensRightSq :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+                   => e -> e -> Int -> Int -> Matrix n n r e -> Matrix n n r e+applyGivensRightSq c s ci ck h =+  makeMatrix @n @n @r $ \i j ->+    if j == ci then+      c * (h ! (i, ci)) - s * (h ! (i, ck))+    else if j == ck then+      s * (h ! (i, ci)) + c * (h ! (i, ck))+    else+      h ! (i, j)++-- | Update Q by applying Givens rotations from the right.+updateQ :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+        => Matrix n n r e -> [(e, e, Int)] -> Int -> Matrix n n r e+updateQ q rots _ = foldl (\qq (c, s, k) ->+  applyGivensRightSq c s k (k+1) qq+  ) q rots++-- | Extract eigenvalues from a (quasi-)upper triangular Schur form \(T\).+--+-- The Schur matrix \(T\) produced by 'schur' has \(1 \times 1\) and+-- \(2 \times 2\) diagonal blocks.  This function walks the diagonal and+-- extracts eigenvalues:+--+--   * A \(1 \times 1\) block \([t_{ii}]\) yields the real eigenvalue+--     \(\lambda = t_{ii}\).+--   * A \(2 \times 2\) block+--     \(\bigl[\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\bigr]\)+--     yields eigenvalues \(\tfrac{a + d}{2} \pm \sqrt{\tfrac{(a+d)^2}{4} -+--     (ad - bc)}\).  When the discriminant is negative (complex conjugate+--     pair) only the real part \(\tfrac{a + d}{2}\) is returned for each+--     eigenvalue, since this module operates over real scalars.+--+-- See GVL4 Section 7.5 for the definition of the real Schur form.+eigenvalues :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+            => Matrix n n r e -> [e]+eigenvalues t =+  let nn = dimVal @n+  in go 0+  where+    nn = dimVal @n+    go i+      | i >= nn = []+      | i == nn - 1 = [t ! (i, i)]  -- Last 1×1 block+      | abs (t ! (i+1, i)) < 1e-12 * (abs (t ! (i, i)) + abs (t ! (i+1, i+1))) =+          -- 1×1 block+          t ! (i, i) : go (i + 1)+      | otherwise =+          -- 2×2 block: eigenvalues of [[a,b],[c,d]]+          let a = t ! (i, i)+              b = t ! (i, i+1)+              c = t ! (i+1, i)+              d = t ! (i+1, i+1)+              tr = a + d+              det_ = a * d - b * c+              disc = tr * tr / 4 - det_+          in if disc >= 0+             then (tr / 2 + sqrt disc) : (tr / 2 - sqrt disc) : go (i + 2)+             else tr / 2 : tr / 2 : go (i + 2)  -- Complex pair, return real parts
+ src/Numeric/LinearAlgebra/Massiv/Eigen/Symmetric.hs view
@@ -0,0 +1,1899 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}+{-# LANGUAGE BangPatterns #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Eigenvalue algorithms specialised to real symmetric matrices, following+-- Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4), Chapter 8,+-- pp. 449--512.+module Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+  ( tridiagonalize+  , tridiagonalizeP+  , symmetricEigen+  , symmetricEigenP+  , symmetricEigenPPar+  , symmetricEigenPDC+  , jacobiEigen+  -- * D&C secular equation infrastructure (for bidiagonal SVD reuse)+  , secularSolve+  , secularSolveOne+  , deflatePartition+  , dcEigenvectors+  , secularFuncSplit+  , secularFuncAndDeriv+  , sumZSq+  , farPoleSum+  , farPoleSumSkip+  , readRawD+  , writeRawD+  , readRawI+  , writeRawI+  , indexRawD+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Ix1, unwrapByteArray, unwrapByteArrayOffset, unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)+import Control.Monad (when, forM_)+import Control.Monad.ST (ST, stToIO)+import Control.Concurrent (forkIO, newEmptyMVar, putMVar, takeMVar)+import System.IO.Unsafe (unsafePerformIO)++import GHC.Exts+import GHC.ST (ST(..))+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..), newByteArray, unsafeFreezeByteArray)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Orthogonal.Givens (givensRotation)+import Numeric.LinearAlgebra.Massiv.Internal.Kernel+  ( rawMutApplyGivensColumns+  , rawMutApplyGivensColumnsCM+  , rawMutSumSqColumn+  , rawMutSymMatvecSub+  , rawMutSymRank2Update+  , rawMutTridiagQAccum+  , rawGemmKernel+  , rawTransposeToColMajor+  , rawTransposeFromColMajor+  , rawZeroDoubles+  , rawCopyDoubles+  , rawNegateDoubles+  , rawCopyColumn+  )++-- | Reduce a symmetric matrix to tridiagonal form (GVL4 Algorithm 8.3.1).+tridiagonalize :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+               => Matrix n n r e+               -> (Matrix n n r e, Vector n r e, Vector n r e)+tridiagonalize a =+  let nn = dimVal @n++      -- Phase 1: In-place tridiagonalisation via symmetric rank-2 updates.+      (betaList, tArr) = M.withMArrayST (unMatrix a) $ \mt -> do+        betas <- mapM (tridiagStep mt nn) [0..nn-3]+        pure betas++      -- Phase 2: Accumulate Q from stored Householder vectors.+      qMat = createMatrix @n @n @r $ \mq -> do+        forM_ [0..nn-1] $ \i -> forM_ [0..nn-1] $ \j ->+          M.write_ mq (i :. j) (if i == j then 1 else 0)+        -- Forward accumulation: Q <- Q · H_k for k = 0..n-3+        forM_ (zip [0..] betaList) $ \(k, beta_k) ->+          when (beta_k /= 0) $+            forM_ [0..nn-1] $ \i -> do+              qik1 <- M.readM mq (i :. (k+1))+              rest <- sumQV mq tArr i (k+1) nn k+              let wi = beta_k * (qik1 + rest)+              M.write_ mq (i :. (k+1)) (qik1 - wi)+              forM_ [k+2..nn-1] $ \l -> do+                let vl = M.index' tArr (l :. k)+                qil <- M.readM mq (i :. l)+                M.write_ mq (i :. l) (qil - wi * vl)++      diag_ = makeVector @n @r $ \i -> M.index' tArr (i :. i)+      subdiag = makeVector @n @r $ \i ->+        if i < nn - 1 then M.index' tArr ((i+1) :. i) else 0++  in (qMat, diag_, subdiag)++-- | One step of Householder tridiagonalisation.+tridiagStep :: (M.Manifest r e, Floating e, Ord e)+            => M.MArray s r Ix2 e -> Int -> Int -> ST s e+tridiagStep mt nn k = do+  x0 <- M.readM mt ((k+1) :. k)+  sigma <- sumSqBelow mt (k+1) nn k+  if sigma == 0 && x0 >= 0+    then pure 0+    else do+      let mu = sqrt (x0 * x0 + sigma)+          v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+          beta = 2 * v0 * v0 / (sigma + v0 * v0)+      -- Build v as a list: v(k+1)=1, v(i)=T(i,k)/v0 for i>k+1+      vList <- mapM (\i -> do+        tik <- M.readM mt (i :. k)+        pure (tik / v0)+        ) [k+2..nn-1]+      let fullV = 1 : vList  -- indices k+1, k+2, ..., n-1+      -- p = beta * T * v (rows k+1..n-1)+      pList <- mapM (\i -> do+        s <- dotTV mt i fullV (k+1) nn+        pure (beta * s)+        ) [k+1..nn-1]+      let ptv = sum $ zipWith (*) pList fullV+          alpha_ = beta * ptv / 2+          wList = zipWith (\pi_ vi -> pi_ - alpha_ * vi) pList fullV+      -- Symmetric rank-2 update: T(i,j) -= v(i)*w(j) + w(i)*v(j)+      forM_ (zip3 [k+1..nn-1] fullV wList) $ \(i, vi, wi) ->+        forM_ (zip3 [k+1..nn-1] fullV wList) $ \(j, vj, wj) -> do+          tij <- M.readM mt (i :. j)+          M.write_ mt (i :. j) (tij - vi * wj - wi * vj)+      -- Store Householder vector in below-subdiagonal of column k+      forM_ (zip [k+2..nn-1] vList) $ \(i, vi) ->+        M.write_ mt (i :. k) vi+      -- Set subdiagonal+      M.write_ mt ((k+1) :. k) mu+      M.write_ mt (k :. (k+1)) mu+      pure beta++-- Helpers for tridiagonalize+sumSqBelow :: (M.Manifest r e, Num e) => M.MArray s r Ix2 e -> Int -> Int -> Int -> ST s e+sumSqBelow mt start end col = go (start + 1) 0+  where go i !acc | i >= end = pure acc+                  | otherwise = do v <- M.readM mt (i :. col); go (i+1) (acc + v*v)++dotTV :: (M.Manifest r e, Num e) => M.MArray s r Ix2 e -> Int -> [e] -> Int -> Int -> ST s e+dotTV mt i vList start end = go start vList 0+  where go _ [] !acc = pure acc+        go j (v:vs) !acc | j >= end = pure acc+                         | otherwise = do t <- M.readM mt (i :. j); go (j+1) vs (acc + t*v)++sumQV :: (M.Manifest r1 e, M.Manifest r2 e, Num e)+      => M.MArray s r1 Ix2 e -> M.Array r2 Ix2 e -> Int -> Int -> Int -> Int -> ST s e+sumQV mq tArr row start end col = go (start + 1) 0+  where go l !acc | l >= end = pure acc+                  | otherwise = do+                      q <- M.readM mq (row :. l)+                      let v = M.index' tArr (l :. col)+                      go (l+1) (acc + q*v)++-- | Raw-primop tridiagonalisation specialised for @P Double@.+-- Two-phase: (1) in-place Householder via raw ByteArray# kernels,+-- (2) Q accumulation using rawMutTridiagQAccum.+tridiagonalizeP :: forall n. KnownNat n+                => Matrix n n M.P Double+                -> (Matrix n n M.P Double, Vector n M.P Double, Vector n M.P Double)+tridiagonalizeP a =+  let nn = dimVal @n++      -- Phase 1: In-place Householder tridiagonalisation+      -- For n < panelCrossover: per-column Level-2 (rank-2 update per step)+      -- For n >= panelCrossover: DLATRD-style panel factorisation (Level-3 SYR2K)+      panelCrossover = 64+      (betaList, tArr) = M.withMArrayST (unMatrix a) $ \mt -> do+        let !mbaT = unwrapMutableByteArray mt+            !offT = unwrapMutableByteArrayOffset mt+        mbaV <- newByteArray (nn * 8)+        mbaP <- newByteArray (nn * 8)+        mbaW <- newByteArray (nn * 8)+        if nn < panelCrossover+          then do+            betas <- mapM (\k -> tridiagStepP mbaT offT nn mbaV mbaP mbaW k) [0..nn-3]+            pure betas+          else do+            let !nb = min 64 (max 16 (nn `div` 3))+                !numRef = nn - 2  -- number of Householder reflectors+            -- V_panel (nn × nb) and W_panel (nn × nb) for deferred rank-2 updates+            mbaVp <- newByteArray (nn * nb * 8)+            mbaWp <- newByteArray (nn * nb * 8)+            -- Temporary for GEMM-based trailing update+            mbaTemp <- newByteArray (nn * nb * 8)+            -- Pre-allocate workspace for panelTridiagP (avoids per-panel allocation)+            wsHvSave <- newByteArray (nb * nn * 8)+            wsVr <- newByteArray (nn * nb * 8)+            wsWr <- newByteArray (nn * nb * 8)+            wsNWrT <- newByteArray (nb * nn * 8)+            wsNVrT <- newByteArray (nb * nn * 8)+            wsRem <- newByteArray (nn * nn * 8)+            let go !k0 !accBetas+                  | k0 > numRef - 1 = pure (reverse accBetas)+                  | otherwise = do+                      let !bs = min nb (numRef - k0)+                      panelBetas <- panelTridiagP mbaT offT nn mbaV mbaP mbaW+                                                  mbaVp mbaWp mbaTemp+                                                  wsHvSave wsVr wsWr wsNWrT wsNVrT wsRem+                                                  k0 bs+                      go (k0 + bs) (reverse panelBetas ++ accBetas)+            go 0 []++      -- Get underlying ByteArray from frozen T for Q accumulation+      !tBA  = unwrapByteArray tArr+      !tOff = unwrapByteArrayOffset tArr++      -- Phase 2: Q accumulation.+      -- For n < 200: per-row Householder updates (minimal work, avoids GEMM overhead).+      -- For n >= 200: blocked WY with Level-3 GEMM (better cache/SIMD utilization).+      qMat = createMatrix @n @n @M.P $ \mq -> do+        let !mbaQ = unwrapMutableByteArray mq+            !offQ = unwrapMutableByteArrayOffset mq+        -- Set Q = I (SIMD zero + diagonal ones)+        rawZeroDoubles mbaQ offQ (nn * nn)+        forM_ [0..nn-1] $ \i -> writeRawD mbaQ offQ (i*nn+i) 1+        if nn < 128+          then+            -- Per-row approach: Q <- Q · H_k for k = 0..n-3+            forM_ (zip [0..] betaList) $ \(k, beta_k) ->+              when (beta_k /= 0) $+                forM_ [0..nn-1] $ \row ->+                  rawMutTridiagQAccum mbaQ offQ nn tBA tOff nn beta_k (k+1) k nn row+          else do+            -- Blocked WY approach: Q <- Q * (I - Y * T * Y^T) per block+            let !numRef = nn - 2+                !nb = min 48 numRef+            mbaBetas <- newByteArray (numRef * 8)+            forM_ (zip [0..] betaList) $ \(i, b) -> writeRawD mbaBetas 0 i b+            mbaY  <- newByteArray (nn * nb * 8)+            mbaTf <- newByteArray (nb * nb * 8)+            mbaW1 <- newByteArray (nn * nb * 8)+            mbaW2 <- newByteArray (nn * nb * 8)+            mbaYT <- newByteArray (nb * nn * 8)+            mbaG  <- newByteArray (nb * nb * 8)  -- Gram matrix Y^T Y++            forM_ [0, nb .. numRef - 1] $ \k0 -> do+              let !bs = min nb (numRef - k0)++              -- Pack Y (n × bs) from stored Householder vectors+              rawZeroDoubles mbaY 0 (nn * bs)+              forM_ [0..bs-1] $ \j -> do+                let !k = k0 + j+                writeRawD mbaY 0 ((k+1) * bs + j) 1.0+                forM_ [k+2..nn-1] $ \l ->+                  writeRawD mbaY 0 (l * bs + j) (indexRawD tBA tOff (l * nn + k))++              -- Transpose Y → Y^T (bs × n) early: reused for T factor and final GEMM+              forM_ [0..nn-1] $ \row ->+                forM_ [0..bs-1] $ \col ->+                  writeRawD mbaYT 0 (col * nn + row) 0+              forM_ [0..bs-1] $ \j -> do+                let !k = k0 + j+                writeRawD mbaYT 0 (j * nn + (k+1)) 1.0+                forM_ [k+2..nn-1] $ \l ->+                  writeRawD mbaYT 0 (j * nn + l) (indexRawD tBA tOff (l * nn + k))++              -- Freeze Y and Y^T for GEMM use+              baY  <- unsafeFreezeByteArray mbaY+              baYT <- unsafeFreezeByteArray mbaYT++              -- Compute G = Y^T × Y (bs × bs) via GEMM for T factor dot products+              rawZeroDoubles mbaG 0 (bs * bs)+              rawGemmKernel baYT 0 baY 0 mbaG 0 bs nn bs++              -- Build T factor (bs × bs upper-triangular) using precomputed G+              rawZeroDoubles mbaTf 0 (bs * bs)+              forM_ [0..bs-1] $ \j -> do+                betaj <- readRawD mbaBetas 0 (k0 + j)+                writeRawD mbaTf 0 (j * bs + j) betaj+                when (j > 0 && betaj /= 0) $ do+                  -- Read G[i,j] = Y[:,i]^T Y[:,j] for all i < j+                  forM_ [0..j-1] $ \i -> do+                    g_ij <- readRawD mbaG 0 (i * bs + j)+                    writeRawD mbaW1 0 i g_ij+                  -- Triangular solve: T[i,j] = -betaj * Σ_l T[i,l] * G[l,j]+                  forM_ [0..j-1] $ \i -> do+                    let triLoop !l !acc+                          | l >= j = pure acc+                          | otherwise = do+                              til <- readRawD mbaTf 0 (i * bs + l)+                              dl  <- readRawD mbaW1 0 l+                              triLoop (l+1) (acc + til * dl)+                    z <- triLoop i 0+                    writeRawD mbaTf 0 (i * bs + j) (negate betaj * z)++              -- W1 = Q · Y (GEMM n×n * n×bs → n×bs)+              baQ <- unsafeFreezeByteArray mbaQ+              rawZeroDoubles mbaW1 0 (nn * bs)+              rawGemmKernel baQ offQ baY 0 mbaW1 0 nn nn bs++              -- W2 = W1 · T (GEMM n×bs * bs×bs → n×bs)+              baW1 <- unsafeFreezeByteArray mbaW1+              baTf <- unsafeFreezeByteArray mbaTf+              rawZeroDoubles mbaW2 0 (nn * bs)+              rawGemmKernel baW1 0 baTf 0 mbaW2 0 nn bs bs++              -- Negate W2 in-place (SIMD)+              rawNegateDoubles mbaW2 0 (nn * bs)++              -- Q += (-W2) · Y^T (GEMM n×bs * bs×n → n×n) — reuses baYT+              baNW2 <- unsafeFreezeByteArray mbaW2+              rawGemmKernel baNW2 0 baYT 0 mbaQ offQ nn bs nn++      -- Read diagonal and subdiagonal from frozen T+      diag_   = makeVector @n @M.P $ \i -> M.index' tArr (i :. i)+      subdiag = makeVector @n @M.P $ \i ->+        if i < nn - 1 then M.index' tArr ((i+1) :. i) else 0++  in (qMat, diag_, subdiag)+{-# NOINLINE tridiagonalizeP #-}++-- | One step of raw-primop Householder tridiagonalisation.+tridiagStepP :: MutableByteArray s -> Int -> Int+             -> MutableByteArray s -> MutableByteArray s -> MutableByteArray s+             -> Int -> ST s Double+tridiagStepP mbaT offT nn mbaV mbaP mbaW k = do+  -- 1. Read x0 = T[k+1,k]+  x0 <- readRawD mbaT offT ((k+1)*nn + k)+  -- 2. Compute sigma = Σ T[i,k]^2 for i=k+2..nn-1+  sigma <- rawMutSumSqColumn mbaT offT nn (k+2) nn k+  if sigma == 0 && x0 >= 0+    then pure 0+    else do+      let mu = sqrt (x0 * x0 + sigma)+          v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+          beta = 2 * v0 * v0 / (sigma + v0 * v0)+          subSize = nn - k - 1++      -- 3. Build v in mbaV: v[0]=1, v[i]=T[k+1+i,k]/v0+      writeRawD mbaV 0 0 1.0+      forM_ [1..subSize-1] $ \i -> do+        tik <- readRawD mbaT offT ((k+1+i)*nn + k)+        writeRawD mbaV 0 i (tik / v0)++      -- 4. p = beta * T_sub * v+      rawMutSymMatvecSub mbaT offT nn mbaV 0 mbaP 0 (k+1) nn+      forM_ [0..subSize-1] $ \i -> do+        pi_ <- readRawD mbaP 0 i+        writeRawD mbaP 0 i (beta * pi_)++      -- 5. Dot product p^T v+      ptv <- mutDotVec mbaP 0 mbaV 0 subSize+      let alpha_ = beta * ptv / 2++      -- 6. w = p - alpha*v+      forM_ [0..subSize-1] $ \i -> do+        pi_ <- readRawD mbaP 0 i+        vi  <- readRawD mbaV 0 i+        writeRawD mbaW 0 i (pi_ - alpha_ * vi)++      -- 7. Rank-2 update: T -= vw^T + wv^T+      rawMutSymRank2Update mbaT offT nn mbaV 0 mbaW 0 (k+1) nn++      -- 8. Store Householder vector in column k subdiagonal+      forM_ [1..subSize-1] $ \i -> do+        vi <- readRawD mbaV 0 i+        writeRawD mbaT offT ((k+1+i)*nn + k) vi++      -- 9. Set subdiagonal element+      writeRawD mbaT offT ((k+1)*nn + k) mu+      writeRawD mbaT offT (k*nn + (k+1)) mu++      pure beta++-- | DLATRD-style panel tridiagonalisation.+-- Processes columns k0..k0+bs-1, building V_panel and W_panel matrices+-- that represent the deferred rank-2 updates. After processing all columns+-- in the panel, applies a single Level-3 SYR2K trailing update.+--+-- Within the panel, column k of T is corrected for deferred updates:+--   T[:,k] -= V_panel * W_panel[k,:] + W_panel * V_panel[k,:]+-- before computing the Householder reflector.+--+-- Returns the list of beta values for the panel columns.+panelTridiagP :: MutableByteArray s -> Int -> Int  -- T matrix, offset, n+              -> MutableByteArray s -> MutableByteArray s -> MutableByteArray s  -- v, p, w temps+              -> MutableByteArray s -> MutableByteArray s -> MutableByteArray s  -- Vp, Wp, temp+              -> MutableByteArray s -> MutableByteArray s -> MutableByteArray s  -- wsHvSave, wsVr, wsWr+              -> MutableByteArray s -> MutableByteArray s -> MutableByteArray s  -- wsNWrT, wsNVrT, wsRem+              -> Int -> Int  -- k0, bs (panel start, panel size)+              -> ST s [Double]+panelTridiagP mbaT offT nn mbaV mbaP mbaW mbaVp mbaWp _mbaTemp+              mbaHvSave mbaVr mbaWr mbaNWrT mbaNVrT mbaRem k0 bs = do+  -- DLATRD-style: NO rank-2 updates to T within the panel.+  -- All corrections computed from V_panel, W_panel.+  -- After the panel, apply SYR2K to the full remaining submatrix.+  betas <- go 0 []++  -- After the panel: apply accumulated rank-2 update to the full remaining+  -- submatrix T[k0+1:nn, k0+1:nn]. This includes both within-panel diagonal+  -- entries and the trailing submatrix.+  --+  -- We must save/restore Householder vectors in columns k0..k0+bs-1+  -- because the SYR2K will overwrite them.+  let !remStart = k0 + 1+      !remSize = nn - remStart+  when (remSize > 0 && bs > 0) $ do+    -- Save Householder vectors from T columns k0..k0+bs-1+    -- These are T[i, k] for i > k+1, k in [k0..k0+bs-1]+    -- Also save subdiagonal entries T[k+1, k] = mu+    forM_ [0..bs-1] $ \l -> do+      let !k = k0 + l+          !startRow = k + 1+      forM_ [startRow..nn-1] $ \i -> do+        val <- readRawD mbaT offT (i * nn + k)+        writeRawD mbaHvSave 0 (l * nn + i) val+      -- Also save T[k, k+1] (the upper subdiagonal)+      when (k + 1 < nn) $ do+        val <- readRawD mbaT offT (k * nn + (k + 1))+        writeRawD mbaHvSave 0 (l * nn + k) val  -- reuse slot k < startRow++    -- Build contiguous V_rem (remSize × bs) and W_rem (remSize × bs)+    -- V_panel and W_panel have stride bs, so V_rem is a contiguous subblock+    rawCopyDoubles mbaVr 0 mbaVp (remStart * bs) (remSize * bs)+    rawCopyDoubles mbaWr 0 mbaWp (remStart * bs) (remSize * bs)++    -- Build -W_rem^T and -V_rem^T (bs × remSize) via transpose + negate+    forM_ [0..remSize-1] $ \i ->+      forM_ [0..bs-1] $ \j -> do+        readRawD mbaWr 0 (i * bs + j) >>= writeRawD mbaNWrT 0 (j * remSize + i)+        readRawD mbaVr 0 (i * bs + j) >>= writeRawD mbaNVrT 0 (j * remSize + i)+    rawNegateDoubles mbaNWrT 0 (bs * remSize)+    rawNegateDoubles mbaNVrT 0 (bs * remSize)++    -- Copy T_rem to contiguous temp (row-by-row bulk copy)+    forM_ [0..remSize-1] $ \i ->+      rawCopyDoubles mbaRem (i * remSize) mbaT (offT + (remStart + i) * nn + remStart) remSize++    -- GEMM: rem += V_rem * (-W_rem^T) + W_rem * (-V_rem^T)+    baVr <- unsafeFreezeByteArray mbaVr+    baNWrT <- unsafeFreezeByteArray mbaNWrT+    rawGemmKernel baVr 0 baNWrT 0 mbaRem 0 remSize bs remSize+    baWr <- unsafeFreezeByteArray mbaWr+    baNVrT <- unsafeFreezeByteArray mbaNVrT+    rawGemmKernel baWr 0 baNVrT 0 mbaRem 0 remSize bs remSize++    -- Copy back to T (row-by-row bulk copy)+    forM_ [0..remSize-1] $ \i ->+      rawCopyDoubles mbaT (offT + (remStart + i) * nn + remStart) mbaRem (i * remSize) remSize++    -- Restore saved Householder vectors and subdiagonal entries+    forM_ [0..bs-1] $ \l -> do+      let !k = k0 + l+          !startRow = k + 1+      forM_ [startRow..nn-1] $ \i -> do+        val <- readRawD mbaHvSave 0 (l * nn + i)+        writeRawD mbaT offT (i * nn + k) val+      when (k + 1 < nn) $ do+        val <- readRawD mbaHvSave 0 (l * nn + k)+        writeRawD mbaT offT (k * nn + (k + 1)) val++  pure betas+  where+    go !j !acc+      | j >= bs = pure (reverse acc)+      | otherwise = do+          let !k = k0 + j+              !subSize = nn - k - 1++          -- Step 1: Read corrected column. T is ORIGINAL (no rank-2 updates applied).+          -- corrected_col[i] = T[i+k+1, k] - Σ_l (V[i+k+1,l]*W[k,l] + W[i+k+1,l]*V[k,l])+          forM_ [0..subSize-1] $ \i -> do+            tik <- readRawD mbaT offT ((k+1+i)*nn + k)+            if j == 0+              then writeRawD mbaP 0 i tik+              else do+                let corrLoop !l !accC+                      | l >= j = pure accC+                      | otherwise = do+                          vp_il <- readRawD mbaVp 0 ((k+1+i) * bs + l)+                          wp_kl <- readRawD mbaWp 0 (k * bs + l)+                          wp_il <- readRawD mbaWp 0 ((k+1+i) * bs + l)+                          vp_kl <- readRawD mbaVp 0 (k * bs + l)+                          corrLoop (l+1) (accC + vp_il * wp_kl + wp_il * vp_kl)+                corr <- corrLoop 0 0+                writeRawD mbaP 0 i (tik - corr)++          -- Step 2: Householder from corrected column+          x0 <- readRawD mbaP 0 0+          sigma <- do+            let sigLoop !i !acc_+                  | i >= subSize = pure acc_+                  | otherwise = do+                      ci <- readRawD mbaP 0 i+                      sigLoop (i+1) (acc_ + ci * ci)+            sigLoop 1 0+          if sigma == 0 && x0 >= 0+            then do+              forM_ [0..nn-1] $ \i -> do+                writeRawD mbaVp 0 (i * bs + j) 0+                writeRawD mbaWp 0 (i * bs + j) 0+              go (j+1) (0 : acc)+            else do+              let mu = sqrt (x0 * x0 + sigma)+                  v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                  beta = 2 * v0 * v0 / (sigma + v0 * v0)++              writeRawD mbaV 0 0 1.0+              forM_ [1..subSize-1] $ \i -> do+                ci <- readRawD mbaP 0 i+                writeRawD mbaV 0 i (ci / v0)++              -- Step 3: p = beta * T * v (using ORIGINAL T, then correct via V,W)+              rawMutSymMatvecSub mbaT offT nn mbaV 0 mbaP 0 (k+1) nn+              forM_ [0..subSize-1] $ \i -> do+                pi_ <- readRawD mbaP 0 i+                writeRawD mbaP 0 i (beta * pi_)++              -- Step 4: Full V,W correction.+              -- p -= beta * (V_sub*(W_sub^T*v) + W_sub*(V_sub^T*v))+              -- where V_sub = V_panel[k+1:nn-1, 0:j-1], W_sub = W_panel[k+1:nn-1, 0:j-1]+              when (j > 0) $ do+                forM_ [0..j-1] $ \l -> do+                  let dotW !idx !accW+                        | idx >= subSize = pure accW+                        | otherwise = do+                            wp <- readRawD mbaWp 0 ((k+1+idx) * bs + l)+                            vi <- readRawD mbaV 0 idx+                            dotW (idx+1) (accW + wp * vi)+                  z1 <- dotW 0 0++                  let dotV !idx !accV+                        | idx >= subSize = pure accV+                        | otherwise = do+                            vp <- readRawD mbaVp 0 ((k+1+idx) * bs + l)+                            vi <- readRawD mbaV 0 idx+                            dotV (idx+1) (accV + vp * vi)+                  z2 <- dotV 0 0++                  forM_ [0..subSize-1] $ \i -> do+                    vp_il <- readRawD mbaVp 0 ((k+1+i) * bs + l)+                    wp_il <- readRawD mbaWp 0 ((k+1+i) * bs + l)+                    pi_ <- readRawD mbaP 0 i+                    writeRawD mbaP 0 i (pi_ - beta * (vp_il * z1 + wp_il * z2))++              -- Step 5: w = p - alpha*v+              ptv <- mutDotVec mbaP 0 mbaV 0 subSize+              let alpha_ = beta * ptv / 2+              forM_ [0..subSize-1] $ \i -> do+                pi_ <- readRawD mbaP 0 i+                vi  <- readRawD mbaV 0 i+                writeRawD mbaW 0 i (pi_ - alpha_ * vi)++              -- Step 6: Store v,w in panels+              forM_ [0..k] $ \i -> do+                writeRawD mbaVp 0 (i * bs + j) 0+                writeRawD mbaWp 0 (i * bs + j) 0+              forM_ [0..subSize-1] $ \i -> do+                readRawD mbaV 0 i >>= writeRawD mbaVp 0 ((k+1+i) * bs + j)+                readRawD mbaW 0 i >>= writeRawD mbaWp 0 ((k+1+i) * bs + j)++              -- Step 7: Store Householder vector in T (for Q accumulation)+              forM_ [1..subSize-1] $ \i ->+                readRawD mbaV 0 i >>= writeRawD mbaT offT ((k+1+i)*nn + k)++              -- Step 8: Set subdiagonal+              writeRawD mbaT offT ((k+1)*nn + k) mu+              writeRawD mbaT offT (k*nn + (k+1)) mu++              go (j+1) (beta : acc)++-- | Dot product of two mutable vectors.+mutDotVec :: MutableByteArray s -> Int -> MutableByteArray s -> Int -> Int -> ST s Double+mutDotVec mbaA offA mbaB offB n = go 0 0+  where+    go !i !acc+      | i >= n = pure acc+      | otherwise = do+          ai <- readRawD mbaA offA i+          bi <- readRawD mbaB offB i+          go (i+1) (acc + ai * bi)++-- | Symmetric eigenvalue decomposition (GVL4 Algorithm 8.3.3).+symmetricEigen :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+               => Matrix n n r e -> Int -> e -> (Vector n r e, Matrix n n r e)+symmetricEigen a maxIter tol =+  let nn = dimVal @n+      (q0, diag_, subdiag) = tridiagonalize a+      (dArr, qArr) = M.withMArrayST (unMatrix q0) $ \mq -> do+        md <- M.thawS (unVector diag_)+        msd <- M.thawS (unVector subdiag)+        tridiagQRLoop md msd mq nn maxIter tol+        dFrozen <- M.freezeS md+        pure (MkVector dFrozen)+  in (dArr, MkMatrix qArr)++-- | In-place QR iteration on tridiagonal (d, sd) with mutable Q.+-- Uses both top and bottom deflation to shrink the active range [lo..hi],+-- effectively achieving divide-and-conquer behaviour.+tridiagQRLoop :: (M.Manifest r e, Floating e, Ord e)+              => M.MArray s r Ix1 e -> M.MArray s r Ix1 e -> M.MArray s r Ix2 e+              -> Int -> Int -> e -> ST s ()+tridiagQRLoop md msd mq nn maxIter tol = go 0 0 (nn - 1)+  where+    go !iter !lo !hi+      | iter >= maxIter = pure ()+      | lo >= hi = pure ()+      | otherwise = do+          -- Bottom deflation+          sdhi <- M.readM msd (hi - 1)+          dhi1 <- M.readM md (hi - 1)+          dhi  <- M.readM md hi+          if abs sdhi <= tol * (abs dhi1 + abs dhi)+            then do+              M.write_ msd (hi - 1) 0+              go iter lo (hi - 1)+            else do+              -- Top deflation+              sdlo <- M.readM msd lo+              dlo  <- M.readM md lo+              dlo1 <- M.readM md (lo + 1)+              if abs sdlo <= tol * (abs dlo + abs dlo1)+                then do+                  M.write_ msd lo 0+                  go iter (lo + 1) hi+                else do+                  -- Interior deflation: find split point+                  split <- findSplit md msd lo hi tol+                  case split of+                    Just q -> do+                      -- Split into two subproblems [lo..q] and [q+1..hi]+                      M.write_ msd q 0+                      go iter lo q+                      go iter (q + 1) hi+                    Nothing -> do+                      -- No split found: apply QR step on [lo..hi]+                      let sp1 = sdhi+                          delta = (dhi1 - dhi) / 2+                          sgn = if delta >= 0 then 1 else -1+                          shift = dhi - sp1*sp1 / (delta + sgn * sqrt (delta*delta + sp1*sp1))+                      implicitQRStepInPlace md msd mq nn shift lo hi+                      go (iter + 1) lo hi++-- | Find an interior split point where the subdiagonal is negligible.+findSplit :: (M.Manifest r e, Floating e, Ord e)+          => M.MArray s r Ix1 e -> M.MArray s r Ix1 e -> Int -> Int -> e -> ST s (Maybe Int)+findSplit md msd lo hi tol = scan (lo + 1)+  where+    scan q+      | q >= hi - 1 = pure Nothing+      | otherwise = do+          sdq <- M.readM msd q+          dq  <- M.readM md q+          dq1 <- M.readM md (q + 1)+          if abs sdq <= tol * (abs dq + abs dq1)+            then pure (Just q)+            else scan (q + 1)++-- | One implicit symmetric QR step via bulge-chasing Givens rotations.+-- Operates on the active sub-range [lo..hi] of the tridiagonal.+implicitQRStepInPlace :: (M.Manifest r e, Floating e, Ord e)+                      => M.MArray s r Ix1 e -> M.MArray s r Ix1 e -> M.MArray s r Ix2 e+                      -> Int -> e -> Int -> Int -> ST s ()+implicitQRStepInPlace md msd mq nn shift lo hi = do+  dlo <- M.readM md lo+  sdlo <- M.readM msd lo+  chase lo (dlo - shift) sdlo+  where+    chase k x z = do+      let (c, s) = givensRotation x z+      when (k > lo) $+        M.write_ msd (k-1) (c * x - s * z)+      dk  <- M.readM md k+      ek  <- M.readM msd k+      dk1 <- M.readM md (k+1)+      M.write_ md k     (c*c*dk - 2*c*s*ek + s*s*dk1)+      M.write_ md (k+1) (s*s*dk + 2*c*s*ek + c*c*dk1)+      M.write_ msd k    (c*s*(dk - dk1) + (c*c - s*s)*ek)+      applyGivensRightQ mq c s k (k+1) nn+      if k + 1 < hi+        then do+          ek1 <- M.readM msd (k+1)+          let z' = -s * ek1+          M.write_ msd (k+1) (c * ek1)+          ek_new <- M.readM msd k+          chase (k+1) ek_new z'+        else pure ()++-- | Apply Givens rotation from the right to Q: Q <- Q · G(ci, ck)+-- For P Double, uses raw ByteArray# primops; generic fallback otherwise.+applyGivensRightQ :: (M.Manifest r e, Num e)+                  => M.MArray s r Ix2 e -> e -> e -> Int -> Int -> Int -> ST s ()+applyGivensRightQ mq c s ci ck nn =+  forM_ [0..nn-1] $ \row -> do+    qrc <- M.readM mq (row :. ci)+    qrk <- M.readM mq (row :. ck)+    M.write_ mq (row :. ci) (c * qrc - s * qrk)+    M.write_ mq (row :. ck) (s * qrc + c * qrk)++-- | Specialised symmetric eigenvalue decomposition for @P Double@.+-- Uses raw ByteArray# primops for the entire QR iteration, including+-- diagonal/subdiagonal reads and writes plus Givens rotation on Q.+symmetricEigenP :: forall n. KnownNat n+                => Matrix n n M.P Double -> Int -> Double -> (Vector n M.P Double, Matrix n n M.P Double)+symmetricEigenP a maxIter tol+  | dimVal @n >= dcCrossover = symmetricEigenPDC a tol+  | otherwise =+  let nn = dimVal @n+      (q0, diag_, subdiag) = tridiagonalizeP a+      (dArr, qArr) = M.withMArrayST (unMatrix q0) $ \mq -> do+        md <- M.thawS (unVector diag_)+        msd <- M.thawS (unVector subdiag)+        let !mbaD  = unwrapMutableByteArray md+            !offD  = unwrapMutableByteArrayOffset md+            !mbaSD = unwrapMutableByteArray msd+            !offSD = unwrapMutableByteArrayOffset msd+            !mbaQ  = unwrapMutableByteArray mq+            !offQ  = unwrapMutableByteArrayOffset mq+        -- For small n: use row-major QR loop (avoids two O(n^2) transposes)+        -- For large n: column-major Q for SIMD Givens rotations+        if nn < 100+          then rawTridiagQRLoop mbaD offD mbaSD offSD mbaQ offQ nn maxIter tol+          else do+            mbaQcm <- newByteArray (nn * nn * 8)+            rawTransposeToColMajor mbaQ offQ mbaQcm 0 nn+            rawTridiagQRLoopCM mbaD offD mbaSD offSD mbaQcm 0 nn maxIter tol+            rawTransposeFromColMajor mbaQcm 0 mbaQ offQ nn+        dFrozen <- M.freezeS md+        pure (MkVector dFrozen)+  in (dArr, MkMatrix qArr)+  where dcCrossover = 1000  -- D&C merge GEMM overhead doesn't amortise below ~1000+{-# NOINLINE symmetricEigenP #-}++-- | Parallel specialised symmetric eigenvalue decomposition for @P Double@.+-- Uses raw-primop tridiagonalisation and forks independent sub-problems+-- when the QR loop finds a split point.+symmetricEigenPPar :: forall n. KnownNat n+                   => Matrix n n M.P Double -> Int -> Double+                   -> (Vector n M.P Double, Matrix n n M.P Double)+symmetricEigenPPar a maxIter tol = unsafePerformIO $ do+  let nn = dimVal @n+      (q0, diag_, subdiag) = tridiagonalizeP a+  -- Thaw into IO (s = RealWorld) for parallel QR iteration+  mq  <- M.thawS (unMatrix q0)+  md  <- M.thawS (unVector diag_)+  msd <- M.thawS (unVector subdiag)+  let !mbaD  = unwrapMutableByteArray md+      !offD  = unwrapMutableByteArrayOffset md+      !mbaSD = unwrapMutableByteArray msd+      !offSD = unwrapMutableByteArrayOffset msd+      !mbaQ  = unwrapMutableByteArray mq+      !offQ  = unwrapMutableByteArrayOffset mq+  -- Transpose Q to column-major for SIMD Givens, run parallel QR, transpose back+  mbaQcm <- stToIO $ newByteArray (nn * nn * 8)+  stToIO $ rawTransposeToColMajor mbaQ offQ mbaQcm 0 nn+  rawTridiagQRLoopParCM mbaD offD mbaSD offSD mbaQcm 0 nn maxIter tol+  stToIO $ rawTransposeFromColMajor mbaQcm 0 mbaQ offQ nn+  dFrozen <- M.freezeS md+  qFrozen <- M.freezeS mq+  pure (MkVector dFrozen, MkMatrix qFrozen)+{-# NOINLINE symmetricEigenPPar #-}++-- | Divide-and-conquer specialised symmetric eigenvalue decomposition for @P Double@.+-- Uses raw-primop tridiagonalisation then D&C eigensolver (GEMM-based merge)+-- instead of QR iteration. Faster than 'symmetricEigenP' at larger sizes (n ≥ 50).+symmetricEigenPDC :: forall n. KnownNat n+                  => Matrix n n M.P Double -> Double+                  -> (Vector n M.P Double, Matrix n n M.P Double)+symmetricEigenPDC a tol =+  let nn = dimVal @n+      (q0, diag_, subdiag) = tridiagonalizeP a+      (dArr, qArr) = M.withMArrayST (unMatrix q0) $ \mq -> do+        md  <- M.thawS (unVector diag_)+        msd <- M.thawS (unVector subdiag)+        let !mbaD  = unwrapMutableByteArray md+            !offD  = unwrapMutableByteArrayOffset md+            !mbaSD = unwrapMutableByteArray msd+            !offSD = unwrapMutableByteArrayOffset msd+            !mbaQ  = unwrapMutableByteArray mq+            !offQ  = unwrapMutableByteArrayOffset mq+        dcEigenTridiagOpt mbaD offD mbaSD offSD mbaQ offQ nn 0 (nn - 1) tol+        dFrozen <- M.freezeS md+        pure (MkVector dFrozen)+  in (dArr, MkMatrix qArr)+{-# NOINLINE symmetricEigenPDC #-}++-- | Parallel QR loop: forks independent sub-problems when a split is found.+-- Operates in IO to enable forkIO for non-overlapping sub-problem ranges.+rawTridiagQRLoopPar :: MutableByteArray RealWorld -> Int+                    -> MutableByteArray RealWorld -> Int+                    -> MutableByteArray RealWorld -> Int+                    -> Int -> Int -> Double -> IO ()+rawTridiagQRLoopPar mbaD offD mbaSD offSD mbaQ offQ nn maxIter tol = go 0 0 (nn - 1)+  where+    rd mba off i = stToIO (readRawD mba off i)+    wr mba off i v = stToIO (writeRawD mba off i v)++    go !iter !lo !hi+      | iter >= maxIter = pure ()+      | lo >= hi = pure ()+      | otherwise = do+          sdhi <- rd mbaSD offSD (hi - 1)+          dhi1 <- rd mbaD offD (hi - 1)+          dhi  <- rd mbaD offD hi+          if abs sdhi <= tol * (abs dhi1 + abs dhi)+            then do wr mbaSD offSD (hi - 1) 0; go iter lo (hi - 1)+            else do+              sdlo <- rd mbaSD offSD lo+              dlo  <- rd mbaD offD lo+              dlo1 <- rd mbaD offD (lo + 1)+              if abs sdlo <= tol * (abs dlo + abs dlo1)+                then do wr mbaSD offSD lo 0; go iter (lo + 1) hi+                else do+                  split <- stToIO $ rawFindSplit mbaD offD mbaSD offSD lo hi tol+                  case split of+                    Just q -> do+                      wr mbaSD offSD q 0+                      done <- newEmptyMVar+                      _ <- forkIO $ do+                        go iter lo q+                        putMVar done ()+                      go iter (q + 1) hi+                      takeMVar done+                    Nothing -> do+                      let sp1 = sdhi+                          delta = (dhi1 - dhi) / 2+                          sgn = if delta >= 0 then 1 else -1+                          shift = dhi - sp1*sp1 / (delta + sgn * sqrt (delta*delta + sp1*sp1))+                      stToIO $ rawImplicitQRStep mbaD offD mbaSD offSD mbaQ offQ nn shift lo hi+                      go (iter + 1) lo hi++-- | Parallel QR loop with column-major Q for SIMD Givens.+rawTridiagQRLoopParCM :: MutableByteArray RealWorld -> Int+                      -> MutableByteArray RealWorld -> Int+                      -> MutableByteArray RealWorld -> Int+                      -> Int -> Int -> Double -> IO ()+rawTridiagQRLoopParCM mbaD offD mbaSD offSD mbaQ offQ nn maxIter tol = go 0 0 (nn - 1)+  where+    rd mba off i = stToIO (readRawD mba off i)+    wr mba off i v = stToIO (writeRawD mba off i v)++    go !iter !lo !hi+      | iter >= maxIter = pure ()+      | lo >= hi = pure ()+      | otherwise = do+          sdhi <- rd mbaSD offSD (hi - 1)+          dhi1 <- rd mbaD offD (hi - 1)+          dhi  <- rd mbaD offD hi+          if abs sdhi <= tol * (abs dhi1 + abs dhi)+            then do wr mbaSD offSD (hi - 1) 0; go iter lo (hi - 1)+            else do+              sdlo <- rd mbaSD offSD lo+              dlo  <- rd mbaD offD lo+              dlo1 <- rd mbaD offD (lo + 1)+              if abs sdlo <= tol * (abs dlo + abs dlo1)+                then do wr mbaSD offSD lo 0; go iter (lo + 1) hi+                else do+                  split <- stToIO $ rawFindSplit mbaD offD mbaSD offSD lo hi tol+                  case split of+                    Just q -> do+                      wr mbaSD offSD q 0+                      done <- newEmptyMVar+                      _ <- forkIO $ do+                        go iter lo q+                        putMVar done ()+                      go iter (q + 1) hi+                      takeMVar done+                    Nothing -> do+                      let sp1 = sdhi+                          delta = (dhi1 - dhi) / 2+                          sgn = if delta >= 0 then 1 else -1+                          shift = dhi - sp1*sp1 / (delta + sgn * sqrt (delta*delta + sp1*sp1))+                      stToIO $ rawImplicitQRStepCM mbaD offD mbaSD offSD mbaQ offQ nn shift lo hi+                      go (iter + 1) lo hi++-- | Read a Double from a raw MutableByteArray at element index.+readRawD :: MutableByteArray s -> Int -> Int -> ST s Double+readRawD (MutableByteArray mba) (I# off) (I# i) = ST $ \s ->+  case readDoubleArray# mba (off +# i) s of+    (# s', v #) -> (# s', D# v #)+{-# INLINE readRawD #-}++-- | Write a Double to a raw MutableByteArray at element index.+writeRawD :: MutableByteArray s -> Int -> Int -> Double -> ST s ()+writeRawD (MutableByteArray mba) (I# off) (I# i) (D# v) = ST $ \s ->+  case writeDoubleArray# mba (off +# i) v s of+    s' -> (# s', () #)+{-# INLINE writeRawD #-}++-- | Read an Int from a raw MutableByteArray at element index.+readRawI :: MutableByteArray s -> Int -> Int -> ST s Int+readRawI (MutableByteArray mba) (I# off) (I# i) = ST $ \s ->+  case readIntArray# mba (off +# i) s of+    (# s', v #) -> (# s', I# v #)+{-# INLINE readRawI #-}++-- | Write an Int to a raw MutableByteArray at element index.+writeRawI :: MutableByteArray s -> Int -> Int -> Int -> ST s ()+writeRawI (MutableByteArray mba) (I# off) (I# i) (I# v) = ST $ \s ->+  case writeIntArray# mba (off +# i) v s of+    s' -> (# s', () #)+{-# INLINE writeRawI #-}++-- | Read a Double from an immutable ByteArray at element index.+indexRawD :: ByteArray -> Int -> Int -> Double+indexRawD (ByteArray ba) (I# off) (I# i) =+  case indexDoubleArray# ba (off +# i) of+    v -> D# v+{-# INLINE indexRawD #-}++-- | Raw primop QR loop: all diagonal/subdiagonal access via raw ByteArray# primops.+rawTridiagQRLoop :: MutableByteArray s -> Int   -- ^ diagonal array + offset+                 -> MutableByteArray s -> Int   -- ^ subdiagonal array + offset+                 -> MutableByteArray s -> Int   -- ^ Q matrix + offset+                 -> Int -> Int -> Double -> ST s ()+rawTridiagQRLoop mbaD offD mbaSD offSD mbaQ offQ nn maxIter tol = go 0 0 (nn - 1) (nn - 1) (0 :: Int)+  where+    go !iter !lo !hi !lastHi !stall+      | iter >= maxIter = pure ()+      | stall >= 20 = pure ()  -- bail if 20 consecutive steps fail to deflate+      | lo >= hi = pure ()+      | otherwise = do+          -- Bottom deflation+          sdhi <- readRawD mbaSD offSD (hi - 1)+          dhi1 <- readRawD mbaD offD (hi - 1)+          dhi  <- readRawD mbaD offD hi+          if abs sdhi <= tol * (abs dhi1 + abs dhi)+            then do writeRawD mbaSD offSD (hi - 1) 0; go iter lo (hi - 1) hi 0+            else do+              -- Top deflation+              sdlo <- readRawD mbaSD offSD lo+              dlo  <- readRawD mbaD offD lo+              dlo1 <- readRawD mbaD offD (lo + 1)+              if abs sdlo <= tol * (abs dlo + abs dlo1)+                then do writeRawD mbaSD offSD lo 0; go iter (lo + 1) hi hi 0+                else do+                  -- Interior deflation: find split point+                  split <- rawFindSplit mbaD offD mbaSD offSD lo hi tol+                  case split of+                    Just q -> do+                      writeRawD mbaSD offSD q 0+                      go iter lo q hi 0+                      go iter (q + 1) hi hi 0+                    Nothing -> do+                      -- AED: scan bottom window before committing to a QR step+                      newHi <- if hi - lo >= 6+                        then rawAEDScan mbaD offD mbaSD offSD tol lo hi+                        else pure hi+                      if newHi < hi+                        then go iter lo newHi hi 0  -- deflated without QR sweep+                        else do+                          -- Compute Wilkinson shift from bottom 2×2 block+                          let !sp1 = sdhi+                              !delta = (dhi1 - dhi) / 2+                              !sgn = if delta >= 0 then 1 else -1+                              !shift1 = dhi - sp1*sp1 / (delta + sgn * sqrt (delta*delta + sp1*sp1))+                          rawImplicitQRStep mbaD offD mbaSD offSD mbaQ offQ nn shift1 lo hi+                          -- Double-shift: apply second shift if active range is large enough+                          when (hi - lo >= 4) $ do+                            sdhi' <- readRawD mbaSD offSD (hi - 1)+                            dhi1' <- readRawD mbaD offD (hi - 1)+                            dhi'  <- readRawD mbaD offD hi+                            when (abs sdhi' > tol * (abs dhi1' + abs dhi')) $ do+                              let !delta' = (dhi1' - dhi') / 2+                                  !sgn' = if delta' >= 0 then 1 else -1+                                  !shift2 = dhi' - sdhi'*sdhi' / (delta' + sgn' * sqrt (delta'*delta' + sdhi'*sdhi'))+                              rawImplicitQRStep mbaD offD mbaSD offSD mbaQ offQ nn shift2 lo hi+                          let !newStall = if hi == lastHi then stall + 1 else 0+                          go (iter + 1) lo hi hi newStall++-- | Aggressive Early Deflation: scan bottom w entries for negligible subdiagonals.+-- Returns the new (possibly lower) hi. Deflates from the bottom up, setting+-- negligible subdiagonal entries to zero.+rawAEDScan :: MutableByteArray s -> Int -> MutableByteArray s -> Int+           -> Double -> Int -> Int -> ST s Int+rawAEDScan mbaD offD mbaSD offSD tol lo hi = scan hi+  where+    !w = min 6 ((hi - lo + 1) `div` 3)+    !bottom = max (lo + 1) (hi - w)+    scan !h+      | h <= bottom = pure h+      | otherwise = do+          sdk <- readRawD mbaSD offSD (h - 1)+          dk1 <- readRawD mbaD offD (h - 1)+          dk  <- readRawD mbaD offD h+          let !absdk1 = abs dk1+              !absdk  = abs dk+              !threshold = tol * (absdk1 + absdk)+          if abs sdk <= threshold+            then do writeRawD mbaSD offSD (h - 1) 0; scan (h - 1)+            else pure h+{-# INLINE rawAEDScan #-}++-- | Raw primop interior split search.+rawFindSplit :: MutableByteArray s -> Int -> MutableByteArray s -> Int+             -> Int -> Int -> Double -> ST s (Maybe Int)+rawFindSplit mbaD offD mbaSD offSD lo hi tol = scan (lo + 1)+  where+    scan q+      | q >= hi - 1 = pure Nothing+      | otherwise = do+          sdq <- readRawD mbaSD offSD q+          dq  <- readRawD mbaD offD q+          dq1 <- readRawD mbaD offD (q + 1)+          if abs sdq <= tol * (abs dq + abs dq1)+            then pure (Just q)+            else scan (q + 1)++-- | Raw primop implicit QR step via bulge-chasing Givens rotations.+rawImplicitQRStep :: MutableByteArray s -> Int -> MutableByteArray s -> Int+                  -> MutableByteArray s -> Int+                  -> Int -> Double -> Int -> Int -> ST s ()+rawImplicitQRStep mbaD offD mbaSD offSD mbaQ offQ nn shift lo hi = do+  dlo <- readRawD mbaD offD lo+  sdlo <- readRawD mbaSD offSD lo+  chase lo (dlo - shift) sdlo+  where+    chase k x z = do+      let (c, s) = givensRotation x z+      when (k > lo) $+        writeRawD mbaSD offSD (k-1) (c * x - s * z)+      dk  <- readRawD mbaD offD k+      ek  <- readRawD mbaSD offSD k+      dk1 <- readRawD mbaD offD (k+1)+      writeRawD mbaD offD k     (c*c*dk - 2*c*s*ek + s*s*dk1)+      writeRawD mbaD offD (k+1) (s*s*dk + 2*c*s*ek + c*c*dk1)+      writeRawD mbaSD offSD k   (c*s*(dk - dk1) + (c*c - s*s)*ek)+      -- Raw primop Givens rotation on Q+      rawMutApplyGivensColumns mbaQ offQ nn c (negate s) k (k+1) nn+      if k + 1 < hi+        then do+          ek1 <- readRawD mbaSD offSD (k+1)+          let z' = -s * ek1+          writeRawD mbaSD offSD (k+1) (c * ek1)+          ek_new <- readRawD mbaSD offSD k+          chase (k+1) ek_new z'+        else pure ()++-- | Column-major QR loop: same as rawTridiagQRLoop but Q is column-major.+-- In column-major layout, Q[i,j] at off + j*nn + i. This enables SIMD+-- Givens column updates (4 rows at a time via DoubleX4#).+rawTridiagQRLoopCM :: MutableByteArray s -> Int   -- ^ diagonal array + offset+                   -> MutableByteArray s -> Int   -- ^ subdiagonal array + offset+                   -> MutableByteArray s -> Int   -- ^ Q matrix (COLUMN-MAJOR) + offset+                   -> Int -> Int -> Double -> ST s ()+rawTridiagQRLoopCM mbaD offD mbaSD offSD mbaQ offQ nn maxIter tol = go 0 0 (nn - 1) (nn - 1) (0 :: Int)+  where+    go !iter !lo !hi !lastHi !stall+      | iter >= maxIter = pure ()+      | stall >= 20 = pure ()  -- bail if 20 consecutive steps fail to deflate+      | lo >= hi = pure ()+      | otherwise = do+          sdhi <- readRawD mbaSD offSD (hi - 1)+          dhi1 <- readRawD mbaD offD (hi - 1)+          dhi  <- readRawD mbaD offD hi+          if abs sdhi <= tol * (abs dhi1 + abs dhi)+            then do writeRawD mbaSD offSD (hi - 1) 0; go iter lo (hi - 1) hi 0+            else do+              sdlo <- readRawD mbaSD offSD lo+              dlo  <- readRawD mbaD offD lo+              dlo1 <- readRawD mbaD offD (lo + 1)+              if abs sdlo <= tol * (abs dlo + abs dlo1)+                then do writeRawD mbaSD offSD lo 0; go iter (lo + 1) hi hi 0+                else do+                  split <- rawFindSplit mbaD offD mbaSD offSD lo hi tol+                  case split of+                    Just q -> do+                      writeRawD mbaSD offSD q 0+                      go iter lo q hi 0+                      go iter (q + 1) hi hi 0+                    Nothing -> do+                      -- AED: scan bottom window before committing to a QR step+                      newHi <- if hi - lo >= 6+                        then rawAEDScan mbaD offD mbaSD offSD tol lo hi+                        else pure hi+                      if newHi < hi+                        then go iter lo newHi hi 0  -- deflated without QR sweep+                        else do+                          -- Compute both eigenvalues of bottom 2×2 block+                          let !sp1 = sdhi+                              !delta = (dhi1 - dhi) / 2+                              !sgn = if delta >= 0 then 1 else -1+                              !shift1 = dhi - sp1*sp1 / (delta + sgn * sqrt (delta*delta + sp1*sp1))+                          rawImplicitQRStepCM mbaD offD mbaSD offSD mbaQ offQ nn shift1 lo hi+                          -- Double-shift: apply second shift if active range is large enough+                          when (hi - lo >= 4) $ do+                            -- Check if first shift already deflated the bottom+                            sdhi' <- readRawD mbaSD offSD (hi - 1)+                            dhi1' <- readRawD mbaD offD (hi - 1)+                            dhi'  <- readRawD mbaD offD hi+                            when (abs sdhi' > tol * (abs dhi1' + abs dhi')) $ do+                              -- Compute the other eigenvalue of the (updated) bottom 2×2+                              let !delta' = (dhi1' - dhi') / 2+                                  !sgn' = if delta' >= 0 then 1 else -1+                                  !shift2 = dhi' - sdhi'*sdhi' / (delta' + sgn' * sqrt (delta'*delta' + sdhi'*sdhi'))+                              rawImplicitQRStepCM mbaD offD mbaSD offSD mbaQ offQ nn shift2 lo hi+                      let !newStall = if hi == lastHi then stall + 1 else 0+                      go (iter + 1) lo hi hi newStall++-- | Column-major implicit QR step: same as rawImplicitQRStep but uses+-- SIMD Givens on column-major Q layout.+rawImplicitQRStepCM :: MutableByteArray s -> Int -> MutableByteArray s -> Int+                    -> MutableByteArray s -> Int+                    -> Int -> Double -> Int -> Int -> ST s ()+rawImplicitQRStepCM mbaD offD mbaSD offSD mbaQ offQ nn shift lo hi = do+  dlo <- readRawD mbaD offD lo+  sdlo <- readRawD mbaSD offSD lo+  chase lo (dlo - shift) sdlo+  where+    chase k x z = do+      let (c, s) = givensRotation x z+      when (k > lo) $+        writeRawD mbaSD offSD (k-1) (c * x - s * z)+      dk  <- readRawD mbaD offD k+      ek  <- readRawD mbaSD offSD k+      dk1 <- readRawD mbaD offD (k+1)+      writeRawD mbaD offD k     (c*c*dk - 2*c*s*ek + s*s*dk1)+      writeRawD mbaD offD (k+1) (s*s*dk + 2*c*s*ek + c*c*dk1)+      writeRawD mbaSD offSD k   (c*s*(dk - dk1) + (c*c - s*s)*ek)+      -- SIMD Givens rotation on column-major Q+      rawMutApplyGivensColumnsCM mbaQ offQ nn c (negate s) k (k+1) nn+      if k + 1 < hi+        then do+          ek1 <- readRawD mbaSD offSD (k+1)+          let z' = -s * ek1+          writeRawD mbaSD offSD (k+1) (c * ek1)+          ek_new <- readRawD mbaSD offSD k+          chase (k+1) ek_new z'+        else pure ()++-- --------------------------------------------------------------------------+-- Divide-and-conquer tridiagonal eigensolver (GVL4 Section 8.4)+-- Optimised: pre-allocated workspace, QR fallback for small subproblems,+-- unsafeFreezeByteArray to avoid O(n²) copies.+-- --------------------------------------------------------------------------++-- | Optimised divide-and-conquer eigensolver for a symmetric tridiagonal matrix.+-- Pre-allocates all temporary arrays once (eliminating per-level GC pressure),+-- falls back to QR for subproblems ≤ 25, and uses unsafeFreezeByteArray for+-- O(1) GEMM input preparation.+--+-- The algorithm maintains a LOCAL eigenvector matrix (maxN × maxN, starting as+-- identity) throughout the D&C recursion. The z-vector for each merge step is+-- extracted from this local matrix (not the global Q), ensuring correctness.+-- At the end, the global Q is updated via a single GEMM: Q_out = Q_in * Q_local.+dcEigenTridiagOpt :: MutableByteArray s -> Int   -- ^ d + offset+                  -> MutableByteArray s -> Int   -- ^ e + offset+                  -> MutableByteArray s -> Int   -- ^ Q + offset (fullN × fullN row-major)+                  -> Int                         -- ^ fullN (Q dimension)+                  -> Int -> Int                  -- ^ lo, hi (active range, inclusive)+                  -> Double                      -- ^ tolerance+                  -> ST s ()+dcEigenTridiagOpt mbaD offD mbaE offE mbaQ offQ fullN lo0 hi0 tol+  | hi0 <= lo0 = pure ()+  | otherwise = do+      let !maxN = hi0 - lo0 + 1+      -- Pre-allocate all workspace at maximum needed size (once, not per-level)+      wsLam    <- newByteArray (maxN * 8)+      wsZ      <- newByteArray (maxN * 8)+      wsDSort  <- newByteArray (maxN * 8)+      wsZSort  <- newByteArray (maxN * 8)+      wsIdx    <- newByteArray (maxN * 8)+      wsPerm   <- newByteArray (maxN * 8)  -- deflation permutation (Int indices)+      wsW      <- newByteArray (maxN * maxN * 8)+      wsQsub   <- newByteArray (maxN * maxN * 8)+      wsResult <- newByteArray (maxN * maxN * 8)+      wsQtemp  <- newByteArray (maxN * maxN * 8)++      -- Local eigenvector accumulator (maxN × maxN), initialised to identity.+      -- Indexed with LOCAL coordinates: row/col in [0..maxN-1].+      -- Local index i corresponds to global index (lo0 + i).+      wsQlocal <- newByteArray (maxN * maxN * 8)+      rawZeroDoubles wsQlocal 0 (maxN * maxN)+      forM_ [0..maxN-1] $ \i -> writeRawD wsQlocal 0 (i * maxN + i) 1++      let !dcThreshold = 25++          -- Convenience: convert global index to local index+          toLocal !g = g - lo0++          -- Main recursive function (captures workspace via closure)+          -- All operations affect wsQlocal (local eigenvector accumulator),+          -- NOT the global Q matrix.+          dcGo !lo !hi+            | lo >= hi = pure ()+            | hi == lo + 1 = do+                -- 2×2 direct eigensolve+                d0 <- readRawD mbaD offD lo+                d1 <- readRawD mbaD offD hi+                e0 <- readRawD mbaE offE lo+                let !tr = d0 + d1+                    !det_ = d0 * d1 - e0 * e0+                    !disc = sqrt (max 0 (tr * tr - 4 * det_))+                    !lam1 = (tr - disc) / 2+                    !lam2 = (tr + disc) / 2+                    (!c, !s) = if abs e0 < tol * (abs d0 + abs d1)+                               then (1, 0)+                               else let !theta = (d1 - d0) / (2 * e0)+                                        !t_ = if theta >= 0+                                              then 1 / (theta + sqrt (1 + theta * theta))+                                              else 1 / (theta - sqrt (1 + theta * theta))+                                        !c_ = 1 / sqrt (1 + t_ * t_)+                                    in (c_, t_ * c_)+                writeRawD mbaD offD lo lam1+                writeRawD mbaD offD hi lam2+                writeRawD mbaE offE lo 0+                -- Apply Givens to LOCAL eigenvector matrix columns+                rawMutApplyGivensColumns wsQlocal 0 maxN c (negate s) (toLocal lo) (toLocal hi) maxN+            | hi - lo + 1 <= dcThreshold = do+                -- QR fallback for small subproblems+                let !k = hi - lo + 1+                -- Initialise wsQtemp as k×k identity+                rawZeroDoubles wsQtemp 0 (k * k)+                forM_ [0..k-1] $ \i -> writeRawD wsQtemp 0 (i * k + i) 1+                -- Run QR iteration on d[lo..hi], e[lo..hi-1]+                rawTridiagQRLoop mbaD (offD + lo) mbaE (offE + lo) wsQtemp 0 k (30 * k) tol+                -- Apply rotation to LOCAL eigenvector matrix:+                -- wsQlocal[:, toLocal(lo)..toLocal(lo)+k-1] *= wsQtemp+                applyRotToQlocal (toLocal lo) k wsQtemp+            | otherwise = do+                -- D&C merge for larger subproblems+                let !k  = (lo + hi) `div` 2+                    !n1 = k - lo + 1+                    !n2 = hi - k+                    !nn = hi - lo + 1+                    -- Local coordinates for the split+                    !kL  = toLocal k+                    !loL = toLocal lo++                -- Read and modify the coupling element+                beta <- readRawD mbaE offE k+                dk   <- readRawD mbaD offD k+                dk1  <- readRawD mbaD offD (k + 1)+                let !absBeta = abs beta+                    !rho = absBeta+                writeRawD mbaD offD k     (dk - absBeta)+                writeRawD mbaD offD (k+1) (dk1 - absBeta)+                writeRawD mbaE offE k 0++                -- Recurse on T1 [lo..k] and T2 [k+1..hi]+                dcGo lo k+                dcGo (k + 1) hi++                -- === Merge phase ===+                -- Extract z vector from LOCAL eigenvector matrix rows.+                -- z[0..n1-1] = last row of Q1 = row kL of wsQlocal, columns loL..loL+n1-1+                -- z[n1..nn-1] = first row of Q2 = row (kL+1) of wsQlocal, columns loL+n1..loL+nn-1+                forM_ [0..n1-1] $ \i -> do+                  qv <- readRawD wsQlocal 0 (kL * maxN + (loL + i))+                  writeRawD wsZ 0 i qv+                forM_ [0..n2-1] $ \i -> do+                  qv <- readRawD wsQlocal 0 ((kL + 1) * maxN + (loL + n1 + i))+                  let !zv = if beta < 0 then negate qv else qv+                  writeRawD wsZ 0 (n1 + i) zv++                -- Copy d[lo..hi] and z into sortable arrays with indices+                forM_ [0..nn-1] $ \i -> do+                  di <- readRawD mbaD offD (lo + i)+                  writeRawD wsDSort 0 i di+                  writeRawD wsZSort 0 i =<< readRawD wsZ 0 i+                  writeRawD wsIdx 0 i (fromIntegral i)++                -- Sort by d values (insertion sort)+                forM_ [1..nn-1] $ \i -> do+                  di   <- readRawD wsDSort 0 i+                  zi   <- readRawD wsZSort 0 i+                  idxi <- readRawD wsIdx 0 i+                  let insertAt !j+                        | j < 0 = do+                            writeRawD wsDSort 0 0 di+                            writeRawD wsZSort 0 0 zi+                            writeRawD wsIdx 0 0 idxi+                        | otherwise = do+                            dj <- readRawD wsDSort 0 j+                            if dj > di+                              then do+                                writeRawD wsDSort 0 (j+1) dj+                                writeRawD wsZSort 0 (j+1) =<< readRawD wsZSort 0 j+                                writeRawD wsIdx 0 (j+1) =<< readRawD wsIdx 0 j+                                insertAt (j - 1)+                              else do+                                writeRawD wsDSort 0 (j+1) di+                                writeRawD wsZSort 0 (j+1) zi+                                writeRawD wsIdx 0 (j+1) idxi+                  insertAt (i - 1)++                -- Close-d deflation (cf. LAPACK dlaed2): when consecutive+                -- sorted d values are nearly equal, a Givens rotation zeros+                -- one z entry, preventing ill-conditioned secular roots.+                dMaxAbs <- readRawD wsDSort 0 (nn - 1)+                dMinAbs <- readRawD wsDSort 0 0+                let !closeDTol = 8 * 2.220446049250313e-16+                              * max (abs dMaxAbs) (abs dMinAbs + rho)+                forM_ [0..nn-2] $ \i -> do+                  di  <- readRawD wsDSort 0 i+                  di1 <- readRawD wsDSort 0 (i + 1)+                  when (abs (di1 - di) <= closeDTol) $ do+                    zi  <- readRawD wsZSort 0 i+                    zi1 <- readRawD wsZSort 0 (i + 1)+                    let !r = sqrt (zi * zi + zi1 * zi1)+                    when (r > 1e-300) $ do+                      let !c = zi1 / r+                          !s = zi / r+                      -- Zero z[i], combine into z[i+1]+                      writeRawD wsZSort 0 i 0+                      writeRawD wsZSort 0 (i + 1) r+                      -- Apply same Givens to eigenvector columns+                      origI  <- readRawD wsIdx 0 i+                      origI1 <- readRawD wsIdx 0 (i + 1)+                      let !colI  = loL + (round origI  :: Int)+                          !colI1 = loL + (round origI1 :: Int)+                      rawMutApplyGivensColumns wsQlocal 0 maxN c s colI colI1 maxN++                -- Deflation: partition into non-deflated and deflated.+                -- The perturbation-based criterion ensures that entries whose+                -- eigenvalue shift rho*z[i]² is below machine precision+                -- relative to |d[i]| are deflated, preventing the eigenvector+                -- formula from dividing by zero (d[j] - lambda == 0 in FP).+                zn2 <- sumZSq wsZSort 0 nn+                let !eps_ = 2.220446049250313e-16+                    !matNorm = max (abs dMaxAbs) (abs dMinAbs) + rho * zn2+                    !basicDeflTol = max (tol * sqrt zn2) (8 * eps_ * matNorm)+                    -- Perturbation deflation: deflate when rho*z²<eps*matNorm,+                    -- i.e. |z| < sqrt(eps*matNorm/rho)+                    !pertDeflTol = sqrt (eps_ * (1 + matNorm)+                                        / max rho 1e-300)+                    !deflTol = max basicDeflTol pertDeflTol+                kND <- deflatePartition wsZSort 0 wsPerm 0 nn deflTol++                -- Extract Qlocal_sub with permuted columns into wsQsub (maxN × nn)+                -- Only rows [loL..loL+nn-1] are relevant, but we copy all maxN rows+                -- to maintain the accumulator's full row structure for the GEMM.+                forM_ [0..nn-1] $ \sortedJ -> do+                  origIdx <- readRawD wsIdx 0 sortedJ+                  let !origJ = round origIdx :: Int+                      !srcCol = loL + origJ+                  rawCopyColumn wsQlocal 0 maxN srcCol wsQsub 0 nn sortedJ maxN++                if kND == 0+                  then do+                    -- All deflated: eigenvalues = sorted d, eigenvectors = sorted Qlocal cols+                    forM_ [0..nn-1] $ \i ->+                      rawCopyColumn wsQsub 0 nn i wsQlocal 0 maxN (loL + i) maxN+                    forM_ [0..nn-1] $ \i -> do+                      di <- readRawD wsDSort 0 i+                      writeRawD mbaD offD (lo + i) di++                  else if kND == nn+                    then do+                      -- No deflation: full secular solve + full GEMM+                      secularSolve wsLam 0 wsDSort 0 wsZSort 0 rho nn deflTol+                      dcEigenvectors wsW 0 wsDSort 0 wsZSort 0 wsLam 0 rho nn++                      baQsub <- unsafeFreezeByteArray wsQsub+                      baW    <- unsafeFreezeByteArray wsW+                      rawZeroDoubles wsResult 0 (maxN * nn)+                      rawGemmKernel baQsub 0 baW 0 wsResult 0 maxN nn nn++                      forM_ [0..nn-1] $ \i ->+                        rawCopyColumn wsResult 0 nn i wsQlocal 0 maxN (loL + i) maxN+                      forM_ [0..nn-1] $ \i -> do+                        lam <- readRawD wsLam 0 i+                        writeRawD mbaD offD (lo + i) lam++                    else do+                      -- Partial deflation: reduced secular solve + reduced GEMM+                      -- Build compressed d_nd[0..kND-1] and z_nd[0..kND-1]+                      -- Store in wsQtemp: d_nd at offset 0, z_nd at offset kND+                      forM_ [0..kND-1] $ \j -> do+                        pi_ <- readRawI wsPerm 0 j+                        dpi <- readRawD wsDSort 0 pi_+                        zpi <- readRawD wsZSort 0 pi_+                        writeRawD wsQtemp 0 j dpi+                        writeRawD wsQtemp 0 (kND + j) zpi++                      -- Solve kND secular equations on compressed system+                      secularSolve wsLam 0 wsQtemp 0 wsQtemp kND rho kND deflTol++                      -- Compute kND×kND eigenvector matrix W_nd+                      dcEigenvectors wsW 0 wsQtemp 0 wsQtemp kND wsLam 0 rho kND++                      -- Copy deflated columns from wsQsub to wsQlocal+                      forM_ [kND..nn-1] $ \j -> do+                        pi_ <- readRawI wsPerm 0 j+                        rawCopyColumn wsQsub 0 nn pi_ wsQlocal 0 maxN (loL + j) maxN++                      -- Extract Q_nd (maxN×kND) from non-deflated columns+                      forM_ [0..kND-1] $ \j -> do+                        pi_ <- readRawI wsPerm 0 j+                        rawCopyColumn wsQsub 0 nn pi_ wsResult 0 kND j maxN++                      -- GEMM: wsQsub(maxN×kND) = Q_nd(maxN×kND) × W_nd(kND×kND)+                      baQnd <- unsafeFreezeByteArray wsResult+                      baW   <- unsafeFreezeByteArray wsW+                      rawZeroDoubles wsQsub 0 (maxN * kND)+                      rawGemmKernel baQnd 0 baW 0 wsQsub 0 maxN kND kND++                      -- Copy GEMM result (non-deflated columns) to wsQlocal+                      forM_ [0..kND-1] $ \j ->+                        rawCopyColumn wsQsub 0 kND j wsQlocal 0 maxN (loL + j) maxN++                      -- Write eigenvalues: non-deflated from wsLam, deflated from wsDSort+                      forM_ [0..kND-1] $ \i -> do+                        lam <- readRawD wsLam 0 i+                        writeRawD mbaD offD (lo + i) lam+                      forM_ [kND..nn-1] $ \j -> do+                        pi_ <- readRawI wsPerm 0 j+                        di <- readRawD wsDSort 0 pi_+                        writeRawD mbaD offD (lo + j) di++          -- Apply a k×k rotation matrix to wsQlocal columns [colOff..colOff+k-1] via GEMM+          -- colOff is in LOCAL coordinates.+          applyRotToQlocal !colOff !k rotMat = do+            -- Extract wsQlocal[:, colOff..colOff+k-1] into wsQsub (maxN × k)+            forM_ [0..k-1] $ \j ->+              rawCopyColumn wsQlocal 0 maxN (colOff + j) wsQsub 0 k j maxN+            -- O(1) freeze for GEMM inputs+            baQsub <- unsafeFreezeByteArray wsQsub+            baRot  <- unsafeFreezeByteArray rotMat+            -- Zero result+            rawZeroDoubles wsResult 0 (maxN * k)+            -- GEMM: result(maxN×k) = Qsub(maxN×k) * Rot(k×k)+            rawGemmKernel baQsub 0 baRot 0 wsResult 0 maxN k k+            -- Copy result back to wsQlocal+            forM_ [0..k-1] $ \j ->+              rawCopyColumn wsResult 0 k j wsQlocal 0 maxN (colOff + j) maxN++      -- Run the D&C recursion (operates on wsQlocal and mbaD/mbaE)+      dcGo lo0 hi0++      -- Final step: apply wsQlocal to global Q via GEMM+      -- Q[:, lo0..hi0] = Q[:, lo0..hi0] * wsQlocal+      forM_ [0..maxN-1] $ \j ->+        rawCopyColumn mbaQ offQ fullN (lo0 + j) wsQsub 0 maxN j fullN++      baQsub   <- unsafeFreezeByteArray wsQsub+      baQlocal <- unsafeFreezeByteArray wsQlocal+      rawZeroDoubles wsResult 0 (fullN * maxN)+      rawGemmKernel baQsub 0 baQlocal 0 wsResult 0 fullN maxN maxN++      forM_ [0..maxN-1] $ \j ->+        rawCopyColumn wsResult 0 maxN j mbaQ offQ fullN (lo0 + j) fullN++-- | Solve the secular equation: f(λ) = 1 + ρ * Σ z[i]² / (d[i] - λ) = 0+-- for all nn roots. Roots are stored in mbaLam.+-- d must be sorted in ascending order.+secularSolve :: MutableByteArray s -> Int    -- ^ output eigenvalues+             -> MutableByteArray s -> Int    -- ^ sorted d (poles)+             -> MutableByteArray s -> Int    -- ^ sorted z+             -> Double -> Int -> Double      -- ^ rho, n, deflation tolerance+             -> ST s ()+secularSolve mbaLam offLam mbaD offD mbaZ offZ rho nn deflTol = do+  forM_ [0..nn-1] $ \i -> do+    zi <- readRawD mbaZ offZ i+    di <- readRawD mbaD offD i+    if abs zi <= deflTol+      then do+        -- Deflated: eigenvalue = d[i]+        writeRawD mbaLam offLam i di+      else do+        -- For small z[i], use first-order perturbation formula directly.+        -- This avoids the iterative solver's difficulty with nearly-flat+        -- secular functions near d[i].+        let !zi2 = zi * zi+            !pertTol = sqrt (2.220446049250313e-16) * (1 + abs di)+        if abs zi < pertTol+          then do+            -- Perturbation: lambda ≈ d[i] + rho * z[i]² / (1 + rho * Σ_{j≠i} z[j]²/(d[j]-d[i]))+            farSum <- farPoleSumSkip mbaD offD mbaZ offZ nn i di+            let !denom = 1 + rho * farSum+                !delta = rho * zi2 / denom+            writeRawD mbaLam offLam i (di + delta)+          else do+            lam <- secularSolveOne mbaD offD mbaZ offZ rho i nn+            writeRawD mbaLam offLam i lam++-- | Solve one root of the secular equation between d[j] and d[j+1]+-- (or between d[n-1] and +infinity for the last root when rho > 0,+-- or between -infinity and d[0] for the first root when rho < 0).+-- Uses the Gragg/Borges fixed-weight quadratic method (cf. LAPACK dlasd4):+-- splits f(λ) at the two closest poles, approximates far terms as constant,+-- and solves the resulting quadratic for rapid convergence (2–4 iterations).+secularSolveOne :: MutableByteArray s -> Int -> MutableByteArray s -> Int+                -> Double -> Int -> Int -> ST s Double+secularSolveOne mbaD offD mbaZ offZ rho j nn = do+  dj <- readRawD mbaD offD j+  zj <- readRawD mbaZ offZ j+  let !zj2 = zj * zj+  -- Determine bracket and second pole+  if rho > 0+    then if j < nn - 1+      then do+        dj1 <- readRawD mbaD offD (j+1)+        zj1 <- readRawD mbaZ offZ (j+1)+        -- Interior root between d[j] and d[j+1]+        let !gap = dj1 - dj+            !mid = dj + gap * 0.5+        lam0 <- fixedWeightLoop 0 mid dj dj1 dj dj1 gap zj2 (zj1 * zj1)+        -- Polish with Newton iterations for higher accuracy+        newtonPolish 0 lam0 dj dj1+      else do+        -- Last root when rho > 0: between d[n-1] and d[n-1] + rho*||z||²+        zn2 <- sumZSq mbaZ offZ nn+        -- Compute better initial guess via perturbation theory:+        -- f(d[nn-1]+δ) = 0 ⟹ δ ≈ rho * z[nn-1]² / (1 + rho * Σ_{i<nn-1} z[i]²/(d[i]-d[nn-1]))+        farSum <- farPoleSum mbaD offD mbaZ offZ (nn - 1) dj+        let !denominator = 1 + rho * farSum+            !delta0 = if abs denominator > 1e-300+                      then rho * zj2 / denominator+                      else rho * zn2+            !hi_ = dj + max (rho * zn2) (2 * delta0)+            !mid = dj + max delta0 (1e-15 * (1 + abs dj))+        newtonLoop 0 mid dj hi_+    else if j > 0+      then do+        dj0 <- readRawD mbaD offD (j-1)+        zj0 <- readRawD mbaZ offZ (j-1)+        -- Interior root between d[j-1] and d[j] (rho < 0)+        let !gap = dj - dj0+            !mid = dj0 + gap * 0.5+        fixedWeightLoop 0 mid dj0 dj dj0 dj gap (zj0 * zj0) zj2+      else do+        -- First root when rho < 0+        zn2 <- sumZSq mbaZ offZ nn+        farSum <- farPoleSum mbaD offD mbaZ offZ nn dj+        let !denominator = 1 + rho * farSum+            !delta0 = if abs denominator > 1e-300+                      then abs rho * zj2 / abs denominator+                      else abs rho * zn2+            !lo_ = dj - max (abs rho * zn2) (2 * delta0)+            !mid = dj - max delta0 (1e-15 * (1 + abs dj))+        newtonLoop 0 mid lo_ dj+  where+    !maxIter_ = 100 :: Int++    -- Newton polishing: 3 Newton steps to refine eigenvalue to machine precision.+    -- Uses the secular function and its derivative for rapid convergence.+    newtonPolish !iter !lam !lb !ub+      | iter >= 3 = pure lam+      | otherwise = do+          (f, fp) <- secularFuncAndDeriv mbaD offD mbaZ offZ rho nn lam+          if abs f < 1e-15 * (1 + abs lam) || abs fp < 1e-300+            then pure lam+            else do+              let !step = f / fp+                  !lamNew = lam - step+                  !clamped = max lb (min ub lamNew)+              if abs (clamped - lam) < 1e-16 * (1 + abs lam)+                then pure clamped+                else newtonPolish (iter + 1) clamped lb ub++    -- Fixed-weight quadratic iteration for interior roots.+    -- dLo, dHi are the two FIXED closest poles (never change during iteration).+    -- lb, ub are the bracket bounds (narrow during iteration).+    -- gap = dHi - dLo.  z2Lo, z2Hi are z²[lo_pole] and z²[hi_pole].+    fixedWeightLoop !iter !lam !lb !ub !dLo !dHi !gap !z2Lo !z2Hi+      | iter >= maxIter_ = pure lam+      | otherwise = do+          -- Evaluate f(λ) with split at the fixed poles dLo and dHi+          (psiSum, phiSum) <- secularFuncSplit mbaD offD mbaZ offZ nn lam dLo dHi+          let !f = 1 + rho * (psiSum + phiSum)+          if abs f < 1e-15 * (1 + abs lam)+            then pure lam+            else do+              -- Extract close-pole contributions using FIXED poles+              let !deltaLo = dLo - lam  -- fixed pole - λ (negative for interior root)+                  !deltaHi = dHi - lam  -- fixed pole - λ (positive for interior root)+                  -- Protect against division by zero near poles+                  !aClose = if abs deltaLo > 1e-300 then z2Lo / deltaLo else 0+                  !bClose = if abs deltaHi > 1e-300 then z2Hi / deltaHi else 0+                  -- "Far" residual: W = f - ρ*(aClose + bClose)+                  !w = f - rho * (aClose + bClose)+                  -- Quadratic in τ = dLo - λ (= deltaLo):+                  -- W*τ² - (W*gap + ρ*z2Lo + ρ*z2Hi)*τ + ρ*z2Lo*gap = 0+                  !qa = w+                  !qb = -(w * gap + rho * z2Lo + rho * z2Hi)+                  !qc = rho * z2Lo * gap+                  !disc = qb * qb - 4 * qa * qc+              if disc < 0 || abs qa < 1e-300+                then do+                  -- Degenerate: fall back to bisection+                  let !(lb', ub') = if f * rho > 0 then (lb, lam) else (lam, ub)+                      !lamNew = (lb' + ub') * 0.5+                  fixedWeightLoop (iter + 1) lamNew lb' ub' dLo dHi gap z2Lo z2Hi+                else do+                  let !sqrtDisc = sqrt disc+                      -- Two roots for τ = dLo - λ, i.e. λ = dLo - τ+                      -- Use the numerically stable form+                      !tauA = if qb <= 0+                              then (-qb + sqrtDisc) / (2 * qa)+                              else 2 * qc / (-qb + sqrtDisc)+                      !tauB = if qb <= 0+                              then 2 * qc / (-qb + sqrtDisc)+                              else (-qb + sqrtDisc) / (2 * qa)+                      -- λ = dLo - τ; pick the root in bracket+                      !lamA = dLo - tauA+                      !lamB = dLo - tauB+                      !lamNew0 = if lamA > lb && lamA < ub then lamA+                                 else if lamB > lb && lamB < ub then lamB+                                 else (lb + ub) * 0.5  -- bisection fallback+                      -- Update bracket+                      !(lb', ub') = if f * rho > 0 then (lb, lam) else (lam, ub)+                      -- Ensure lamNew is in updated bracket+                      !lamNew = if lamNew0 > lb' && lamNew0 < ub'+                                then lamNew0+                                else (lb' + ub') * 0.5+                  if abs (lamNew - lam) < 1e-15 * (1 + abs lam)+                    then pure lamNew+                    else fixedWeightLoop (iter + 1) lamNew lb' ub' dLo dHi gap z2Lo z2Hi++    -- Newton+bisection fallback for edge roots (first/last eigenvalue).+    newtonLoop !iter !lam !lb !ub+      | iter >= maxIter_ = pure lam+      | otherwise = do+          (f, fp) <- secularFuncAndDeriv mbaD offD mbaZ offZ rho nn lam+          if abs f < 1e-15 * (1 + abs lam)+            then pure lam+            else do+              let !(lb', ub') = if f > 0 then (lb, lam) else (lam, ub)+                  !step = f / fp+                  !lamNew0 = lam - step+                  !lamNew = if lamNew0 <= lb' || lamNew0 >= ub'+                            then (lb' + ub') * 0.5+                            else lamNew0+              if abs (lamNew - lam) < 1e-15 * (1 + abs lam)+                then pure lamNew+                else newtonLoop (iter + 1) lamNew lb' ub'++-- | Evaluate the secular function split at the two bracket poles.+-- Returns (ψ, φ) where f(λ) = 1 + ρ*(ψ + φ).+-- ψ = Σ_{d[i] ≤ dLo} z[i]²/(d[i] - λ), φ = Σ_{d[i] ≥ dHi} z[i]²/(d[i] - λ)+secularFuncSplit :: MutableByteArray s -> Int -> MutableByteArray s -> Int+                 -> Int -> Double -> Double -> Double -> ST s (Double, Double)+secularFuncSplit mbaD offD mbaZ offZ nn lam dLo _dHi = go 0 0 0+  where+    go i !psiAcc !phiAcc+      | i >= nn = pure (psiAcc, phiAcc)+      | otherwise = do+          di <- readRawD mbaD offD i+          zi <- readRawD mbaZ offZ i+          let !diff = di - lam+              !zi2 = zi * zi+          if abs diff < 1e-300+            then go (i+1) psiAcc phiAcc+            else let !term = zi2 / diff+                 in if di <= dLo+                    then go (i+1) (psiAcc + term) phiAcc+                    else go (i+1) psiAcc (phiAcc + term)++-- | Evaluate the secular function f(λ) = 1 + ρ * Σ z[i]² / (d[i] - λ)+-- and its derivative f'(λ) = ρ * Σ z[i]² / (d[i] - λ)².+secularFuncAndDeriv :: MutableByteArray s -> Int -> MutableByteArray s -> Int+                    -> Double -> Int -> Double -> ST s (Double, Double)+secularFuncAndDeriv mbaD offD mbaZ offZ rho nn lam = do+  (fSum, fpSum) <- go 0 0 0+  pure (1 + rho * fSum, rho * fpSum)+  where+    go i !fAcc !fpAcc+      | i >= nn = pure (fAcc, fpAcc)+      | otherwise = do+          di <- readRawD mbaD offD i+          zi <- readRawD mbaZ offZ i+          let diff = di - lam+              zi2 = zi * zi+          if abs diff < 1e-300+            then go (i+1) fAcc fpAcc  -- skip near-pole+            else go (i+1) (fAcc + zi2 / diff) (fpAcc + zi2 / (diff * diff))++-- | Sum of squares of z vector.  SIMD-accelerated with DoubleX4# accumulator.+sumZSq :: MutableByteArray s -> Int -> Int -> ST s Double+sumZSq mbaZ offZ nn+  | nn < 4    = goScalar 0 0.0+  | otherwise = do+      baZ <- unsafeFreezeByteArray mbaZ+      let !(ByteArray baZ#) = baZ+          !(I# offZ#) = offZ+          !nn4 = nn - (nn `rem` 4)+          z4 = broadcastDoubleX4# 0.0##+          goSimd !i acc4+            | i >= nn4 = do+                let !(# a, b, c, d #) = unpackDoubleX4# acc4+                goScalar nn4 (D# a + D# b + D# c + D# d)+            | otherwise =+                let !(I# ii) = i+                    zv = indexDoubleArrayAsDoubleX4# baZ# (offZ# +# ii)+                    !p = timesDoubleX4# zv zv+                in goSimd (i + 4) (plusDoubleX4# acc4 p)+      goSimd (0 :: Int) z4+  where+    goScalar i !acc+      | i >= nn = pure acc+      | otherwise = do+          zi <- readRawD mbaZ offZ i+          goScalar (i+1) (acc + zi * zi)++-- | Sum of z[i]^2 / (d[i] - dj) for i in [0..skip-1], skipping near-zero denominators.+farPoleSum :: MutableByteArray s -> Int -> MutableByteArray s -> Int+           -> Int -> Double -> ST s Double+farPoleSum mbaD offD mbaZ offZ skip dj = go 0 0+  where+    go i !acc+      | i >= skip = pure acc+      | otherwise = do+          di <- readRawD mbaD offD i+          zi <- readRawD mbaZ offZ i+          let !diff = di - dj+          if abs diff < 1e-300+            then go (i+1) acc+            else go (i+1) (acc + zi * zi / diff)++-- | Sum of z[k]^2 / (d[k] - dj) for all k in [0..nn-1] except k == skip.+farPoleSumSkip :: MutableByteArray s -> Int -> MutableByteArray s -> Int+               -> Int -> Int -> Double -> ST s Double+farPoleSumSkip mbaD offD mbaZ offZ nn skip dj = go 0 0+  where+    go i !acc+      | i >= nn = pure acc+      | i == skip = go (i+1) acc+      | otherwise = do+          di <- readRawD mbaD offD i+          zi <- readRawD mbaZ offZ i+          let !diff = di - dj+          if abs diff < 1e-300+            then go (i+1) acc+            else go (i+1) (acc + zi * zi / diff)++-- | Partition sorted indices into non-deflated (|z[i]| > deflTol) and deflated.+-- Returns k (non-deflated count).+-- perm[0..k-1] = sorted indices of non-deflated entries (in sorted order).+-- perm[k..nn-1] = sorted indices of deflated entries (in sorted order).+deflatePartition :: MutableByteArray s -> Int    -- ^ sorted z + offset+                 -> MutableByteArray s -> Int    -- ^ output perm (Int array) + offset+                 -> Int -> Double                -- ^ nn, deflTol+                 -> ST s Int+deflatePartition mbaZ offZ mbaPerm offPerm nn deflTol = do+    k <- goND 0 0+    goDF 0 k+    pure k+  where+    goND !i !kND+      | i >= nn = pure kND+      | otherwise = do+          zi <- readRawD mbaZ offZ i+          if abs zi > deflTol+            then do+              writeRawI mbaPerm offPerm kND i+              goND (i+1) (kND+1)+            else goND (i+1) kND+    goDF !i !pos+      | i >= nn = pure ()+      | otherwise = do+          zi <- readRawD mbaZ offZ i+          if abs zi <= deflTol+            then do+              writeRawI mbaPerm offPerm pos i+              goDF (i+1) (pos+1)+            else goDF (i+1) pos++-- | Compute eigenvector matrix W from secular equation solutions.+-- W[j,i] = z[j] / (d[j] - lambda[i]), each column normalised.+-- Single-pass: writes unnormalised entries and accumulates norm² simultaneously,+-- then normalises each column with SIMD.+-- | Compute eigenvector matrix W for the D&C merge step using the+-- Gu-Eisenstat formula (GVL4 Theorem 8.4.4, p. 469) for improved+-- numerical stability.+--+-- Instead of the naive W[j,i] = z[j]/(d[j]-λ[i]) which suffers from+-- catastrophic cancellation when d[j] ≈ λ[i], we first compute:+--+--   z_new[j]² = ∏_k (λ[k] - d[j]) / ∏_{k≠j} (d[k] - d[j])+--+-- This is an algebraic identity but computes z_new to full relative accuracy+-- because all factors in numerator and denominator are well-separated.+-- Then W[j,i] = z_new[j] / (d[j] - λ[i]) with column normalization.+dcEigenvectors :: MutableByteArray s -> Int     -- ^ W (nn × nn output)+               -> MutableByteArray s -> Int     -- ^ d (sorted poles)+               -> MutableByteArray s -> Int     -- ^ z+               -> MutableByteArray s -> Int     -- ^ lambda (eigenvalues)+               -> Double -> Int                 -- ^ rho, nn+               -> ST s ()+dcEigenvectors mbaW offW mbaD offD mbaZ offZ mbaLam offLam _rho nn = do+  -- Phase 1: Compute z_new via Gu-Eisenstat formula in log space+  mbaZnew <- newByteArray (nn * 8)+  forM_ [0..nn-1] $ \j -> do+    zj <- readRawD mbaZ offZ j+    dj <- readRawD mbaD offD j+    -- log|z_new[j]²| = Σ_k log|λ[k] - d[j]| - Σ_{k≠j} log|d[k] - d[j]|+    logNumer <- goLogSum mbaLam offLam 0 nn dj 0 (-1) -- sum all k+    logDenom <- goLogSum mbaD offD 0 nn dj 0 j         -- sum all k ≠ j+    let !logZ2 = logNumer - logDenom+        !absZnew = exp (logZ2 * 0.5)+        !znew = if zj >= 0 then absZnew else negate absZnew+    writeRawD mbaZnew 0 j znew++  -- Phase 2: Build W[j,i] = z_new[j] / (d[j] - λ[i]), normalise columns+  forM_ [0..nn-1] $ \i -> do+    lami <- readRawD mbaLam offLam i+    norm2 <- writeAndNorm mbaZnew lami i 0 0+    let !invNorm = if norm2 > 0 then 1 / sqrt norm2 else 1+    forM_ [0..nn-1] $ \j -> do+      wji <- readRawD mbaW offW (j * nn + i)+      writeRawD mbaW offW (j * nn + i) (wji * invNorm)+  where+    -- Sum of log|arr[k] - val| for k in [lo..hi-1], skipping index 'skip' (-1 = skip none)+    goLogSum !arr !off !lo !hi !val !acc !skip+      | lo >= hi = pure acc+      | lo == skip = goLogSum arr off (lo + 1) hi val acc skip+      | otherwise = do+          ak <- readRawD arr off lo+          let !diff = abs (ak - val)+              !logDiff = if diff < 1e-300 then -690.7755 else log diff  -- log(1e-300)+          goLogSum arr off (lo + 1) hi val (acc + logDiff) skip++    writeAndNorm !mbaZnew !lami !i !j !acc+      | j >= nn = pure acc+      | otherwise = do+          znewj <- readRawD mbaZnew 0 j+          dj <- readRawD mbaD offD j+          let !diff = dj - lami+              !w = if abs diff < 1e-300 then 0 else znewj / diff+          writeRawD mbaW offW (j * nn + i) w+          writeAndNorm mbaZnew lami i (j + 1) (acc + w * w)++-- | Classical Jacobi eigenvalue method (GVL4 Section 8.5).+jacobiEigen :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+            => Matrix n n r e -> Int -> e -> (Vector n r e, Matrix n n r e)+jacobiEigen a maxSweeps tol =+  let nn = dimVal @n+      (eigvals, qArr) = M.withMArrayST (unMatrix (identityMatrix @n @r)) $ \mq -> do+        ma <- M.thawS (unMatrix a)+        jacobiLoop ma mq nn maxSweeps tol+        evs <- mapM (\i -> M.readM ma (i :. i)) [0..nn-1]+        pure (makeVector @n @r $ \i -> evs !! i)+  in (eigvals, MkMatrix qArr)++jacobiLoop :: (M.Manifest r e, Floating e, Ord e)+           => M.MArray s r Ix2 e -> M.MArray s r Ix2 e -> Int -> Int -> e -> ST s ()+jacobiLoop ma mq nn maxSweeps tol = go 0+  where+    go !sweep+      | sweep >= maxSweeps = pure ()+      | otherwise = do+          offNorm <- offDiagNormST ma nn+          if offNorm < tol then pure ()+          else do+            forM_ [(p_, q_) | p_ <- [0..nn-2], q_ <- [p_+1..nn-1]] $ \(p_, q_) -> do+              apq <- M.readM ma (p_ :. q_)+              when (abs apq > tol * 1e-3) $ do+                app <- M.readM ma (p_ :. p_)+                aqq <- M.readM ma (q_ :. q_)+                let (c, s) = jacobiRotation app apq aqq+                applyJacobiInPlace ma c s p_ q_ nn+                applyGivensRightQ mq c s p_ q_ nn+            go (sweep + 1)++offDiagNormST :: (M.Manifest r e, Floating e) => M.MArray s r Ix2 e -> Int -> ST s e+offDiagNormST ma nn = do+  s <- go 0 0 0+  pure (sqrt s)+  where go !i !j !acc+          | i >= nn = pure acc+          | j >= nn = go (i+1) 0 acc+          | i == j = go i (j+1) acc+          | otherwise = do v <- M.readM ma (i :. j); go i (j+1) (acc + v*v)++jacobiRotation :: (Floating e, Ord e) => e -> e -> e -> (e, e)+jacobiRotation app apq aqq+  | apq == 0 = (1, 0)+  | otherwise =+    let tau = (aqq - app) / (2 * apq)+        t = if tau >= 0+            then 1 / (tau + sqrt (1 + tau * tau))+            else 1 / (tau - sqrt (1 + tau * tau))+        c = 1 / sqrt (1 + t * t)+        s = t * c+    in (c, s)++applyJacobiInPlace :: (M.Manifest r e, Num e)+                   => M.MArray s r Ix2 e -> e -> e -> Int -> Int -> Int -> ST s ()+applyJacobiInPlace ma c s p q nn = do+  app <- M.readM ma (p :. p)+  apq_ <- M.readM ma (p :. q)+  aqq <- M.readM ma (q :. q)+  M.write_ ma (p :. p) (c*c*app - 2*s*c*apq_ + s*s*aqq)+  M.write_ ma (q :. q) (s*s*app + 2*s*c*apq_ + c*c*aqq)+  M.write_ ma (p :. q) 0+  M.write_ ma (q :. p) 0+  forM_ [0..nn-1] $ \i -> when (i /= p && i /= q) $ do+    aip <- M.readM ma (i :. p)+    aiq <- M.readM ma (i :. q)+    let aip_new = c * aip - s * aiq+        aiq_new = s * aip + c * aiq+    M.write_ ma (i :. p) aip_new+    M.write_ ma (p :. i) aip_new+    M.write_ ma (i :. q) aiq_new+    M.write_ ma (q :. i) aiq_new
+ src/Numeric/LinearAlgebra/Massiv/Internal.hs view
@@ -0,0 +1,261 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Internal+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Internal module providing unsafe constructors, dimension reification helpers,+-- and array creation utilities. This module is re-exported by+-- "Numeric.LinearAlgebra.Massiv" for convenience, but the unsafe constructors+-- ('unsafeMatrix', 'unsafeVector') bypass the dimension checks provided by the+-- smart constructors in "Numeric.LinearAlgebra.Massiv.Types".+--+-- = Array creation patterns+--+-- Two primary patterns are provided for constructing dimensioned arrays:+--+-- 1. __Pure indexed construction__ via 'makeMatrix' and 'makeVector': supply a+--    pure function @(Int -> Int -> e)@ or @(Int -> e)@ that computes each element+--    from its indices. These use massiv's delayed (@D@) intermediate representation+--    and then @'Data.Massiv.Array.compute'@ to materialise the result.+--+-- 2. __Mutable ST construction__ via 'createMatrix' and 'createVector': supply an+--    @ST@ action operating on a mutable array. This is essential for algorithms+--    that require in-place updates (e.g., LU factorization, Cholesky).+--+-- Both patterns produce arrays with the sequential ('Data.Massiv.Array.Seq')+-- computation strategy by default. Use the @Comp@ variants ('makeMatrixComp',+-- 'makeVectorComp') for parallel construction.+module Numeric.LinearAlgebra.Massiv.Internal+  ( -- * Unsafe constructors+    unsafeMatrix+  , unsafeVector+    -- * Dimension reification+  , dimVal+  , dimVal2+  , reifyDim+  , reifyDim2+    -- * Array creation helpers (pure, sequential)+  , makeMatrix+  , makeVector+    -- * Array creation helpers (pure, with Comp)+  , makeMatrixComp+  , makeVectorComp+    -- * Array creation helpers (mutable ST)+  , createMatrix+  , createVector+  , createMatrixComp+  , createVectorComp+    -- * Mutable modification helpers+  , withMutableMatrix+  , withMutableVector+  , withMutableMatrix_+  , withMutableVector_+    -- * Indexing helpers+  , (!)+  , (!.)+    -- * Identity and zero+  , identityMatrix+  , zeroMatrix+  , zeroVector+  ) where++import Data.Massiv.Array (Array, Ix2(..), Sz(..), Ix1, Comp(..), D)+import qualified Data.Massiv.Array as M+import GHC.TypeNats (Nat, KnownNat, natVal, SomeNat(..), someNatVal)+import Data.Proxy (Proxy(..))+import Control.Monad.ST (ST)++import Numeric.LinearAlgebra.Massiv.Types++-- | Unsafe matrix constructor — wraps a massiv array with /no/ dimension check.+--+-- __Precondition__: the array must have exactly \(m\) rows and \(n\) columns.+-- Violating this precondition leads to index-out-of-bounds errors at runtime.+--+-- Prefer the safe 'matrix' constructor from "Numeric.LinearAlgebra.Massiv.Types"+-- unless you can guarantee correctness (e.g., the array was just constructed+-- with the correct dimensions).+unsafeMatrix :: Array r Ix2 e -> Matrix m n r e+unsafeMatrix = MkMatrix++-- | Unsafe vector constructor — wraps a massiv array with /no/ size check.+--+-- __Precondition__: the array must have exactly \(n\) elements.+unsafeVector :: Array r Ix1 e -> Vector n r e+unsafeVector = MkVector++-- | Get the runtime value of a type-level dimension.+--+-- @+-- dimVal \@3  ==  3+-- dimVal \@100  ==  100+-- @+dimVal :: forall n. KnownNat n => Int+dimVal = fromIntegral (natVal (Proxy @n))++-- | Get both dimensions of a matrix type as a tuple.+dimVal2 :: forall m n. (KnownNat m, KnownNat n) => (Int, Int)+dimVal2 = (dimVal @m, dimVal @n)++-- | Index into a matrix (0-based, unchecked).+--+-- @mat '!' (i, j)@ returns the element at row \(i\), column \(j\).+--+-- __Warning__: No bounds checking is performed. Out-of-bounds access+-- results in undefined behaviour for unboxed\/primitive representations.+(!) :: M.Manifest r e => Matrix m n r e -> (Int, Int) -> e+(!) (MkMatrix arr) (i, j) = M.index' arr (i :. j)++-- | Index into a vector (0-based, unchecked).+--+-- @vec '!.' i@ returns the element at position \(i\).+(!.) :: M.Manifest r e => Vector n r e -> Int -> e+(!.) (MkVector arr) i = M.index' arr i++-- | Reify a runtime 'Int' as a type-level 'GHC.TypeNats.Nat'.+--+-- The continuation receives a 'Proxy' carrying the reified type.+-- This is useful for working with matrices of runtime-determined size.+reifyDim :: Int -> (forall n. KnownNat n => Proxy n -> a) -> a+reifyDim n f = case someNatVal (fromIntegral n) of+  SomeNat p -> f p++-- | Reify two runtime 'Int's as type-level 'GHC.TypeNats.Nat's.+reifyDim2 :: Int -> Int -> (forall m n. (KnownNat m, KnownNat n) => Proxy m -> Proxy n -> a) -> a+reifyDim2 m n f = reifyDim m $ \pm -> reifyDim n $ \pn -> f pm pn++-- | Create a matrix using a pure indexing function (sequential computation).+--+-- @+-- makeMatrix \@3 \@3 \@P $ \\i j -> fromIntegral (i * 3 + j)+-- @+--+-- Internally uses massiv's @'Data.Massiv.Array.Delayed'@ representation as+-- an intermediate before computing to the target representation @r@.+makeMatrix :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+           => (Int -> Int -> e) -> Matrix m n r e+makeMatrix f =+  let r = dimVal @m+      c = dimVal @n+  in MkMatrix $ M.compute @r $ M.makeArray @D Seq (M.Sz2 r c) (\(i :. j) -> f i j)++-- | Create a vector using a pure indexing function (sequential computation).+makeVector :: forall n r e. (KnownNat n, M.Manifest r e)+           => (Int -> e) -> Vector n r e+makeVector f =+  let sz = dimVal @n+  in MkVector $ M.compute @r $ M.makeArray @D Seq (M.Sz1 sz) f++-- | Create a matrix using a pure indexing function with specified+-- 'Data.Massiv.Array.Comp' strategy.+--+-- Use @Par@ for parallel construction of large matrices:+--+-- @+-- makeMatrixComp \@1000 \@1000 \@P Par $ \\i j -> ...+-- @+makeMatrixComp :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+               => Comp -> (Int -> Int -> e) -> Matrix m n r e+makeMatrixComp comp f =+  let r = dimVal @m+      c = dimVal @n+  in MkMatrix $ M.compute @r $ M.makeArray @D comp (M.Sz2 r c) (\(i :. j) -> f i j)++-- | Create a vector using a pure indexing function with specified 'Comp'.+makeVectorComp :: forall n r e. (KnownNat n, M.Manifest r e)+               => Comp -> (Int -> e) -> Vector n r e+makeVectorComp comp f =+  let sz = dimVal @n+  in MkVector $ M.compute @r $ M.makeArray @D comp (M.Sz1 sz) f++-- | Create a matrix using a mutable 'ST' computation.+--+-- The action receives a pre-allocated mutable array of the correct size.+-- All writes must be within bounds. The mutable array is frozen after+-- the action completes.+--+-- This is the primary mechanism for implementing algorithms with in-place+-- updates (e.g., LU factorization, Cholesky decomposition).+createMatrix :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+             => (forall s. M.MArray s r Ix2 e -> ST s ()) -> Matrix m n r e+createMatrix action =+  let r = dimVal @m+      c = dimVal @n+      arr = M.createArrayST_ (M.Sz2 r c) action+  in MkMatrix arr++-- | Create a vector using a mutable 'ST' computation.+createVector :: forall n r e. (KnownNat n, M.Manifest r e)+             => (forall s. M.MArray s r Ix1 e -> ST s ()) -> Vector n r e+createVector action =+  let sz = dimVal @n+      arr = M.createArrayST_ (M.Sz1 sz) action+  in MkVector arr++-- | Create a matrix using a mutable computation with specified 'Comp'.+createMatrixComp :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+                 => Comp -> (forall s. M.MArray s r Ix2 e -> ST s ()) -> Matrix m n r e+createMatrixComp _comp action =+  -- Note: createArrayST_ always runs sequentially; Comp is for delayed computations+  createMatrix @m @n action++-- | Create a vector using a mutable computation with specified 'Comp'.+createVectorComp :: forall n r e. (KnownNat n, M.Manifest r e)+                 => Comp -> (forall s. M.MArray s r Ix1 e -> ST s ()) -> Vector n r e+createVectorComp _comp action = createVector @n action++-- | Run a mutable operation on a /copy/ of the matrix, returning both the+-- action's result and the modified matrix. The original matrix is not modified.+withMutableMatrix :: (M.Manifest r e)+                  => Matrix m n r e+                  -> (forall s. M.MArray s r Ix2 e -> ST s a)+                  -> (a, Matrix m n r e)+withMutableMatrix (MkMatrix arr) action =+  let (result, arr') = M.withMArrayST arr action+  in (result, MkMatrix arr')++-- | Run a mutable operation on a /copy/ of the vector.+withMutableVector :: (M.Manifest r e)+                  => Vector n r e+                  -> (forall s. M.MArray s r Ix1 e -> ST s a)+                  -> (a, Vector n r e)+withMutableVector (MkVector arr) action =+  let (result, arr') = M.withMArrayST arr action+  in (result, MkVector arr')++-- | Like 'withMutableMatrix' but discards the action's result.+withMutableMatrix_ :: (M.Manifest r e)+                   => Matrix m n r e+                   -> (forall s. M.MArray s r Ix2 e -> ST s ())+                   -> Matrix m n r e+withMutableMatrix_ mat action = snd $ withMutableMatrix mat action++-- | Like 'withMutableVector' but discards the action's result.+withMutableVector_ :: (M.Manifest r e)+                   => Vector n r e+                   -> (forall s. M.MArray s r Ix1 e -> ST s ())+                   -> Vector n r e+withMutableVector_ vec action = snd $ withMutableVector vec action++-- | The \(n \times n\) identity matrix \(I_n\).+--+-- \[+--   I_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{otherwise} \end{cases}+-- \]+identityMatrix :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+               => Matrix n n r e+identityMatrix = makeMatrix @n @n @r $ \i j -> if i == j then 1 else 0++-- | The \(m \times n\) zero matrix.+zeroMatrix :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+           => Matrix m n r e+zeroMatrix = makeMatrix @m @n @r $ \_ _ -> 0++-- | The zero vector of dimension \(n\).+zeroVector :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+           => Vector n r e+zeroVector = makeVector @n @r $ const 0
+ src/Numeric/LinearAlgebra/Massiv/Internal/Kernel.hs view
@@ -0,0 +1,2440 @@+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}+{-# LANGUAGE BangPatterns #-}++-- | Raw ByteArray# + AVX2 SIMD kernels for hot inner loops.+--+-- These functions bypass massiv's per-element abstraction layer+-- (M.readM / M.write_ / mapM_) and operate directly on the underlying+-- ByteArray# / MutableByteArray# storage, using GHC's DoubleX4# primops+-- for 256-bit AVX2 SIMD where possible.+module Numeric.LinearAlgebra.Massiv.Internal.Kernel+  ( -- * BLAS-1: dot product+    rawDot+    -- * BLAS-2: matrix-vector multiply+  , rawGemv+    -- * BLAS-3: matrix multiply (GEMM)+  , rawGemmKernel+  , rawGemmBISlice+  , rawGemmBIBJSlice+  , rawGemmBISlicePackedBK+    -- * BLAS-3: symmetric rank-k update (SYRK)+  , rawSyrkLowerKernel+    -- * QR helpers (immutable)+  , rawSumSqRange+  , rawSumProdRange+  , rawHouseholderApplyCol+  , rawQAccumCol+    -- * Eigen / tridiag helpers+  , rawApplyGivensRows+  , rawSymRank2Update+    -- * LU kernels+  , rawLUEliminateColumn+  , rawLUEliminateColumnTo+  , rawSwapRows+  , rawPivotSearch+  , rawForwardSubUnitPacked+  , rawBackSubPacked+  , rawForwardSubUnitPackedSIMD+  , rawBackSubPackedSIMD+    -- * Cholesky kernels+  , rawCholColumn+  , rawCholColumnSIMD+  , rawCholColumnSIMDFrom+  , rawForwardSubCholPacked+  , rawBackSubCholTPacked+  , rawForwardSubCholPackedSIMD+  , rawBackSubCholTPackedSIMD+    -- * QR mutable kernels+  , rawMutSumSqColumn+  , rawMutSumProdColumns+  , rawMutHouseholderApply+  , rawMutQAccum+    -- * Tridiagonalisation mutable kernels+  , rawMutSymMatvecSub+  , rawMutSymRank2Update+  , rawMutTridiagQAccum+    -- * Eigen mutable kernels+  , rawMutApplyGivensColumns+  , rawMutApplyGivensColumnsCM+    -- * Matrix transpose+  , rawTransposeToColMajor+  , rawTransposeFromColMajor+    -- * Bulk memory operations+  , rawZeroDoubles+  , rawCopyDoubles+  , rawNegateDoubles+  , rawCopyColumn+    -- * SVD / bidiagonalisation kernels+  , rawMutHouseholderApplyRow+  , rawMutSumSqRow+  ) where++import GHC.Exts+import GHC.Prim+import GHC.ST (ST(..))+import GHC.Types (Double(..), Int(..))+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..))++-- --------------------------------------------------------------------------+-- BLAS-1: dot product  (DoubleX4# FMA, scalar cleanup)+-- --------------------------------------------------------------------------++-- | Dot product of two Double vectors stored in ByteArrays.+-- @rawDot ba1 off1 ba2 off2 n@ computes Σ ba1[off1+i] * ba2[off2+i] for i in [0..n-1].+rawDot :: ByteArray -> Int -> ByteArray -> Int -> Int -> Double+rawDot (ByteArray ba1) (I# off1) (ByteArray ba2) (I# off2) (I# n) =+  D# (rawDot# ba1 off1 ba2 off2 n)+{-# INLINE rawDot #-}++rawDot# :: ByteArray# -> Int# -> ByteArray# -> Int# -> Int# -> Double#+rawDot# ba1 off1 ba2 off2 n =+  let n4 = n -# (n `remInt#` 4#)+      -- SIMD phase: accumulate 4 doubles at a time+      goSimd i acc+        | isTrue# (i >=# n4) = acc+        | otherwise =+            let va = indexDoubleArrayAsDoubleX4# ba1 (off1 +# i)+                vb = indexDoubleArrayAsDoubleX4# ba2 (off2 +# i)+            in goSimd (i +# 4#) (fmaddDoubleX4# va vb acc)+      acc4 = goSimd 0# (broadcastDoubleX4# 0.0##)+      !(# a, b, c, d #) = unpackDoubleX4# acc4+      simdSum = a +## b +## c +## d+      -- Scalar cleanup for remainder+      goScalar i acc+        | isTrue# (i >=# n) = acc+        | otherwise =+            let x = indexDoubleArray# ba1 (off1 +# i)+                y = indexDoubleArray# ba2 (off2 +# i)+            in goScalar (i +# 1#) (acc +## x *## y)+  in goScalar n4 simdSum+{-# NOINLINE rawDot# #-}++-- --------------------------------------------------------------------------+-- BLAS-2: matrix-vector multiply+-- --------------------------------------------------------------------------++-- | @rawGemv ba_a off_a n_cols ba_x off_x mba_y off_y n_rows@ computes+-- y[i] = Σ_j A[i,j] * x[j] for i in [0..n_rows-1], j in [0..n_cols-1].+-- A is row-major with stride = n_cols.+rawGemv :: ByteArray -> Int -> Int+        -> ByteArray -> Int+        -> MutableByteArray s -> Int -> Int+        -> ST s ()+rawGemv (ByteArray ba_a) (I# off_a) (I# ncols)+        (ByteArray ba_x) (I# off_x)+        (MutableByteArray mba_y) (I# off_y) (I# nrows) = ST $ \s0 ->+  let go i s+        | isTrue# (i >=# nrows) = s+        | otherwise =+            let rowOff = off_a +# i *# ncols+                dot = rawDot# ba_a rowOff ba_x off_x ncols+            in case writeDoubleArray# mba_y (off_y +# i) dot s of+                 s' -> go (i +# 1#) s'+  in (# go 0# s0, () #)+{-# INLINE rawGemv #-}++-- --------------------------------------------------------------------------+-- BLAS-3: tiled ikj GEMM kernel+-- --------------------------------------------------------------------------++-- | @rawGemmKernel ba_a off_a ba_b off_b mba_c off_c m k n@ computes+-- C += A * B where A is m×k, B is k×n, C is m×n (all row-major).+-- C must be pre-initialised (e.g. to zero, or to β*C for gemm).+-- Uses packed-B variant (BK-outer with panel packing) for large matrices+-- where cache/TLB effects dominate, unpacked variant for smaller matrices.+rawGemmKernel :: ByteArray -> Int -> ByteArray -> Int+              -> MutableByteArray s -> Int+              -> Int -> Int -> Int -> ST s ()+rawGemmKernel ba offA bb offB mc offC m k n+  | min m (min k n) >= gemmPackCrossover =+      rawGemmBISlicePackedBK ba offA bb offB mc offC 0 m m k n+  | otherwise =+      rawGemmBISlice ba offA bb offB mc offC 0 m m k n+{-# INLINE rawGemmKernel #-}++-- | Crossover threshold for GEMM packing.  Below this, the unpacked+-- BI-outer kernel is used; above it, the packed-B BK-outer kernel.+gemmPackCrossover :: Int+gemmPackCrossover = 96+{-# INLINE gemmPackCrossover #-}++-- | @rawGemmBISlice@ computes C[biStart..biEnd-1, :] += A[biStart..biEnd-1, :] * B.+-- Delegates to 'rawGemmBIBJSlice' with full column range.+rawGemmBISlice :: ByteArray -> Int -> ByteArray -> Int+               -> MutableByteArray s -> Int+               -> Int -> Int -> Int -> Int -> Int -> ST s ()+rawGemmBISlice ba offA bb offB mc offC biStart biEnd m k n =+  rawGemmBIBJSlice ba offA bb offB mc offC biStart biEnd 0 n m k n+{-# INLINE rawGemmBISlice #-}++-- | @rawGemmBIBJSlice ba_a off_a ba_b off_b mba_c off_c biStart biEnd bjStart bjEnd m k n@+-- computes C[biStart..biEnd-1, bjStart..bjEnd-1] += A[biStart..biEnd-1, :] * B[:, bjStart..bjEnd-1]+-- where A is m×k, B is k×n, C is m×n (all row-major).+-- Only the rows [biStart, biEnd) and columns [bjStart, bjEnd) of C are written.+-- This enables 2D parallel GEMM by partitioning both row and column ranges.+rawGemmBIBJSlice :: ByteArray -> Int -> ByteArray -> Int+                 -> MutableByteArray s -> Int+                 -> Int -> Int -> Int -> Int -> Int -> Int -> Int -> ST s ()+rawGemmBIBJSlice (ByteArray ba_a) (I# off_a) (ByteArray ba_b) (I# off_b)+                 (MutableByteArray mba_c) (I# off_c)+                 (I# biStart) (I# biEnd) (I# bjStart) (I# bjEnd)+                 (I# _m) (I# k) (I# n) = ST $ \s0 ->+  let bs = 64#++      goBI bi s+        | isTrue# (bi >=# biEnd) = s+        | otherwise =+            let iEnd = minI (bi +# bs) biEnd+            in goBI (bi +# bs) (goBK bi iEnd 0# s)++      goBK bi iEnd bk s+        | isTrue# (bk >=# k) = s+        | otherwise =+            let kEnd = minI (bk +# bs) k+            in goBK bi iEnd (bk +# bs) (goBJ bi iEnd bk kEnd bjStart s)++      goBJ bi iEnd bk kEnd bj s+        | isTrue# (bj >=# bjEnd) = s+        | otherwise =+            let jEnd = minI (bj +# bs) bjEnd+            in goBJ bi iEnd bk kEnd (bj +# bs) (innerBlock bi iEnd bk kEnd bj jEnd s)++      -- Register-blocked micro-kernel: process 4 rows × 8 columns of C+      -- in SIMD registers across the full k-range, writing back once.+      innerBlock bi iEnd bk kEnd bj jEnd s0_ =+        let !jSpan = jEnd -# bj+            !j8End = bj +# (jSpan -# (jSpan `remInt#` 8#))+            !j4End = bj +# (jSpan -# (jSpan `remInt#` 4#))+            !iSpan = iEnd -# bi+            !i4End = bi +# (iSpan -# (iSpan `remInt#` 4#))+        in goI4 bi i4End iEnd j8End j4End bk kEnd bj jEnd s0_++      -- Process 4 rows at a time+      goI4 i i4End iEnd_ j8End j4End bk kEnd bj jEnd s+        | isTrue# (i >=# i4End) = goI1 i iEnd_ j8End j4End bk kEnd bj jEnd s+        | otherwise =+            goI4 (i +# 4#) i4End iEnd_ j8End j4End bk kEnd bj jEnd+              (goJ8_4x8 i bk kEnd bj j8End+                (goJ4_4x4 i bk kEnd j8End j4End+                  (goJScalar4 i bk kEnd j4End jEnd s)))++      -- Process remaining 1 row at a time (up to tile boundary iEnd_, not biEnd)+      goI1 i iEnd_ j8End j4End bk kEnd bj jEnd s+        | isTrue# (i >=# iEnd_) = s+        | otherwise =+            goI1 (i +# 1#) iEnd_ j8End j4End bk kEnd bj jEnd+              (goJ8_1x8 i bk kEnd bj j8End+                (goJ4_1x4 i bk kEnd j8End j4End+                  (goJScalar1 i bk kEnd j4End jEnd s)))++      -- 4×8 micro-kernel: 4 rows of i, 8 columns of j (2× DoubleX4#)+      -- Load 8 C accumulators, sweep k, write back+      goJ8_4x8 i bk kEnd j j8End s+        | isTrue# (j >=# j8End) = s+        | otherwise =+          let !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0a, c00 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) s0a of { (# s0b, c01 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff1 s0b of { (# s1a, c10 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) s1a of { (# s1b, c11 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff2 s1b of { (# s2a, c20 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) s2a of { (# s2b, c21 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff3 s2b of { (# s3a, c30 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) s3a of { (# s3b, c31 #) ->+             -- Now sweep k, accumulating in registers+             case goK4x8 i j bk kEnd c00 c01 c10 c11 c20 c21 c30 c31 of+               (# r00, r01, r10, r11, r20, r21, r30, r31 #) ->+                 -- Write back all 8 SIMD registers+                 case writeDoubleArrayAsDoubleX4# mba_c cOff0 r00 s3b of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) r01 sw0 of { sw1 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff1 r10 sw1 of { sw2 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) r11 sw2 of { sw3 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff2 r20 sw3 of { sw4 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) r21 sw4 of { sw5 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff3 r30 sw5 of { sw6 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) r31 sw6 of { sw7 ->+                 goJ8_4x8 i bk kEnd (j +# 8#) j8End sw7+                 }}}}}}}}+             }}}}}}}}++      -- Pure k-loop for 4×8: no state threading needed (immutable A, B reads)+      goK4x8 i j kk kEnd c00 c01 c10 c11 c20 c21 c30 c31+        | isTrue# (kk >=# kEnd) =+            (# c00, c01, c10, c11, c20, c21, c30, c31 #)+        | otherwise =+            let !bOff = off_b +# kk *# n +# j+                !bv0 = indexDoubleArrayAsDoubleX4# ba_b bOff+                !bv1 = indexDoubleArrayAsDoubleX4# ba_b (bOff +# 4#)+                !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+                !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk))+                !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk))+                !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk))+            in goK4x8 i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# a0 bv0 c00) (fmaddDoubleX4# a0 bv1 c01)+                 (fmaddDoubleX4# a1 bv0 c10) (fmaddDoubleX4# a1 bv1 c11)+                 (fmaddDoubleX4# a2 bv0 c20) (fmaddDoubleX4# a2 bv1 c21)+                 (fmaddDoubleX4# a3 bv0 c30) (fmaddDoubleX4# a3 bv1 c31)++      -- 4×4 cleanup: 4 rows, 4 columns (1× DoubleX4#)+      goJ4_4x4 i bk kEnd j j4End s+        | isTrue# (j >=# j4End) = s+        | otherwise =+          let !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0, c0 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff1 s0 of { (# s1, c1 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff2 s1 of { (# s2, c2 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff3 s2 of { (# s3, c3 #) ->+             case goK4x4 i j bk kEnd c0 c1 c2 c3 of+               (# r0, r1, r2, r3 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff0 r0 s3 of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff1 r1 sw0 of { sw1 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff2 r2 sw1 of { sw2 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff3 r3 sw2 of { sw3 ->+                 goJ4_4x4 i bk kEnd (j +# 4#) j4End sw3+                 }}}}+             }}}}++      goK4x4 i j kk kEnd c0 c1 c2 c3+        | isTrue# (kk >=# kEnd) = (# c0, c1, c2, c3 #)+        | otherwise =+            let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+                !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk))+                !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk))+                !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk))+            in goK4x4 i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# a0 bv c0) (fmaddDoubleX4# a1 bv c1)+                 (fmaddDoubleX4# a2 bv c2) (fmaddDoubleX4# a3 bv c3)++      -- Scalar cleanup for 4 rows (columns not a multiple of 4)+      goJScalar4 i bk kEnd j jEnd_ s+        | isTrue# (j >=# jEnd_) = s+        | otherwise =+          let goK_s4 kk acc0 acc1 acc2 acc3+                | isTrue# (kk >=# kEnd) = (# acc0, acc1, acc2, acc3 #)+                | otherwise =+                    let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                        !a0_ = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                        !a1_ = indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk)+                        !a2_ = indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk)+                        !a3_ = indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk)+                    in goK_s4 (kk +# 1#)+                         (acc0 +## a0_ *## bkj) (acc1 +## a1_ *## bkj)+                         (acc2 +## a2_ *## bkj) (acc3 +## a3_ *## bkj)+              !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case (# goK_s4 bk 0.0## 0.0## 0.0## 0.0## #) of+               (# (# d0, d1, d2, d3 #) #) ->+                 case readDoubleArray# mba_c cOff0 s of { (# s0, v0 #) ->+                 case writeDoubleArray# mba_c cOff0 (v0 +## d0) s0 of { s1 ->+                 case readDoubleArray# mba_c cOff1 s1 of { (# s2, v1 #) ->+                 case writeDoubleArray# mba_c cOff1 (v1 +## d1) s2 of { s3 ->+                 case readDoubleArray# mba_c cOff2 s3 of { (# s4, v2 #) ->+                 case writeDoubleArray# mba_c cOff2 (v2 +## d2) s4 of { s5 ->+                 case readDoubleArray# mba_c cOff3 s5 of { (# s6, v3 #) ->+                 case writeDoubleArray# mba_c cOff3 (v3 +## d3) s6 of { s7 ->+                 goJScalar4 i bk kEnd (j +# 1#) jEnd_ s7+                 }}}}}}}}++      -- 1×8 micro-kernel: 1 row of i, 8 columns of j+      goJ8_1x8 i bk kEnd j j8End s+        | isTrue# (j >=# j8End) = s+        | otherwise =+          let !cOff = off_c +# i *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) s0 of { (# s1, c1 #) ->+             case goK1x8 i j bk kEnd c0 c1 of+               (# r0, r1 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s1 of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) r1 sw0 of { sw1 ->+                 goJ8_1x8 i bk kEnd (j +# 8#) j8End sw1+                 }}+             }}++      goK1x8 i j kk kEnd c0 c1+        | isTrue# (kk >=# kEnd) = (# c0, c1 #)+        | otherwise =+            let !bOff = off_b +# kk *# n +# j+                !bv0 = indexDoubleArrayAsDoubleX4# ba_b bOff+                !bv1 = indexDoubleArrayAsDoubleX4# ba_b (bOff +# 4#)+                !av = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+            in goK1x8 i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# av bv0 c0) (fmaddDoubleX4# av bv1 c1)++      -- 1×4 cleanup+      goJ4_1x4 i bk kEnd j j4End s+        | isTrue# (j >=# j4End) = s+        | otherwise =+          let !cOff = off_c +# i *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+             case goK1x4 i j bk kEnd c0 of { r0 ->+             case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s0 of { s1 ->+             goJ4_1x4 i bk kEnd (j +# 4#) j4End s1+             }}}++      goK1x4 i j kk kEnd c0+        | isTrue# (kk >=# kEnd) = c0+        | otherwise =+            let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                !av = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+            in goK1x4 i j (kk +# 1#) kEnd (fmaddDoubleX4# av bv c0)++      -- Scalar cleanup for 1 row+      goJScalar1 i bk kEnd j jEnd_ s+        | isTrue# (j >=# jEnd_) = s+        | otherwise =+          let goK_s1 kk acc+                | isTrue# (kk >=# kEnd) = acc+                | otherwise =+                    let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                        !aik = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                    in goK_s1 (kk +# 1#) (acc +## aik *## bkj)+          in case readDoubleArray# mba_c (off_c +# i *# n +# j) s of+               (# s', cij #) ->+                 case writeDoubleArray# mba_c (off_c +# i *# n +# j) (cij +## goK_s1 bk 0.0##) s' of+                   s'' -> goJScalar1 i bk kEnd (j +# 1#) jEnd_ s''++  in (# goBI biStart s0, () #)+{-# INLINE rawGemmBIBJSlice #-}++-- | Packed-B GEMM with BK-outer loop ordering.+-- Packs B into 8-column panels for sequential cache access (stride 8 instead+-- of stride n).  Only the rows [biStart, biEnd) of C are written.+-- Beneficial for large matrices where stride-n B access causes cache/TLB misses.+rawGemmBISlicePackedBK :: ByteArray -> Int -> ByteArray -> Int+                       -> MutableByteArray s -> Int+                       -> Int -> Int -> Int -> Int -> Int -> ST s ()+rawGemmBISlicePackedBK (ByteArray ba_a) (I# off_a) (ByteArray ba_b) (I# off_b)+                       (MutableByteArray mba_c) (I# off_c)+                       (I# biStart) (I# biEnd) (I# _m) (I# k) (I# n) = ST $ \s0 ->+  let bs = 64#+      !nPanels = n `quotInt#` 8#+      !j8End = nPanels *# 8#+      !j4End = n -# (n `remInt#` 4#)+      -- Packed buffer: nPanels * bs * 8 doubles (each panel: kc × 8)+      !packDoubles = nPanels *# bs *# 8#++      -- Fallback unpacked path (when j8End == 0, i.e. n < 8)+      goBI_unpacked bi s+        | isTrue# (bi >=# biEnd) = s+        | otherwise =+            let iEnd = minI (bi +# bs) biEnd+            in goBI_unpacked (bi +# bs) (goBK_u bi iEnd 0# s)++      goBK_u bi iEnd bk s+        | isTrue# (bk >=# k) = s+        | otherwise =+            let kEnd = minI (bk +# bs) k+                iSpan = iEnd -# bi+                i4End_ = bi +# (iSpan -# (iSpan `remInt#` 4#))+            in goBK_u bi iEnd (bk +# bs)+                 (goI4U_fb bi i4End_ iEnd bk kEnd+                   (goI1U_fb i4End_ iEnd bk kEnd s))++      goI4U_fb i i4End _iEnd bk kEnd s+        | isTrue# (i >=# i4End) = s+        | otherwise =+            goI4U_fb (i +# 4#) i4End _iEnd bk kEnd+              (goJ4Ufb4 i bk kEnd 0# j4End+                (goJSUfb4 i bk kEnd j4End n s))++      goI1U_fb i iEnd bk kEnd s+        | isTrue# (i >=# iEnd) = s+        | otherwise =+            goI1U_fb (i +# 1#) iEnd bk kEnd+              (goJ4Ufb1 i bk kEnd 0# j4End+                (goJSUfb1 i bk kEnd j4End n s))++      -- Unpacked 4×4 and scalar for fallback+      goJ4Ufb4 i bk kEnd j j4EndV s+        | isTrue# (j >=# j4EndV) = s+        | otherwise =+          let !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0, c0 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff1 s0 of { (# s1, c1 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff2 s1 of { (# s2, c2 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff3 s2 of { (# s3, c3 #) ->+             case goKUfb4x4 i j bk kEnd c0 c1 c2 c3 of+               (# r0, r1, r2, r3 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff0 r0 s3 of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff1 r1 sw0 of { sw1 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff2 r2 sw1 of { sw2 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff3 r3 sw2 of { sw3 ->+                 goJ4Ufb4 i bk kEnd (j +# 4#) j4EndV sw3+                 }}}}+             }}}}++      goKUfb4x4 i j kk kEnd c0 c1 c2 c3+        | isTrue# (kk >=# kEnd) = (# c0, c1, c2, c3 #)+        | otherwise =+            let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+                !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk))+                !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk))+                !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk))+            in goKUfb4x4 i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# a0 bv c0) (fmaddDoubleX4# a1 bv c1)+                 (fmaddDoubleX4# a2 bv c2) (fmaddDoubleX4# a3 bv c3)++      goJSUfb4 i bk kEnd j jEndV s+        | isTrue# (j >=# jEndV) = s+        | otherwise =+          let goKSfb4 kk acc0 acc1 acc2 acc3+                | isTrue# (kk >=# kEnd) = (# acc0, acc1, acc2, acc3 #)+                | otherwise =+                    let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                        !a0_ = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                        !a1_ = indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk)+                        !a2_ = indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk)+                        !a3_ = indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk)+                    in goKSfb4 (kk +# 1#)+                         (acc0 +## a0_ *## bkj) (acc1 +## a1_ *## bkj)+                         (acc2 +## a2_ *## bkj) (acc3 +## a3_ *## bkj)+              !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case (# goKSfb4 bk 0.0## 0.0## 0.0## 0.0## #) of+               (# (# d0, d1, d2, d3 #) #) ->+                 case readDoubleArray# mba_c cOff0 s of { (# s0, v0 #) ->+                 case writeDoubleArray# mba_c cOff0 (v0 +## d0) s0 of { s1 ->+                 case readDoubleArray# mba_c cOff1 s1 of { (# s2, v1 #) ->+                 case writeDoubleArray# mba_c cOff1 (v1 +## d1) s2 of { s3 ->+                 case readDoubleArray# mba_c cOff2 s3 of { (# s4, v2 #) ->+                 case writeDoubleArray# mba_c cOff2 (v2 +## d2) s4 of { s5 ->+                 case readDoubleArray# mba_c cOff3 s5 of { (# s6, v3 #) ->+                 case writeDoubleArray# mba_c cOff3 (v3 +## d3) s6 of { s7 ->+                 goJSUfb4 i bk kEnd (j +# 1#) jEndV s7+                 }}}}}}}}++      goJ4Ufb1 i bk kEnd j j4EndV s+        | isTrue# (j >=# j4EndV) = s+        | otherwise =+          let !cOff = off_c +# i *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+             case goKUfb1x4 i j bk kEnd c0 of { r0 ->+             case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s0 of { s1 ->+             goJ4Ufb1 i bk kEnd (j +# 4#) j4EndV s1+             }}}++      goKUfb1x4 i j kk kEnd c0+        | isTrue# (kk >=# kEnd) = c0+        | otherwise =+            let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                !av = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+            in goKUfb1x4 i j (kk +# 1#) kEnd (fmaddDoubleX4# av bv c0)++      goJSUfb1 i bk kEnd j jEndV s+        | isTrue# (j >=# jEndV) = s+        | otherwise =+          let goKSfb1 kk acc+                | isTrue# (kk >=# kEnd) = acc+                | otherwise =+                    let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                        !aik = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                    in goKSfb1 (kk +# 1#) (acc +## aik *## bkj)+          in case readDoubleArray# mba_c (off_c +# i *# n +# j) s of+               (# s', cij #) ->+                 case writeDoubleArray# mba_c (off_c +# i *# n +# j) (cij +## goKSfb1 bk 0.0##) s' of+                   s'' -> goJSUfb1 i bk kEnd (j +# 1#) jEndV s''++  in if isTrue# (j8End ==# 0#)+     -- No full 8-column panels: fall back to unpacked scalar path+     then (# goBI_unpacked biStart s0, () #)+     else++     -- BK outer loop: allocate pack buffer, pack B, freeze, then run BI/BJ loops+     -- with pure k-loop reads (indexDoubleArrayAsDoubleX4# on immutable ByteArray#).+     let goBKO bk s+           | isTrue# (bk >=# k) = s+           | otherwise =+               let !kEnd = minI (bk +# bs) k+                   !kc = kEnd -# bk+                   !packBufBytes = nPanels *# kc *# 8# *# 8#+               in case newByteArray# packBufBytes s of { (# s1, mba_bp #) ->+                  -- Pack B[bk:bk+kc, 0:j8End] into panel-major layout+                  let s2 = packB mba_bp bk kc s1+                  in case unsafeFreezeByteArray# mba_bp s2 of { (# s3, ba_bp #) ->+                     goBKO (bk +# bs) (goBIO bk kEnd kc ba_bp biStart s3)+                  }}++         -- Pack B into mba_bp: panel p (cols 8p..8p+7):+         --   bp[p * kc * 8 + kLocal * 8 + jLocal] = B[bk+kLocal, 8p+jLocal]+         packB mba_bp bk kc sp =+           let goPnl p sp0+                 | isTrue# (p >=# nPanels) = sp0+                 | otherwise =+                     let !jBase = p *# 8#+                         !pOff = p *# kc *# 8#+                         goKP kl sk+                           | isTrue# (kl >=# kc) = sk+                           | otherwise =+                               let !srcOff = off_b +# (bk +# kl) *# n +# jBase+                                   !dstOff = pOff +# kl *# 8#+                                   !v0 = indexDoubleArrayAsDoubleX4# ba_b srcOff+                                   !v1 = indexDoubleArrayAsDoubleX4# ba_b (srcOff +# 4#)+                               in case writeDoubleArrayAsDoubleX4# mba_bp dstOff v0 sk of+                                    sk' -> case writeDoubleArrayAsDoubleX4# mba_bp (dstOff +# 4#) v1 sk' of+                                             sk'' -> goKP (kl +# 1#) sk''+                     in goPnl (p +# 1#) (goKP 0# sp0)+           in goPnl 0# sp++         -- BI middle loop (ba_bp is frozen ByteArray# for this BK tile)+         goBIO bk kEnd kc ba_bp bi s+           | isTrue# (bi >=# biEnd) = s+           | otherwise =+               let !iEnd = minI (bi +# bs) biEnd+                   !iSpan = iEnd -# bi+                   !i4End_ = bi +# (iSpan -# (iSpan `remInt#` 4#))+               in goBIO bk kEnd kc ba_bp (bi +# bs)+                    (goI4P bi i4End_ iEnd bk kEnd kc ba_bp+                      (goI1P i4End_ iEnd bk kEnd kc ba_bp s))++         -- 4 rows: packed panels then unpacked tail+         goI4P i i4End _iEnd bk kEnd kc ba_bp s+           | isTrue# (i >=# i4End) = s+           | otherwise =+               goI4P (i +# 4#) i4End _iEnd bk kEnd kc ba_bp+                 (goJ8P4 i bk kc ba_bp 0# j8End+                   (goJ4U4 i bk kEnd j8End j4End+                     (goJSU4 i bk kEnd j4End n s)))++         -- 1 row: packed panels then unpacked tail+         goI1P i iEnd bk kEnd kc ba_bp s+           | isTrue# (i >=# iEnd) = s+           | otherwise =+               goI1P (i +# 1#) iEnd bk kEnd kc ba_bp+                 (goJ8P1 i bk kc ba_bp 0# j8End+                   (goJ4U1 i bk kEnd j8End j4End+                     (goJSU1 i bk kEnd j4End n s)))++         -- ----------------------------------------------------------------+         -- Packed 4×8 j-loop: step by 8, pure reads from frozen ByteArray#+         -- ----------------------------------------------------------------+         goJ8P4 i bk kc ba_bp j jEnd s+           | isTrue# (j >=# jEnd) = s+           | otherwise =+             let !cOff0 = off_c +# i *# n +# j+                 !cOff1 = off_c +# (i +# 1#) *# n +# j+                 !cOff2 = off_c +# (i +# 2#) *# n +# j+                 !cOff3 = off_c +# (i +# 3#) *# n +# j+                 !panOff = (j `quotInt#` 8#) *# kc *# 8#+             in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0a, c00 #) ->+                case readDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) s0a of { (# s0b, c01 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff1 s0b of { (# s1a, c10 #) ->+                case readDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) s1a of { (# s1b, c11 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff2 s1b of { (# s2a, c20 #) ->+                case readDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) s2a of { (# s2b, c21 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff3 s2b of { (# s3a, c30 #) ->+                case readDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) s3a of { (# s3b, c31 #) ->+                case goKP4x8 i panOff bk 0# kc ba_bp c00 c01 c10 c11 c20 c21 c30 c31 of+                  (# r00, r01, r10, r11, r20, r21, r30, r31 #) ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff0 r00 s3b of { sw0 ->+                    case writeDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) r01 sw0 of { sw1 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff1 r10 sw1 of { sw2 ->+                    case writeDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) r11 sw2 of { sw3 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff2 r20 sw3 of { sw4 ->+                    case writeDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) r21 sw4 of { sw5 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff3 r30 sw5 of { sw6 ->+                    case writeDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) r31 sw6 of { sw7 ->+                    goJ8P4 i bk kc ba_bp (j +# 8#) jEnd sw7+                    }}}}}}}}+                }}}}}}}}++         -- Pure k-loop for packed 4×8: reads from frozen ByteArray# (no State#)+         goKP4x8 i panOff bk kl kc ba_bp c00 c01 c10 c11 c20 c21 c30 c31+           | isTrue# (kl >=# kc) =+               (# c00, c01, c10, c11, c20, c21, c30, c31 #)+           | otherwise =+               let !bOff = panOff +# kl *# 8#+                   !bv0 = indexDoubleArrayAsDoubleX4# ba_bp bOff+                   !bv1 = indexDoubleArrayAsDoubleX4# ba_bp (bOff +# 4#)+                   !kk = bk +# kl+                   !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+                   !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk))+                   !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk))+                   !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk))+               in goKP4x8 i panOff bk (kl +# 1#) kc ba_bp+                    (fmaddDoubleX4# a0 bv0 c00) (fmaddDoubleX4# a0 bv1 c01)+                    (fmaddDoubleX4# a1 bv0 c10) (fmaddDoubleX4# a1 bv1 c11)+                    (fmaddDoubleX4# a2 bv0 c20) (fmaddDoubleX4# a2 bv1 c21)+                    (fmaddDoubleX4# a3 bv0 c30) (fmaddDoubleX4# a3 bv1 c31)++         -- ----------------------------------------------------------------+         -- Packed 1×8 j-loop+         -- ----------------------------------------------------------------+         goJ8P1 i bk kc ba_bp j jEnd s+           | isTrue# (j >=# jEnd) = s+           | otherwise =+             let !cOff = off_c +# i *# n +# j+                 !panOff = (j `quotInt#` 8#) *# kc *# 8#+             in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+                case readDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) s0 of { (# s1, c1 #) ->+                case goKP1x8 i panOff bk 0# kc ba_bp c0 c1 of+                  (# r0, r1 #) ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s1 of { sw0 ->+                    case writeDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) r1 sw0 of { sw1 ->+                    goJ8P1 i bk kc ba_bp (j +# 8#) jEnd sw1+                    }}+                }}++         -- Pure k-loop for packed 1×8+         goKP1x8 i panOff bk kl kc ba_bp c0 c1+           | isTrue# (kl >=# kc) = (# c0, c1 #)+           | otherwise =+               let !bOff = panOff +# kl *# 8#+                   !bv0 = indexDoubleArrayAsDoubleX4# ba_bp bOff+                   !bv1 = indexDoubleArrayAsDoubleX4# ba_bp (bOff +# 4#)+                   !kk = bk +# kl+                   !av = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+               in goKP1x8 i panOff bk (kl +# 1#) kc ba_bp+                    (fmaddDoubleX4# av bv0 c0) (fmaddDoubleX4# av bv1 c1)++         -- ----------------------------------------------------------------+         -- Unpacked j-tail: 4×4 cleanup (columns j8End..j4End)+         -- ----------------------------------------------------------------+         goJ4U4 i bk kEnd j j4EndV s+           | isTrue# (j >=# j4EndV) = s+           | otherwise =+             let !cOff0 = off_c +# i *# n +# j+                 !cOff1 = off_c +# (i +# 1#) *# n +# j+                 !cOff2 = off_c +# (i +# 2#) *# n +# j+                 !cOff3 = off_c +# (i +# 3#) *# n +# j+             in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0, c0 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff1 s0 of { (# s1, c1 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff2 s1 of { (# s2, c2 #) ->+                case readDoubleArrayAsDoubleX4# mba_c cOff3 s2 of { (# s3, c3 #) ->+                case goKU4x4 i j bk kEnd c0 c1 c2 c3 of+                  (# r0, r1, r2, r3 #) ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff0 r0 s3 of { sw0 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff1 r1 sw0 of { sw1 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff2 r2 sw1 of { sw2 ->+                    case writeDoubleArrayAsDoubleX4# mba_c cOff3 r3 sw2 of { sw3 ->+                    goJ4U4 i bk kEnd (j +# 4#) j4EndV sw3+                    }}}}+                }}}}++         goKU4x4 i j kk kEnd c0 c1 c2 c3+           | isTrue# (kk >=# kEnd) = (# c0, c1, c2, c3 #)+           | otherwise =+               let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                   !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+                   !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk))+                   !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk))+                   !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk))+               in goKU4x4 i j (kk +# 1#) kEnd+                    (fmaddDoubleX4# a0 bv c0) (fmaddDoubleX4# a1 bv c1)+                    (fmaddDoubleX4# a2 bv c2) (fmaddDoubleX4# a3 bv c3)++         -- Unpacked scalar cleanup for 4 rows (columns j4End..n)+         goJSU4 i bk kEnd j jEndV s+           | isTrue# (j >=# jEndV) = s+           | otherwise =+             let goKS4 kk acc0 acc1 acc2 acc3+                   | isTrue# (kk >=# kEnd) = (# acc0, acc1, acc2, acc3 #)+                   | otherwise =+                       let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                           !a0_ = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                           !a1_ = indexDoubleArray# ba_a (off_a +# (i +# 1#) *# k +# kk)+                           !a2_ = indexDoubleArray# ba_a (off_a +# (i +# 2#) *# k +# kk)+                           !a3_ = indexDoubleArray# ba_a (off_a +# (i +# 3#) *# k +# kk)+                       in goKS4 (kk +# 1#)+                            (acc0 +## a0_ *## bkj) (acc1 +## a1_ *## bkj)+                            (acc2 +## a2_ *## bkj) (acc3 +## a3_ *## bkj)+                 !cOff0 = off_c +# i *# n +# j+                 !cOff1 = off_c +# (i +# 1#) *# n +# j+                 !cOff2 = off_c +# (i +# 2#) *# n +# j+                 !cOff3 = off_c +# (i +# 3#) *# n +# j+             in case (# goKS4 bk 0.0## 0.0## 0.0## 0.0## #) of+                  (# (# d0, d1, d2, d3 #) #) ->+                    case readDoubleArray# mba_c cOff0 s of { (# s0, v0 #) ->+                    case writeDoubleArray# mba_c cOff0 (v0 +## d0) s0 of { s1 ->+                    case readDoubleArray# mba_c cOff1 s1 of { (# s2, v1 #) ->+                    case writeDoubleArray# mba_c cOff1 (v1 +## d1) s2 of { s3 ->+                    case readDoubleArray# mba_c cOff2 s3 of { (# s4, v2 #) ->+                    case writeDoubleArray# mba_c cOff2 (v2 +## d2) s4 of { s5 ->+                    case readDoubleArray# mba_c cOff3 s5 of { (# s6, v3 #) ->+                    case writeDoubleArray# mba_c cOff3 (v3 +## d3) s6 of { s7 ->+                    goJSU4 i bk kEnd (j +# 1#) jEndV s7+                    }}}}}}}}++         -- Unpacked 1×4 cleanup+         goJ4U1 i bk kEnd j j4EndV s+           | isTrue# (j >=# j4EndV) = s+           | otherwise =+             let !cOff = off_c +# i *# n +# j+             in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+                case goKU1x4 i j bk kEnd c0 of { r0 ->+                case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s0 of { s1 ->+                goJ4U1 i bk kEnd (j +# 4#) j4EndV s1+                }}}++         goKU1x4 i j kk kEnd c0+           | isTrue# (kk >=# kEnd) = c0+           | otherwise =+               let !bv = indexDoubleArrayAsDoubleX4# ba_b (off_b +# kk *# n +# j)+                   !av = broadcastDoubleX4# (indexDoubleArray# ba_a (off_a +# i *# k +# kk))+               in goKU1x4 i j (kk +# 1#) kEnd (fmaddDoubleX4# av bv c0)++         -- Unpacked scalar cleanup for 1 row+         goJSU1 i bk kEnd j jEndV s+           | isTrue# (j >=# jEndV) = s+           | otherwise =+             let goKS1 kk acc+                   | isTrue# (kk >=# kEnd) = acc+                   | otherwise =+                       let !bkj = indexDoubleArray# ba_b (off_b +# kk *# n +# j)+                           !aik = indexDoubleArray# ba_a (off_a +# i *# k +# kk)+                       in goKS1 (kk +# 1#) (acc +## aik *## bkj)+             in case readDoubleArray# mba_c (off_c +# i *# n +# j) s of+                  (# s', cij #) ->+                    case writeDoubleArray# mba_c (off_c +# i *# n +# j) (cij +## goKS1 bk 0.0##) s' of+                      s'' -> goJSU1 i bk kEnd (j +# 1#) jEndV s''++     in (# goBKO 0# s0, () #)+{-# INLINE rawGemmBISlicePackedBK #-}++-- | @rawSyrkLowerKernel ba_a off_a mba_c off_c m n@+-- computes C = A^T * A where A is m×n (row-major).+-- Only the lower triangle of C (n×n) is computed via tiled SIMD micro-kernels,+-- then mirrored to the upper triangle.  C must be pre-zeroed.+-- This avoids materialising A^T and halves the flop count vs full GEMM.+rawSyrkLowerKernel :: ByteArray -> Int+                   -> MutableByteArray s -> Int+                   -> Int -> Int -> ST s ()+rawSyrkLowerKernel (ByteArray ba_a) (I# off_a)+                   (MutableByteArray mba_c) (I# off_c)+                   (I# m) (I# n) = ST $ \s0 ->+  let bs = 64#++      -- Tile rows of C (i dimension)+      goBI bi s+        | isTrue# (bi >=# n) = s+        | otherwise =+            let iEnd = minI (bi +# bs) n+            in goBI (bi +# bs) (goBK bi iEnd 0# s)++      -- Tile inner dimension (k = rows of A, 0..m-1)+      goBK bi iEnd bk s+        | isTrue# (bk >=# m) = s+        | otherwise =+            let kEnd = minI (bk +# bs) m+            in goBK bi iEnd (bk +# bs) (goBJ bi iEnd bk kEnd 0# s)++      -- Tile columns of C (j dimension) — lower triangle only: bj <= bi+      goBJ bi iEnd bk kEnd bj s+        | isTrue# (bj ># bi) = s  -- stop past diagonal+        | otherwise =+            let jEnd = minI (bj +# bs) n+            in goBJ bi iEnd bk kEnd (bj +# bs) (innerBlock bi iEnd bk kEnd bj jEnd s)++      -- Micro-kernel dispatch (same structure as GEMM)+      innerBlock bi iEnd bk kEnd bj jEnd s0_ =+        let !jSpan = jEnd -# bj+            !j8End = bj +# (jSpan -# (jSpan `remInt#` 8#))+            !j4End = bj +# (jSpan -# (jSpan `remInt#` 4#))+            !iSpan = iEnd -# bi+            !i4End = bi +# (iSpan -# (iSpan `remInt#` 4#))+        in goI4 bi i4End iEnd j8End j4End bk kEnd bj jEnd s0_++      goI4 i i4End iEnd_ j8End j4End bk kEnd bj jEnd s+        | isTrue# (i >=# i4End) = goI1 i iEnd_ j8End j4End bk kEnd bj jEnd s+        | otherwise =+            goI4 (i +# 4#) i4End iEnd_ j8End j4End bk kEnd bj jEnd+              (goJ8s i bk kEnd bj j8End+                (goJ4s i bk kEnd j8End j4End+                  (goJSs4 i bk kEnd j4End jEnd s)))++      goI1 i iEnd_ j8End j4End bk kEnd bj jEnd s+        | isTrue# (i >=# iEnd_) = s+        | otherwise =+            goI1 (i +# 1#) iEnd_ j8End j4End bk kEnd bj jEnd+              (goJ8s1 i bk kEnd bj j8End+                (goJ4s1 i bk kEnd j8End j4End+                  (goJSs1 i bk kEnd j4End jEnd s)))++      -- 4×8 micro-kernel for SYRK+      -- A^T[i,kk] = A[kk,i] at off_a + kk*n + i+      -- A[kk,j]   at off_a + kk*n + j+      goJ8s i bk kEnd j j8End s+        | isTrue# (j >=# j8End) = s+        | otherwise =+          let !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0a, c00 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) s0a of { (# s0b, c01 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff1 s0b of { (# s1a, c10 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) s1a of { (# s1b, c11 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff2 s1b of { (# s2a, c20 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) s2a of { (# s2b, c21 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff3 s2b of { (# s3a, c30 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) s3a of { (# s3b, c31 #) ->+             case goK8s i j bk kEnd c00 c01 c10 c11 c20 c21 c30 c31 of+               (# r00, r01, r10, r11, r20, r21, r30, r31 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff0 r00 s3b of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff0 +# 4#) r01 sw0 of { sw1 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff1 r10 sw1 of { sw2 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff1 +# 4#) r11 sw2 of { sw3 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff2 r20 sw3 of { sw4 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff2 +# 4#) r21 sw4 of { sw5 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff3 r30 sw5 of { sw6 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff3 +# 4#) r31 sw6 of { sw7 ->+                 goJ8s i bk kEnd (j +# 8#) j8End sw7+                 }}}}}}}}+             }}}}}}}}++      -- k-loop for 4×8 SYRK: A^T[i,kk]=A[kk,i], A[kk,j..j+7]+      goK8s i j kk kEnd c00 c01 c10 c11 c20 c21 c30 c31+        | isTrue# (kk >=# kEnd) =+            (# c00, c01, c10, c11, c20, c21, c30, c31 #)+        | otherwise =+            let !rowOff = off_a +# kk *# n+                !bv0 = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j)+                !bv1 = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j +# 4#)+                !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i))+                !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 1#))+                !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 2#))+                !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 3#))+            in goK8s i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# a0 bv0 c00) (fmaddDoubleX4# a0 bv1 c01)+                 (fmaddDoubleX4# a1 bv0 c10) (fmaddDoubleX4# a1 bv1 c11)+                 (fmaddDoubleX4# a2 bv0 c20) (fmaddDoubleX4# a2 bv1 c21)+                 (fmaddDoubleX4# a3 bv0 c30) (fmaddDoubleX4# a3 bv1 c31)++      -- 4×4 cleanup for SYRK+      goJ4s i bk kEnd j j4End s+        | isTrue# (j >=# j4End) = s+        | otherwise =+          let !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff0 s of { (# s0, c0 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff1 s0 of { (# s1, c1 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff2 s1 of { (# s2, c2 #) ->+             case readDoubleArrayAsDoubleX4# mba_c cOff3 s2 of { (# s3, c3 #) ->+             case goK4s i j bk kEnd c0 c1 c2 c3 of+               (# r0, r1, r2, r3 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff0 r0 s3 of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff1 r1 sw0 of { sw1 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff2 r2 sw1 of { sw2 ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff3 r3 sw2 of { sw3 ->+                 goJ4s i bk kEnd (j +# 4#) j4End sw3+                 }}}}+             }}}}++      goK4s i j kk kEnd c0 c1 c2 c3+        | isTrue# (kk >=# kEnd) = (# c0, c1, c2, c3 #)+        | otherwise =+            let !rowOff = off_a +# kk *# n+                !bv = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j)+                !a0 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i))+                !a1 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 1#))+                !a2 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 2#))+                !a3 = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i +# 3#))+            in goK4s i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# a0 bv c0) (fmaddDoubleX4# a1 bv c1)+                 (fmaddDoubleX4# a2 bv c2) (fmaddDoubleX4# a3 bv c3)++      -- Scalar cleanup for 4 rows (SYRK)+      goJSs4 i bk kEnd j jEnd_ s+        | isTrue# (j >=# jEnd_) = s+        | otherwise =+          let goK_s4 kk acc0 acc1 acc2 acc3+                | isTrue# (kk >=# kEnd) = (# acc0, acc1, acc2, acc3 #)+                | otherwise =+                    let !rowOff = off_a +# kk *# n+                        !bkj = indexDoubleArray# ba_a (rowOff +# j)+                        !a0_ = indexDoubleArray# ba_a (rowOff +# i)+                        !a1_ = indexDoubleArray# ba_a (rowOff +# i +# 1#)+                        !a2_ = indexDoubleArray# ba_a (rowOff +# i +# 2#)+                        !a3_ = indexDoubleArray# ba_a (rowOff +# i +# 3#)+                    in goK_s4 (kk +# 1#)+                         (acc0 +## a0_ *## bkj) (acc1 +## a1_ *## bkj)+                         (acc2 +## a2_ *## bkj) (acc3 +## a3_ *## bkj)+              !cOff0 = off_c +# i *# n +# j+              !cOff1 = off_c +# (i +# 1#) *# n +# j+              !cOff2 = off_c +# (i +# 2#) *# n +# j+              !cOff3 = off_c +# (i +# 3#) *# n +# j+          in case (# goK_s4 bk 0.0## 0.0## 0.0## 0.0## #) of+               (# (# d0, d1, d2, d3 #) #) ->+                 case readDoubleArray# mba_c cOff0 s of { (# s0, v0 #) ->+                 case writeDoubleArray# mba_c cOff0 (v0 +## d0) s0 of { s1 ->+                 case readDoubleArray# mba_c cOff1 s1 of { (# s2, v1 #) ->+                 case writeDoubleArray# mba_c cOff1 (v1 +## d1) s2 of { s3 ->+                 case readDoubleArray# mba_c cOff2 s3 of { (# s4, v2 #) ->+                 case writeDoubleArray# mba_c cOff2 (v2 +## d2) s4 of { s5 ->+                 case readDoubleArray# mba_c cOff3 s5 of { (# s6, v3 #) ->+                 case writeDoubleArray# mba_c cOff3 (v3 +## d3) s6 of { s7 ->+                 goJSs4 i bk kEnd (j +# 1#) jEnd_ s7+                 }}}}}}}}++      -- 1×8 micro-kernel for SYRK+      goJ8s1 i bk kEnd j j8End s+        | isTrue# (j >=# j8End) = s+        | otherwise =+          let !cOff = off_c +# i *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+             case readDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) s0 of { (# s1, c1 #) ->+             case goK8s1 i j bk kEnd c0 c1 of+               (# r0, r1 #) ->+                 case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s1 of { sw0 ->+                 case writeDoubleArrayAsDoubleX4# mba_c (cOff +# 4#) r1 sw0 of { sw1 ->+                 goJ8s1 i bk kEnd (j +# 8#) j8End sw1+                 }}+             }}++      goK8s1 i j kk kEnd c0 c1+        | isTrue# (kk >=# kEnd) = (# c0, c1 #)+        | otherwise =+            let !rowOff = off_a +# kk *# n+                !bv0 = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j)+                !bv1 = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j +# 4#)+                !av = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i))+            in goK8s1 i j (kk +# 1#) kEnd+                 (fmaddDoubleX4# av bv0 c0) (fmaddDoubleX4# av bv1 c1)++      -- 1×4 cleanup for SYRK+      goJ4s1 i bk kEnd j j4End s+        | isTrue# (j >=# j4End) = s+        | otherwise =+          let !cOff = off_c +# i *# n +# j+          in case readDoubleArrayAsDoubleX4# mba_c cOff s of { (# s0, c0 #) ->+             case goK4s1 i j bk kEnd c0 of { r0 ->+             case writeDoubleArrayAsDoubleX4# mba_c cOff r0 s0 of { s1 ->+             goJ4s1 i bk kEnd (j +# 4#) j4End s1+             }}}++      goK4s1 i j kk kEnd c0+        | isTrue# (kk >=# kEnd) = c0+        | otherwise =+            let !rowOff = off_a +# kk *# n+                !bv = indexDoubleArrayAsDoubleX4# ba_a (rowOff +# j)+                !av = broadcastDoubleX4# (indexDoubleArray# ba_a (rowOff +# i))+            in goK4s1 i j (kk +# 1#) kEnd (fmaddDoubleX4# av bv c0)++      -- Scalar cleanup for 1 row (SYRK)+      goJSs1 i bk kEnd j jEnd_ s+        | isTrue# (j >=# jEnd_) = s+        | otherwise =+          let goK_s1 kk acc+                | isTrue# (kk >=# kEnd) = acc+                | otherwise =+                    let !rowOff = off_a +# kk *# n+                        !bkj = indexDoubleArray# ba_a (rowOff +# j)+                        !aik = indexDoubleArray# ba_a (rowOff +# i)+                    in goK_s1 (kk +# 1#) (acc +## aik *## bkj)+          in case readDoubleArray# mba_c (off_c +# i *# n +# j) s of+               (# s', cij #) ->+                 case writeDoubleArray# mba_c (off_c +# i *# n +# j) (cij +## goK_s1 bk 0.0##) s' of+                   s'' -> goJSs1 i bk kEnd (j +# 1#) jEnd_ s''++      -- Mirror lower triangle to upper: C[j,i] = C[i,j] for j < i+      mirror i s+        | isTrue# (i >=# n) = s+        | otherwise = mirror (i +# 1#) (mirrorRow i 0# s)++      mirrorRow i j s+        | isTrue# (j >=# i) = s+        | otherwise =+          case readDoubleArray# mba_c (off_c +# i *# n +# j) s of { (# s0, cij #) ->+          case writeDoubleArray# mba_c (off_c +# j *# n +# i) cij s0 of { s1 ->+          mirrorRow i (j +# 1#) s1+          }}++  in (# mirror 0# (goBI 0# s0), () #)+{-# INLINE rawSyrkLowerKernel #-}++-- --------------------------------------------------------------------------+-- QR helpers+-- --------------------------------------------------------------------------++-- | Sum of squares: Σ arr[off+i]^2 for i in [from..to-1]+rawSumSqRange :: ByteArray -> Int -> Int -> Int -> Double+rawSumSqRange (ByteArray ba) (I# off) (I# from_) (I# to) =+  D# (goSumSq from_ 0.0##)+  where+    goSumSq i acc+      | isTrue# (i >=# to) = acc+      | otherwise =+          let x = indexDoubleArray# ba (off +# i)+          in goSumSq (i +# 1#) (acc +## x *## x)+{-# INLINE rawSumSqRange #-}++-- | Dot product of a column slice: Σ arr1[off1+i*stride1] * arr2[off2+i*stride2]+-- for i in [from..to-1]. Used for column-wise access patterns in QR.+rawSumProdRange :: ByteArray -> Int -> Int+                -> ByteArray -> Int -> Int+                -> Int -> Int -> Double+rawSumProdRange (ByteArray ba1) (I# off1) (I# stride1)+                (ByteArray ba2) (I# off2) (I# stride2)+                (I# from_) (I# to) =+  D# (goSumProd from_ 0.0##)+  where+    goSumProd i acc+      | isTrue# (i >=# to) = acc+      | otherwise =+          let x = indexDoubleArray# ba1 (off1 +# i *# stride1)+              y = indexDoubleArray# ba2 (off2 +# i *# stride2)+          in goSumProd (i +# 1#) (acc +## x *## y)+{-# INLINE rawSumProdRange #-}++-- | Apply Householder reflector to one column of a mutable matrix.+-- @rawHouseholderApplyCol mba_r off_r ncols ba_v off_v beta from to col@+-- computes w = β * Σ_{i=from}^{to-1} v[i] * R[i, col], then+-- R[i, col] -= v[i] * w for i in [from..to-1].+rawHouseholderApplyCol :: MutableByteArray s -> Int -> Int+                       -> ByteArray -> Int -> Double+                       -> Int -> Int -> Int -> ST s ()+rawHouseholderApplyCol (MutableByteArray mba_r) (I# off_r) (I# ncols)+                       (ByteArray ba_v) (I# off_v) (D# beta)+                       (I# from_) (I# to) (I# col) = ST $ \s0 ->+  -- Phase 1: compute w = β * Σ v[i] * R[i, col]+  let goSum i acc s+        | isTrue# (i >=# to) = (# s, beta *## acc #)+        | otherwise =+            let vi = indexDoubleArray# ba_v (off_v +# i)+            in case readDoubleArray# mba_r (off_r +# i *# ncols +# col) s of+                 (# s', rij #) -> goSum (i +# 1#) (acc +## vi *## rij) s'+  in case goSum from_ 0.0## s0 of+       (# s1, w #) ->+         -- Phase 2: R[i, col] -= v[i] * w+         let goUpdate i s+               | isTrue# (i >=# to) = s+               | otherwise =+                   let vi = indexDoubleArray# ba_v (off_v +# i)+                   in case readDoubleArray# mba_r (off_r +# i *# ncols +# col) s of+                        (# s', rij #) ->+                          case writeDoubleArray# mba_r (off_r +# i *# ncols +# col) (rij -## vi *## w) s' of+                            s'' -> goUpdate (i +# 1#) s''+         in (# goUpdate from_ s1, () #)+{-# INLINE rawHouseholderApplyCol #-}++-- | Update one row of Q during backward accumulation.+-- @rawQAccumCol mba_q off_q ncols ba_v off_v beta from to row@+-- computes qi = β * Σ_{k=from}^{to-1} Q[row, k] * v[k], then+-- Q[row, k] -= qi * v[k] for all k in [from..to-1].+rawQAccumCol :: MutableByteArray s -> Int -> Int+             -> ByteArray -> Int -> Double+             -> Int -> Int -> Int -> ST s ()+rawQAccumCol (MutableByteArray mba_q) (I# off_q) (I# ncols)+             (ByteArray ba_v) (I# off_v) (D# beta)+             (I# from_) (I# to) (I# row) = ST $ \s0 ->+  -- Phase 1: compute qi = β * Σ Q[row, k] * v[k]+  let goSum k acc s+        | isTrue# (k >=# to) = (# s, beta *## acc #)+        | otherwise =+            let vk = indexDoubleArray# ba_v (off_v +# k)+            in case readDoubleArray# mba_q (off_q +# row *# ncols +# k) s of+                 (# s', qrk #) -> goSum (k +# 1#) (acc +## vk *## qrk) s'+  in case goSum from_ 0.0## s0 of+       (# s1, qi #) ->+         -- Phase 2: Q[row, k] -= qi * v[k]+         let goUpdate k s+               | isTrue# (k >=# to) = s+               | otherwise =+                   let vk = indexDoubleArray# ba_v (off_v +# k)+                   in case readDoubleArray# mba_q (off_q +# row *# ncols +# k) s of+                        (# s', qrk #) ->+                          case writeDoubleArray# mba_q (off_q +# row *# ncols +# k) (qrk -## qi *## vk) s' of+                            s'' -> goUpdate (k +# 1#) s''+         in (# goUpdate from_ s1, () #)+{-# INLINE rawQAccumCol #-}++-- --------------------------------------------------------------------------+-- Eigen / tridiag helpers+-- --------------------------------------------------------------------------++-- | Apply Givens rotation to two rows of a mutable matrix.+-- @rawApplyGivensRows mba off ncols cosθ sinθ row_p row_q from to@+-- For each column j in [from..to-1]:+--   tmp        =  c * M[row_p, j] + s * M[row_q, j]+--   M[row_q, j] = -s * M[row_p, j] + c * M[row_q, j]+--   M[row_p, j] = tmp+rawApplyGivensRows :: MutableByteArray s -> Int -> Int+                   -> Double -> Double -> Int -> Int+                   -> Int -> Int -> ST s ()+rawApplyGivensRows (MutableByteArray mba) (I# off) (I# ncols)+                   (D# c_) (D# s_) (I# row_p) (I# row_q)+                   (I# from_) (I# to) = ST $ \s0 ->+  let pOff = off +# row_p *# ncols+      qOff = off +# row_q *# ncols+      jSpan = to -# from_+      j4End = from_ +# (jSpan -# (jSpan `remInt#` 4#))+      cV = broadcastDoubleX4# c_+      sV = broadcastDoubleX4# s_+      nsV = negateDoubleX4# sV++      goSimd j s+        | isTrue# (j >=# j4End) = s+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (pOff +# j) s of+              (# s1, pv #) ->+                case readDoubleArrayAsDoubleX4# mba (qOff +# j) s1 of+                  (# s2, qv #) ->+                    let tmp = fmaddDoubleX4# cV pv (timesDoubleX4# sV qv)+                        q'  = fmaddDoubleX4# nsV pv (timesDoubleX4# cV qv)+                    in case writeDoubleArrayAsDoubleX4# mba (pOff +# j) tmp s2 of+                         s3 -> case writeDoubleArrayAsDoubleX4# mba (qOff +# j) q' s3 of+                                 s4 -> goSimd (j +# 4#) s4++      goScalar j s+        | isTrue# (j >=# to) = s+        | otherwise =+            case readDoubleArray# mba (pOff +# j) s of+              (# s1, pj #) ->+                case readDoubleArray# mba (qOff +# j) s1 of+                  (# s2, qj #) ->+                    let tmp = c_ *## pj +## s_ *## qj+                        qj' = negateDouble# s_ *## pj +## c_ *## qj+                    in case writeDoubleArray# mba (pOff +# j) tmp s2 of+                         s3 -> case writeDoubleArray# mba (qOff +# j) qj' s3 of+                                 s4 -> goScalar (j +# 1#) s4++  in (# goScalar j4End (goSimd from_ s0), () #)+{-# INLINE rawApplyGivensRows #-}++-- | Symmetric rank-2 update on a mutable matrix.+-- @rawSymRank2Update mba off n ba_v off_v ba_w off_w from to@+-- For i in [from..to-1], j in [from..i]:+--   T[i,j] -= v[i]*w[j] + w[i]*v[j]+--   T[j,i] = T[i,j]   (maintain symmetry)+rawSymRank2Update :: MutableByteArray s -> Int -> Int+                  -> ByteArray -> Int -> ByteArray -> Int+                  -> Int -> Int -> ST s ()+rawSymRank2Update (MutableByteArray mba) (I# off) (I# n)+                  (ByteArray ba_v) (I# off_v) (ByteArray ba_w) (I# off_w)+                  (I# from_) (I# to) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# to) = s+        | otherwise =+            let vi = indexDoubleArray# ba_v (off_v +# i)+                wi = indexDoubleArray# ba_w (off_w +# i)+            in goI (i +# 1#) (goJ i vi wi from_ s)++      goJ i vi wi j s+        | isTrue# (j ># i) = s+        | otherwise =+            let vj = indexDoubleArray# ba_v (off_v +# j)+                wj = indexDoubleArray# ba_w (off_w +# j)+                delta = vi *## wj +## wi *## vj+                ij = off +# i *# n +# j+                ji = off +# j *# n +# i+            in case readDoubleArray# mba ij s of+                 (# s1, tij #) ->+                   let tij' = tij -## delta+                   in case writeDoubleArray# mba ij tij' s1 of+                        s2 | isTrue# (i ==# j) -> goJ i vi wi (j +# 1#) s2+                           | otherwise ->+                               case writeDoubleArray# mba ji tij' s2 of+                                 s3 -> goJ i vi wi (j +# 1#) s3++  in (# goI from_ s0, () #)+{-# INLINE rawSymRank2Update #-}++-- --------------------------------------------------------------------------+-- Tridiagonalisation mutable kernels+-- --------------------------------------------------------------------------++-- | Symmetric submatrix-vector product for tridiagonalisation.+-- Computes p[i-from] = Σ_{j=from}^{to-1} T[i,j] * v[j-from]+-- for i in [from..to-1].+-- T is read from MutableByteArray (being modified in-place),+-- v is read from MutableByteArray (temporary vector),+-- p is written to MutableByteArray (temporary vector).+rawMutSymMatvecSub :: MutableByteArray s -> Int -> Int+                   -> MutableByteArray s -> Int+                   -> MutableByteArray s -> Int+                   -> Int -> Int -> ST s ()+rawMutSymMatvecSub (MutableByteArray mba_t) (I# off_t) (I# ncols)+                   (MutableByteArray mba_v) (I# off_v)+                   (MutableByteArray mba_p) (I# off_p)+                   (I# from_) (I# to) = ST $ \s0 ->+  let !len = to -# from_+      !len8 = len -# (len `remInt#` 8#)+      !len4 = len -# (len `remInt#` 4#)++      goI i s+        | isTrue# (i >=# to) = s+        | otherwise =+            let !rowBase = off_t +# i *# ncols +# from_+            -- 8-wide SIMD phase: two independent accumulators+            in case goJ8 rowBase 0#+                      (broadcastDoubleX4# 0.0##)+                      (broadcastDoubleX4# 0.0##) s of+                (# s1, accV0, accV1 #) ->+                  -- 4-wide cleanup phase+                  case goJ4 rowBase len8 accV0 s1 of+                    (# s2, accV0' #) ->+                      -- Reduce both SIMD accumulators to scalar+                      let !combined = plusDoubleX4# accV0' accV1+                          !(# a0, a1, a2, a3 #) = unpackDoubleX4# combined+                          !simdSum = a0 +## a1 +## a2 +## a3+                      -- Scalar tail+                      in case goJTail rowBase len4 simdSum s2 of+                          (# s3, acc #) ->+                            case writeDoubleArray# mba_p (off_p +# (i -# from_)) acc s3 of+                              s4 -> goI (i +# 1#) s4++      -- Process 8 doubles (2× DoubleX4#) per iteration+      goJ8 rowBase j accV0 accV1 s+        | isTrue# (j >=# len8) = (# s, accV0, accV1 #)+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba_t (rowBase +# j) s of+              (# s1, tv0 #) ->+                case readDoubleArrayAsDoubleX4# mba_v (off_v +# j) s1 of+                  (# s2, vv0 #) ->+                    case readDoubleArrayAsDoubleX4# mba_t (rowBase +# j +# 4#) s2 of+                      (# s3, tv1 #) ->+                        case readDoubleArrayAsDoubleX4# mba_v (off_v +# j +# 4#) s3 of+                          (# s4, vv1 #) ->+                            goJ8 rowBase (j +# 8#)+                              (fmaddDoubleX4# tv0 vv0 accV0)+                              (fmaddDoubleX4# tv1 vv1 accV1) s4++      -- 4-wide cleanup for elements between len8 and len4+      goJ4 rowBase j accV s+        | isTrue# (j >=# len4) = (# s, accV #)+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba_t (rowBase +# j) s of+              (# s1, tv #) ->+                case readDoubleArrayAsDoubleX4# mba_v (off_v +# j) s1 of+                  (# s2, vv #) ->+                    goJ4 rowBase (j +# 4#) (fmaddDoubleX4# tv vv accV) s2++      goJTail rowBase j acc s+        | isTrue# (j >=# len) = (# s, acc #)+        | otherwise =+            case readDoubleArray# mba_t (rowBase +# j) s of+              (# s1, tij #) ->+                case readDoubleArray# mba_v (off_v +# j) s1 of+                  (# s2, vj #) -> goJTail rowBase (j +# 1#) (acc +## tij *## vj) s2++  in (# goI from_ s0, () #)+{-# INLINE rawMutSymMatvecSub #-}++-- | Symmetric rank-2 update reading v, w from MutableByteArrays.+-- For i in [from..to-1], j in [from..i]:+--   T[i,j] -= v[i-from]*w[j-from] + w[i-from]*v[j-from]+--   T[j,i] = T[i,j]   (maintain symmetry)+-- v and w are indexed relative to from (i.e., v[0] corresponds to row 'from').+rawMutSymRank2Update :: MutableByteArray s -> Int -> Int+                     -> MutableByteArray s -> Int+                     -> MutableByteArray s -> Int+                     -> Int -> Int -> ST s ()+rawMutSymRank2Update (MutableByteArray mba_t) (I# off_t) (I# n)+                     (MutableByteArray mba_v) (I# off_v)+                     (MutableByteArray mba_w) (I# off_w)+                     (I# from_) (I# to) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# to) = s+        | otherwise =+            case readDoubleArray# mba_v (off_v +# (i -# from_)) s of+              (# s1, vi #) ->+                case readDoubleArray# mba_w (off_w +# (i -# from_)) s1 of+                  (# s2, wi #) -> goI (i +# 1#) (goJ i vi wi from_ s2)++      goJ i vi wi j s+        | isTrue# (j ># i) = s+        | otherwise =+            case readDoubleArray# mba_v (off_v +# (j -# from_)) s of+              (# s1, vj #) ->+                case readDoubleArray# mba_w (off_w +# (j -# from_)) s1 of+                  (# s2, wj #) ->+                    let delta = vi *## wj +## wi *## vj+                        ij = off_t +# i *# n +# j+                        ji = off_t +# j *# n +# i+                    in case readDoubleArray# mba_t ij s2 of+                         (# s3, tij #) ->+                           let tij' = tij -## delta+                           in case writeDoubleArray# mba_t ij tij' s3 of+                                s4 | isTrue# (i ==# j) -> goJ i vi wi (j +# 1#) s4+                                   | otherwise ->+                                       case writeDoubleArray# mba_t ji tij' s4 of+                                         s5 -> goJ i vi wi (j +# 1#) s5++  in (# goI from_ s0, () #)+{-# INLINE rawMutSymRank2Update #-}++-- | Q accumulation for tridiagonalisation.+-- Householder vectors are stored in column hvCol of frozen T,+-- with implicit v[qCol] = 1.0.+-- Phase 1: wi = beta * (Q[row,qCol] + Σ_{l=qCol+1}^{endRow-1} Q[row,l] * T[l,hvCol])+-- Phase 2: Q[row,qCol] -= wi; Q[row,l] -= wi * T[l,hvCol]+rawMutTridiagQAccum :: MutableByteArray s -> Int -> Int+                    -> ByteArray -> Int -> Int+                    -> Double -> Int -> Int -> Int -> Int -> ST s ()+rawMutTridiagQAccum (MutableByteArray mba_q) (I# off_q) (I# qcols)+                    (ByteArray ba_t) (I# off_t) (I# tcols)+                    (D# beta) (I# qCol) (I# hvCol) (I# endRow) (I# row) = ST $ \s0 ->+  -- Phase 1: wi = beta * (Q[row,qCol] + Σ Q[row,l] * T[l,hvCol])+  case readDoubleArray# mba_q (off_q +# row *# qcols +# qCol) s0 of+    (# s1, qrk #) ->+      let goSum l acc s+            | isTrue# (l >=# endRow) = (# s, beta *## (qrk +## acc) #)+            | otherwise =+                let vl = indexDoubleArray# ba_t (off_t +# l *# tcols +# hvCol)+                in case readDoubleArray# mba_q (off_q +# row *# qcols +# l) s of+                     (# s', qrl #) -> goSum (l +# 1#) (acc +## qrl *## vl) s'+      in case goSum (qCol +# 1#) 0.0## s1 of+           (# s2, wi #) ->+             -- Phase 2: Q[row,qCol] -= wi+             case writeDoubleArray# mba_q (off_q +# row *# qcols +# qCol) (qrk -## wi) s2 of+               s3 ->+                 let goUpdate l s+                       | isTrue# (l >=# endRow) = s+                       | otherwise =+                           let vl = indexDoubleArray# ba_t (off_t +# l *# tcols +# hvCol)+                           in case readDoubleArray# mba_q (off_q +# row *# qcols +# l) s of+                                (# s', qrl #) ->+                                  case writeDoubleArray# mba_q (off_q +# row *# qcols +# l) (qrl -## wi *## vl) s' of+                                    s'' -> goUpdate (l +# 1#) s''+                 in (# goUpdate (qCol +# 1#) s3, () #)+{-# INLINE rawMutTridiagQAccum #-}++-- --------------------------------------------------------------------------+-- LU kernels+-- --------------------------------------------------------------------------++-- | In-place LU elimination for column k of an n×n row-major matrix.+-- Computes multipliers and updates the trailing submatrix.+-- Inner j-loop uses DoubleX4# SIMD (contiguous row access).+rawLUEliminateColumn :: MutableByteArray s -> Int -> Int -> Int -> ST s ()+rawLUEliminateColumn (MutableByteArray mba) (I# off) (I# n) (I# k) = ST $ \s0 ->+  -- Read A[k,k] (the pivot)+  case readDoubleArray# mba (off +# k *# n +# k) s0 of+    (# s1, akk #) ->+      let goI i s+            | isTrue# (i >=# n) = s+            | otherwise =+                -- Read A[i,k], compute multiplier+                case readDoubleArray# mba (off +# i *# n +# k) s of+                  (# s', aik #) ->+                    let mult = aik /## akk+                        iRowOff = off +# i *# n+                        kRowOff = off +# k *# n+                        jSpan = n -# k -# 1#+                        jStart = k +# 1#+                        j4End = jStart +# (jSpan -# (jSpan `remInt#` 4#))+                        negMultV = broadcastDoubleX4# (negateDouble# mult)+                    -- Store multiplier at A[i,k]+                    in case writeDoubleArray# mba (off +# i *# n +# k) mult s' of+                         s'' ->+                           -- SIMD j-loop: A[i,j] -= mult * A[k,j]  =  A[i,j] + (-mult)*A[k,j]+                           let goJSimd j s_+                                 | isTrue# (j >=# j4End) = s_+                                 | otherwise =+                                     case readDoubleArrayAsDoubleX4# mba (iRowOff +# j) s_ of+                                       (# s1_, aij #) ->+                                         case readDoubleArrayAsDoubleX4# mba (kRowOff +# j) s1_ of+                                              (# s2_, akjV_ #) ->+                                                let aij' = fmaddDoubleX4# negMultV akjV_ aij+                                                in case writeDoubleArrayAsDoubleX4# mba (iRowOff +# j) aij' s2_ of+                                                     s3_ -> goJSimd (j +# 4#) s3_+                               -- Scalar cleanup+                               goJScalar j s_+                                 | isTrue# (j >=# n) = s_+                                 | otherwise =+                                     case readDoubleArray# mba (iRowOff +# j) s_ of+                                       (# s1_, aij #) ->+                                         case readDoubleArray# mba (kRowOff +# j) s1_ of+                                           (# s2_, akj #) ->+                                             case writeDoubleArray# mba (iRowOff +# j) (aij -## mult *## akj) s2_ of+                                               s3_ -> goJScalar (j +# 1#) s3_+                           in goI (i +# 1#) (goJScalar j4End (goJSimd jStart s''))+      in (# goI (k +# 1#) s1, () #)+{-# INLINE rawLUEliminateColumn #-}++-- | @rawLUEliminateColumnTo mba off n k colEnd@ — like 'rawLUEliminateColumn'+-- but the trailing update only touches columns @k+1 .. colEnd-1@ (not @k+1 .. n-1@).+-- Multipliers are still computed for ALL rows @k+1 .. n-1@.+-- Used by panel LU to restrict updates to within the current panel.+rawLUEliminateColumnTo :: MutableByteArray s -> Int -> Int -> Int -> Int -> ST s ()+rawLUEliminateColumnTo (MutableByteArray mba) (I# off) (I# n) (I# k) (I# colEnd) = ST $ \s0 ->+  case readDoubleArray# mba (off +# k *# n +# k) s0 of+    (# s1, akk #) ->+      let goI i s+            | isTrue# (i >=# n) = s+            | otherwise =+                case readDoubleArray# mba (off +# i *# n +# k) s of+                  (# s', aik #) ->+                    let mult = aik /## akk+                        iRowOff = off +# i *# n+                        kRowOff = off +# k *# n+                        jSpan = colEnd -# k -# 1#+                        jStart = k +# 1#+                        j4End = jStart +# (jSpan -# (jSpan `remInt#` 4#))+                        negMultV = broadcastDoubleX4# (negateDouble# mult)+                    in case writeDoubleArray# mba (off +# i *# n +# k) mult s' of+                         s'' ->+                           let goJSimd j s_+                                 | isTrue# (j >=# j4End) = s_+                                 | otherwise =+                                     case readDoubleArrayAsDoubleX4# mba (iRowOff +# j) s_ of+                                       (# s1_, aij #) ->+                                         case readDoubleArrayAsDoubleX4# mba (kRowOff +# j) s1_ of+                                              (# s2_, akjV_ #) ->+                                                let aij' = fmaddDoubleX4# negMultV akjV_ aij+                                                in case writeDoubleArrayAsDoubleX4# mba (iRowOff +# j) aij' s2_ of+                                                     s3_ -> goJSimd (j +# 4#) s3_+                               goJScalar j s_+                                 | isTrue# (j >=# colEnd) = s_+                                 | otherwise =+                                     case readDoubleArray# mba (iRowOff +# j) s_ of+                                       (# s1_, aij #) ->+                                         case readDoubleArray# mba (kRowOff +# j) s1_ of+                                           (# s2_, akj #) ->+                                             case writeDoubleArray# mba (iRowOff +# j) (aij -## mult *## akj) s2_ of+                                               s3_ -> goJScalar (j +# 1#) s3_+                           in goI (i +# 1#) (goJScalar j4End (goJSimd jStart s''))+      in (# goI (k +# 1#) s1, () #)+{-# INLINE rawLUEliminateColumnTo #-}++-- | Swap elements in columns [fromCol..n-1] between two rows of an n-wide matrix.+-- Uses DoubleX4# SIMD for the contiguous row data.+rawSwapRows :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> ST s ()+rawSwapRows (MutableByteArray mba) (I# off) (I# n) (I# row1) (I# row2) (I# fromCol) = ST $ \s0 ->+  let r1Off = off +# row1 *# n+      r2Off = off +# row2 *# n+      jSpan = n -# fromCol+      j4End = fromCol +# (jSpan -# (jSpan `remInt#` 4#))++      goSimd j s+        | isTrue# (j >=# j4End) = s+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (r1Off +# j) s of+              (# s1, v1 #) ->+                case readDoubleArrayAsDoubleX4# mba (r2Off +# j) s1 of+                  (# s2, v2 #) ->+                    case writeDoubleArrayAsDoubleX4# mba (r1Off +# j) v2 s2 of+                      s3 -> case writeDoubleArrayAsDoubleX4# mba (r2Off +# j) v1 s3 of+                              s4 -> goSimd (j +# 4#) s4++      goScalar j s+        | isTrue# (j >=# n) = s+        | otherwise =+            case readDoubleArray# mba (r1Off +# j) s of+              (# s1, v1 #) ->+                case readDoubleArray# mba (r2Off +# j) s1 of+                  (# s2, v2 #) ->+                    case writeDoubleArray# mba (r1Off +# j) v2 s2 of+                      s3 -> case writeDoubleArray# mba (r2Off +# j) v1 s3 of+                              s4 -> goScalar (j +# 1#) s4++  in (# goScalar j4End (goSimd fromCol s0), () #)+{-# INLINE rawSwapRows #-}++-- | Find row with maximum |A[i,k]| for i in [fromRow..n-1].+-- Returns the row index of the pivot.+rawPivotSearch :: MutableByteArray s -> Int -> Int -> Int -> Int -> ST s Int+rawPivotSearch (MutableByteArray mba) (I# off) (I# n) (I# k) (I# fromRow) = ST $ \s0 ->+  let go i bestIdx bestVal s+        | isTrue# (i >=# n) = (# s, I# bestIdx #)+        | otherwise =+            case readDoubleArray# mba (off +# i *# n +# k) s of+              (# s', v #) ->+                let av = if isTrue# (v >=## 0.0##) then v else negateDouble# v+                in if isTrue# (av >## bestVal)+                   then go (i +# 1#) i av s'+                   else go (i +# 1#) bestIdx bestVal s'+  in case readDoubleArray# mba (off +# fromRow *# n +# k) s0 of+       (# s1, v0 #) ->+         let av0 = if isTrue# (v0 >=## 0.0##) then v0 else negateDouble# v0+         in go (fromRow +# 1#) fromRow av0 s1+{-# INLINE rawPivotSearch #-}++-- | In-place forward substitution using packed LU matrix (unit lower triangular).+-- Solves Ly = b where L is stored in the strictly lower part of ba_lu.+-- The solution overwrites mba_x.+rawForwardSubUnitPacked :: ByteArray -> Int -> Int -> MutableByteArray s -> Int -> ST s ()+rawForwardSubUnitPacked (ByteArray ba_lu) (I# off_lu) (I# n)+                        (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j >=# n) = s+        | otherwise =+            case readDoubleArray# mba_x (off_x +# j) s of+              (# s', xj #) ->+                let goI i s_+                      | isTrue# (i >=# n) = s_+                      | otherwise =+                          let lij = indexDoubleArray# ba_lu (off_lu +# i *# n +# j)+                          in case readDoubleArray# mba_x (off_x +# i) s_ of+                               (# s1, xi #) ->+                                 case writeDoubleArray# mba_x (off_x +# i) (xi -## lij *## xj) s1 of+                                   s2 -> goI (i +# 1#) s2+                in goJ (j +# 1#) (goI (j +# 1#) s')+  in (# goJ 0# s0, () #)+{-# INLINE rawForwardSubUnitPacked #-}++-- | In-place back substitution using packed LU matrix (upper triangular).+-- Solves Ux = y where U is stored in the upper part of ba_lu.+-- The solution overwrites mba_x.+rawBackSubPacked :: ByteArray -> Int -> Int -> MutableByteArray s -> Int -> ST s ()+rawBackSubPacked (ByteArray ba_lu) (I# off_lu) (I# n)+                 (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j <# 0#) = s+        | otherwise =+            -- x[j] /= U[j,j]+            let ujj = indexDoubleArray# ba_lu (off_lu +# j *# n +# j)+            in case readDoubleArray# mba_x (off_x +# j) s of+                 (# s', xj_ #) ->+                   let xj = xj_ /## ujj+                   in case writeDoubleArray# mba_x (off_x +# j) xj s' of+                        s'' ->+                          -- for i = 0..j-1: x[i] -= U[i,j] * x[j]+                          let goI i s_+                                | isTrue# (i >=# j) = s_+                                | otherwise =+                                    let uij = indexDoubleArray# ba_lu (off_lu +# i *# n +# j)+                                    in case readDoubleArray# mba_x (off_x +# i) s_ of+                                         (# s1, xi #) ->+                                           case writeDoubleArray# mba_x (off_x +# i) (xi -## uij *## xj) s1 of+                                             s2 -> goI (i +# 1#) s2+                          in goJ (j -# 1#) (goI 0# s'')+  in (# goJ (n -# 1#) s0, () #)+{-# INLINE rawBackSubPacked #-}++-- --------------------------------------------------------------------------+-- Cholesky kernels+-- --------------------------------------------------------------------------++-- | Process one column j of Cholesky factorisation in-place.+-- For k in [0..j-1]: subtract G[i,k]*G[j,k] from G[i,j] for i in [j..n-1].+-- Then scale: G[j,j] = sqrt(G[j,j]); G[i,j] /= G[j,j] for i > j.+rawCholColumn :: MutableByteArray s -> Int -> Int -> Int -> ST s ()+rawCholColumn (MutableByteArray mba) (I# off) (I# n) (I# j) = ST $ \s0 ->+  -- Phase 1: subtract contributions from previous columns+  let goK k s+        | isTrue# (k >=# j) = s+        | otherwise =+            -- Read G[j,k]+            case readDoubleArray# mba (off +# j *# n +# k) s of+              (# s', gjk #) ->+                -- For i in [j..n-1]: G[i,j] -= G[i,k] * gjk+                let goI i s_+                      | isTrue# (i >=# n) = s_+                      | otherwise =+                          case readDoubleArray# mba (off +# i *# n +# j) s_ of+                            (# s1, gij #) ->+                              case readDoubleArray# mba (off +# i *# n +# k) s1 of+                                (# s2, gik #) ->+                                  case writeDoubleArray# mba (off +# i *# n +# j) (gij -## gik *## gjk) s2 of+                                    s3 -> goI (i +# 1#) s3+                in goK (k +# 1#) (goI j s')+  in case goK 0# s0 of+       s1 ->+         -- Phase 2: scale column+         case readDoubleArray# mba (off +# j *# n +# j) s1 of+           (# s2, gjj #) ->+             let sjj = sqrtDouble# gjj+             in case writeDoubleArray# mba (off +# j *# n +# j) sjj s2 of+                  s3 ->+                    let goScale i s_+                          | isTrue# (i >=# n) = s_+                          | otherwise =+                              case readDoubleArray# mba (off +# i *# n +# j) s_ of+                                (# s4, gij #) ->+                                  case writeDoubleArray# mba (off +# i *# n +# j) (gij /## sjj) s4 of+                                    s5 -> goScale (i +# 1#) s5+                    in (# goScale (j +# 1#) s3, () #)+{-# INLINE rawCholColumn #-}++-- | Forward substitution with Cholesky factor G (lower triangular, non-unit diagonal).+-- Solves Gy = b, overwrites mba_x with y.+rawForwardSubCholPacked :: ByteArray -> Int -> Int -> MutableByteArray s -> Int -> ST s ()+rawForwardSubCholPacked (ByteArray ba_g) (I# off_g) (I# n)+                        (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j >=# n) = s+        | otherwise =+            let gjj = indexDoubleArray# ba_g (off_g +# j *# n +# j)+            in case readDoubleArray# mba_x (off_x +# j) s of+                 (# s', xj_ #) ->+                   let xj = xj_ /## gjj+                   in case writeDoubleArray# mba_x (off_x +# j) xj s' of+                        s'' ->+                          let goI i s_+                                | isTrue# (i >=# n) = s_+                                | otherwise =+                                    let gij = indexDoubleArray# ba_g (off_g +# i *# n +# j)+                                    in case readDoubleArray# mba_x (off_x +# i) s_ of+                                         (# s1, xi #) ->+                                           case writeDoubleArray# mba_x (off_x +# i) (xi -## gij *## xj) s1 of+                                             s2 -> goI (i +# 1#) s2+                          in goJ (j +# 1#) (goI (j +# 1#) s'')+  in (# goJ 0# s0, () #)+{-# INLINE rawForwardSubCholPacked #-}++-- | Back substitution with G^T (upper triangular) WITHOUT forming G^T.+-- Solves G^T x = y, overwrites mba_x with x.+-- Uses G^T[i,j] = G[j,i] to read from the lower triangle.+rawBackSubCholTPacked :: ByteArray -> Int -> Int -> MutableByteArray s -> Int -> ST s ()+rawBackSubCholTPacked (ByteArray ba_g) (I# off_g) (I# n)+                      (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j <# 0#) = s+        | otherwise =+            -- G^T[j,j] = G[j,j]+            let gjj = indexDoubleArray# ba_g (off_g +# j *# n +# j)+            in case readDoubleArray# mba_x (off_x +# j) s of+                 (# s', xj_ #) ->+                   let xj = xj_ /## gjj+                   in case writeDoubleArray# mba_x (off_x +# j) xj s' of+                        s'' ->+                          -- for i = 0..j-1: x[i] -= G^T[i,j] * x[j] = G[j,i] * x[j]+                          let goI i s_+                                | isTrue# (i >=# j) = s_+                                | otherwise =+                                    -- G^T[i,j] = G[j,i]+                                    let gji = indexDoubleArray# ba_g (off_g +# j *# n +# i)+                                    in case readDoubleArray# mba_x (off_x +# i) s_ of+                                         (# s1, xi #) ->+                                           case writeDoubleArray# mba_x (off_x +# i) (xi -## gji *## xj) s1 of+                                             s2 -> goI (i +# 1#) s2+                          in goJ (j -# 1#) (goI 0# s'')+  in (# goJ (n -# 1#) s0, () #)+{-# INLINE rawBackSubCholTPacked #-}++-- --------------------------------------------------------------------------+-- QR mutable kernels+-- --------------------------------------------------------------------------++-- | Sum of squares of a column slice in a mutable row-major matrix.+-- Σ A[i,col]² for i in [startRow..endRow-1].+rawMutSumSqColumn :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> ST s Double+rawMutSumSqColumn (MutableByteArray mba) (I# off) (I# ncols) (I# startRow) (I# endRow) (I# col) = ST $ \s0 ->+  let go i acc s+        | isTrue# (i >=# endRow) = (# s, D# acc #)+        | otherwise =+            case readDoubleArray# mba (off +# i *# ncols +# col) s of+              (# s', v #) -> go (i +# 1#) (acc +## v *## v) s'+  in go startRow 0.0## s0+{-# INLINE rawMutSumSqColumn #-}++-- | Dot product of two column slices in a mutable row-major matrix.+-- Σ A[i,col1] * A[i,col2] for i in [startRow..endRow-1].+rawMutSumProdColumns :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> Int -> ST s Double+rawMutSumProdColumns (MutableByteArray mba) (I# off) (I# ncols) (I# startRow) (I# endRow) (I# col1) (I# col2) = ST $ \s0 ->+  let go i acc s+        | isTrue# (i >=# endRow) = (# s, D# acc #)+        | otherwise =+            case readDoubleArray# mba (off +# i *# ncols +# col1) s of+              (# s1, v1 #) ->+                case readDoubleArray# mba (off +# i *# ncols +# col2) s1 of+                  (# s2, v2 #) -> go (i +# 1#) (acc +## v1 *## v2) s2+  in go startRow 0.0## s0+{-# INLINE rawMutSumProdColumns #-}++-- | Apply Householder reflector stored in column k (rows k+1..endRow-1,+-- with v[k]=1 implicit) to targetCol of a mutable row-major matrix.+-- Phase 1: w = beta * (R[k,targetCol] + Σ_{i=k+1}^{endRow-1} v[i]*R[i,targetCol])+-- Phase 2: R[k,targetCol] -= w; R[i,targetCol] -= v[i]*w+rawMutHouseholderApply :: MutableByteArray s -> Int -> Int -> Double+                       -> Int -> Int -> Int -> ST s ()+rawMutHouseholderApply (MutableByteArray mba) (I# off) (I# ncols) (D# beta)+                       (I# k) (I# endRow) (I# targetCol) = ST $ \s0 ->+  -- Phase 1: compute dot product+  case readDoubleArray# mba (off +# k *# ncols +# targetCol) s0 of+    (# s1, rkj #) ->+      let goSum i acc s+            | isTrue# (i >=# endRow) = (# s, beta *## (rkj +## acc) #)+            | otherwise =+                case readDoubleArray# mba (off +# i *# ncols +# k) s of+                  (# s', vi #) ->+                    case readDoubleArray# mba (off +# i *# ncols +# targetCol) s' of+                      (# s'', rij #) -> goSum (i +# 1#) (acc +## vi *## rij) s''+      in case goSum (k +# 1#) 0.0## s1 of+           (# s2, w #) ->+             -- Phase 2: update R[k,targetCol]+             case writeDoubleArray# mba (off +# k *# ncols +# targetCol) (rkj -## w) s2 of+               s3 ->+                 let goUpdate i s+                       | isTrue# (i >=# endRow) = s+                       | otherwise =+                           case readDoubleArray# mba (off +# i *# ncols +# k) s of+                             (# s', vi #) ->+                               case readDoubleArray# mba (off +# i *# ncols +# targetCol) s' of+                                 (# s'', rij #) ->+                                   case writeDoubleArray# mba (off +# i *# ncols +# targetCol) (rij -## vi *## w) s'' of+                                     s''' -> goUpdate (i +# 1#) s'''+                 in (# goUpdate (k +# 1#) s3, () #)+{-# INLINE rawMutHouseholderApply #-}++-- | Apply stored Householder reflector to one row of Q during accumulation.+-- v is stored in the subdiagonal of frozen R (column k, rows k+1..endRow-1).+-- Phase 1: wi = beta * (Q[row,k] + Σ_{l=k+1}^{endRow-1} Q[row,l] * v[l])+-- Phase 2: Q[row,k] -= wi; Q[row,l] -= wi * v[l]+rawMutQAccum :: MutableByteArray s -> Int -> Int+             -> ByteArray -> Int -> Int+             -> Double -> Int -> Int -> Int -> ST s ()+rawMutQAccum (MutableByteArray mba_q) (I# off_q) (I# qcols)+             (ByteArray ba_r) (I# off_r) (I# rcols)+             (D# beta) (I# k) (I# endRow) (I# row) = ST $ \s0 ->+  -- Phase 1: compute wi = beta * (Q[row,k] + Σ Q[row,l] * v[l])+  case readDoubleArray# mba_q (off_q +# row *# qcols +# k) s0 of+    (# s1, qrk #) ->+      let goSum l acc s+            | isTrue# (l >=# endRow) = (# s, beta *## (qrk +## acc) #)+            | otherwise =+                let vl = indexDoubleArray# ba_r (off_r +# l *# rcols +# k)+                in case readDoubleArray# mba_q (off_q +# row *# qcols +# l) s of+                     (# s', qrl #) -> goSum (l +# 1#) (acc +## qrl *## vl) s'+      in case goSum (k +# 1#) 0.0## s1 of+           (# s2, wi #) ->+             -- Phase 2: Q[row,k] -= wi+             case writeDoubleArray# mba_q (off_q +# row *# qcols +# k) (qrk -## wi) s2 of+               s3 ->+                 let goUpdate l s+                       | isTrue# (l >=# endRow) = s+                       | otherwise =+                           let vl = indexDoubleArray# ba_r (off_r +# l *# rcols +# k)+                           in case readDoubleArray# mba_q (off_q +# row *# qcols +# l) s of+                                (# s', qrl #) ->+                                  case writeDoubleArray# mba_q (off_q +# row *# qcols +# l) (qrl -## wi *## vl) s' of+                                    s'' -> goUpdate (l +# 1#) s''+                 in (# goUpdate (k +# 1#) s3, () #)+{-# INLINE rawMutQAccum #-}++-- --------------------------------------------------------------------------+-- Eigen mutable kernels+-- --------------------------------------------------------------------------++-- | Apply Givens rotation to two columns of a mutable matrix.+-- For each row in [0..nrows-1]:+--   tmp = c * M[row,col_p] + s * M[row,col_q]+--   M[row,col_q] = -s * M[row,col_p] + c * M[row,col_q]+--   M[row,col_p] = tmp+rawMutApplyGivensColumns :: MutableByteArray s -> Int -> Int+                         -> Double -> Double -> Int -> Int -> Int -> ST s ()+rawMutApplyGivensColumns (MutableByteArray mba) (I# off) (I# ncols)+                         (D# c_) (D# s_) (I# col_p) (I# col_q) (I# nrows) = ST $ \s0 ->+  let go row s+        | isTrue# (row >=# nrows) = s+        | otherwise =+            let pIdx = off +# row *# ncols +# col_p+                qIdx = off +# row *# ncols +# col_q+            in case readDoubleArray# mba pIdx s of+                 (# s1, mp #) ->+                   case readDoubleArray# mba qIdx s1 of+                     (# s2, mq #) ->+                       let tmp = c_ *## mp +## s_ *## mq+                           qnew = negateDouble# s_ *## mp +## c_ *## mq+                       in case writeDoubleArray# mba pIdx tmp s2 of+                            s3 -> case writeDoubleArray# mba qIdx qnew s3 of+                                    s4 -> go (row +# 1#) s4+  in (# go 0# s0, () #)+{-# INLINE rawMutApplyGivensColumns #-}++-- | Apply Givens rotation to two columns of a COLUMN-MAJOR mutable matrix.+-- In column-major layout, Q[i,j] is at off + j*nrows + i.+-- Column col_p occupies contiguous memory, enabling SIMD vectorisation.+-- For each row in [0..nrows-1]:+--   tmp = c * M[row,col_p] + s * M[row,col_q]+--   M[row,col_q] = -s * M[row,col_p] + c * M[row,col_q]+--   M[row,col_p] = tmp+rawMutApplyGivensColumnsCM :: MutableByteArray s -> Int -> Int+                           -> Double -> Double -> Int -> Int -> Int -> ST s ()+rawMutApplyGivensColumnsCM (MutableByteArray mba) (I# off) (I# nrows)+                           (D# c_) (D# s_) (I# col_p) (I# col_q) (I# _ncols) = ST $ \s0 ->+  let pBase = off +# col_p *# nrows+      qBase = off +# col_q *# nrows+      nrows4 = nrows -# (nrows `remInt#` 4#)+      cV = broadcastDoubleX4# c_+      sV = broadcastDoubleX4# s_+      nsV = negateDoubleX4# sV++      goSimd i s+        | isTrue# (i >=# nrows4) = s+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (pBase +# i) s of+              (# s1, pv #) ->+                case readDoubleArrayAsDoubleX4# mba (qBase +# i) s1 of+                  (# s2, qv #) ->+                    let tmp = fmaddDoubleX4# cV pv (timesDoubleX4# sV qv)+                        q'  = fmaddDoubleX4# nsV pv (timesDoubleX4# cV qv)+                    in case writeDoubleArrayAsDoubleX4# mba (pBase +# i) tmp s2 of+                         s3 -> case writeDoubleArrayAsDoubleX4# mba (qBase +# i) q' s3 of+                                 s4 -> goSimd (i +# 4#) s4++      goScalar i s+        | isTrue# (i >=# nrows) = s+        | otherwise =+            case readDoubleArray# mba (pBase +# i) s of+              (# s1, mp #) ->+                case readDoubleArray# mba (qBase +# i) s1 of+                  (# s2, mq #) ->+                    let tmp = c_ *## mp +## s_ *## mq+                        qnew = negateDouble# s_ *## mp +## c_ *## mq+                    in case writeDoubleArray# mba (pBase +# i) tmp s2 of+                         s3 -> case writeDoubleArray# mba (qBase +# i) qnew s3 of+                                 s4 -> goScalar (i +# 1#) s4++  in (# goScalar nrows4 (goSimd 0# s0), () #)+{-# INLINE rawMutApplyGivensColumnsCM #-}++-- --------------------------------------------------------------------------+-- Matrix transpose (row-major <-> column-major)+-- --------------------------------------------------------------------------++-- | Transpose an n×n row-major matrix to column-major layout.+-- src[i,j] at offS + i*n + j  ->  dst[i,j] at offD + j*n + i+rawTransposeToColMajor :: MutableByteArray s -> Int -> MutableByteArray s -> Int -> Int -> ST s ()+rawTransposeToColMajor (MutableByteArray src) (I# offS)+                       (MutableByteArray dst) (I# offD) (I# n) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let goJ j s'+                  | isTrue# (j >=# n) = s'+                  | otherwise =+                      case readDoubleArray# src (offS +# i *# n +# j) s' of+                        (# s1, v #) ->+                          case writeDoubleArray# dst (offD +# j *# n +# i) v s1 of+                            s2 -> goJ (j +# 1#) s2+            in goI (i +# 1#) (goJ 0# s)+  in (# goI 0# s0, () #)+{-# INLINE rawTransposeToColMajor #-}++-- | Transpose an n×n column-major matrix back to row-major layout.+-- src[i,j] at offS + j*n + i  ->  dst[i,j] at offD + i*n + j+rawTransposeFromColMajor :: MutableByteArray s -> Int -> MutableByteArray s -> Int -> Int -> ST s ()+rawTransposeFromColMajor (MutableByteArray src) (I# offS)+                         (MutableByteArray dst) (I# offD) (I# n) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j >=# n) = s+        | otherwise =+            let goI i s'+                  | isTrue# (i >=# n) = s'+                  | otherwise =+                      case readDoubleArray# src (offS +# j *# n +# i) s' of+                        (# s1, v #) ->+                          case writeDoubleArray# dst (offD +# i *# n +# j) v s1 of+                            s2 -> goI (i +# 1#) s2+            in goJ (j +# 1#) (goI 0# s)+  in (# goJ 0# s0, () #)+{-# INLINE rawTransposeFromColMajor #-}++-- --------------------------------------------------------------------------+-- Bulk memory operations+-- --------------------------------------------------------------------------++-- | Zero n consecutive doubles in a MutableByteArray starting at element offset.+-- Uses SIMD (DoubleX4#) for the main loop with scalar cleanup.+rawZeroDoubles :: MutableByteArray s -> Int -> Int -> ST s ()+rawZeroDoubles (MutableByteArray mba) (I# off) (I# n) = ST $ \s0 ->+  let n4 = n -# (n `remInt#` 4#)+      zeroV = broadcastDoubleX4# 0.0##++      goSimd i s+        | isTrue# (i >=# n4) = s+        | otherwise =+            case writeDoubleArrayAsDoubleX4# mba (off +# i) zeroV s of+              s1 -> goSimd (i +# 4#) s1++      goScalar i s+        | isTrue# (i >=# n) = s+        | otherwise =+            case writeDoubleArray# mba (off +# i) 0.0## s of+              s1 -> goScalar (i +# 1#) s1++  in (# goScalar n4 (goSimd 0# s0), () #)+{-# INLINE rawZeroDoubles #-}++-- | Copy n consecutive doubles from src to dst using memcpy (copyMutableByteArray#).+-- @rawCopyDoubles dst dstOff src srcOff n@ copies src[srcOff..srcOff+n-1] to dst[dstOff..dstOff+n-1].+-- All offsets are in element (Double) units.+rawCopyDoubles :: MutableByteArray s -> Int -> MutableByteArray s -> Int -> Int -> ST s ()+rawCopyDoubles (MutableByteArray dst) (I# dstOff) (MutableByteArray src) (I# srcOff) (I# n) = ST $ \s ->+  case copyMutableByteArray# src (srcOff *# 8#) dst (dstOff *# 8#) (n *# 8#) s of+    s' -> (# s', () #)+{-# INLINE rawCopyDoubles #-}++-- | Negate n consecutive doubles in-place using SIMD.+rawNegateDoubles :: MutableByteArray s -> Int -> Int -> ST s ()+rawNegateDoubles (MutableByteArray mba) (I# off) (I# n) = ST $ \s0 ->+  let n4 = n -# (n `remInt#` 4#)+      negOneV = broadcastDoubleX4# (negateDouble# 1.0##)++      goSimd i s+        | isTrue# (i >=# n4) = s+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (off +# i) s of+              (# s1, v #) ->+                case writeDoubleArrayAsDoubleX4# mba (off +# i) (timesDoubleX4# negOneV v) s1 of+                  s2 -> goSimd (i +# 4#) s2++      goScalar i s+        | isTrue# (i >=# n) = s+        | otherwise =+            case readDoubleArray# mba (off +# i) s of+              (# s1, v #) ->+                case writeDoubleArray# mba (off +# i) (negateDouble# v) s1 of+                  s2 -> goScalar (i +# 1#) s2++  in (# goScalar n4 (goSimd 0# s0), () #)+{-# INLINE rawNegateDoubles #-}++-- | Copy a column from one matrix to another (both row-major).+-- Copies src[row, srcCol] to dst[row, dstCol] for row in [0..nrows-1].+-- Parameters: srcMBA srcOff srcStride srcCol -> dstMBA dstOff dstStride dstCol -> nrows+rawCopyColumn :: MutableByteArray s -> Int -> Int -> Int+              -> MutableByteArray s -> Int -> Int -> Int -> Int -> ST s ()+rawCopyColumn (MutableByteArray src) (I# offS) (I# strideS) (I# colS)+              (MutableByteArray dst) (I# offD) (I# strideD) (I# colD) (I# nrows) = ST $ \s0 ->+  let go i s+        | isTrue# (i >=# nrows) = s+        | otherwise =+            case readDoubleArray# src (offS +# i *# strideS +# colS) s of+              (# s1, v #) ->+                case writeDoubleArray# dst (offD +# i *# strideD +# colD) v s1 of+                  s2 -> go (i +# 1#) s2+  in (# go 0# s0, () #)+{-# INLINE rawCopyColumn #-}++-- --------------------------------------------------------------------------+-- SVD / bidiagonalisation kernels+-- --------------------------------------------------------------------------++-- | Apply a right Householder reflector to one row of a mutable matrix.+-- The Householder vector v is stored in row hvRow of the matrix,+-- columns [hvStart..hvEnd-1], with implicit v[hvStart] = 1.0.+-- Updates row targetRow: R[targetRow, hvStart..hvEnd-1] -= w * v+-- where w = beta * (R[targetRow,hvStart] + Σ_{l=hvStart+1}^{hvEnd-1} R[targetRow,l] * R[hvRow,l])+rawMutHouseholderApplyRow :: MutableByteArray s -> Int -> Int+                          -> Double -> Int -> Int -> Int -> Int -> ST s ()+rawMutHouseholderApplyRow (MutableByteArray mba) (I# off) (I# ncols) (D# beta)+                          (I# hvRow) (I# hvStart) (I# hvEnd) (I# targetRow) = ST $ \s0 ->+  let trOff = off +# targetRow *# ncols+      hvOff = off +# hvRow *# ncols+  -- Phase 1: w = beta * (R[targetRow,hvStart] + Σ R[targetRow,l] * R[hvRow,l])+  in case readDoubleArray# mba (trOff +# hvStart) s0 of+       (# s1, r0 #) ->+         let goSum l acc s+               | isTrue# (l >=# hvEnd) = (# s, beta *## (r0 +## acc) #)+               | otherwise =+                   case readDoubleArray# mba (trOff +# l) s of+                     (# s', rl #) ->+                       case readDoubleArray# mba (hvOff +# l) s' of+                         (# s'', vl #) -> goSum (l +# 1#) (acc +## rl *## vl) s''+         in case goSum (hvStart +# 1#) 0.0## s1 of+              (# s2, w #) ->+                -- Phase 2: R[targetRow,hvStart] -= w (implicit v[hvStart]=1)+                case writeDoubleArray# mba (trOff +# hvStart) (r0 -## w) s2 of+                  s3 ->+                    let goUpdate l s+                          | isTrue# (l >=# hvEnd) = s+                          | otherwise =+                              case readDoubleArray# mba (hvOff +# l) s of+                                (# s', vl #) ->+                                  case readDoubleArray# mba (trOff +# l) s' of+                                    (# s'', rl #) ->+                                      case writeDoubleArray# mba (trOff +# l) (rl -## w *## vl) s'' of+                                        s''' -> goUpdate (l +# 1#) s'''+                    in (# goUpdate (hvStart +# 1#) s3, () #)+{-# INLINE rawMutHouseholderApplyRow #-}++-- | Sum of squares of a row slice in a mutable row-major matrix.+-- Σ A[row,j]² for j in [startCol..endCol-1].+rawMutSumSqRow :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> ST s Double+rawMutSumSqRow (MutableByteArray mba) (I# off) (I# ncols) (I# row) (I# startCol) (I# endCol) = ST $ \s0 ->+  let rowOff = off +# row *# ncols+      go j acc s+        | isTrue# (j >=# endCol) = (# s, D# acc #)+        | otherwise =+            case readDoubleArray# mba (rowOff +# j) s of+              (# s', v #) -> go (j +# 1#) (acc +## v *## v) s'+  in go startCol 0.0## s0+{-# INLINE rawMutSumSqRow #-}++-- --------------------------------------------------------------------------+-- Cholesky SIMD kernel+-- --------------------------------------------------------------------------++-- | SIMD-vectorised Cholesky column kernel.+-- Restructures the inner loop as a dot product of contiguous row segments:+--   G[i,j] -= Σ_{k=0}^{j-1} G[i,k] * G[j,k]+-- which is a dot product of row[i][0..j-1] and row[j][0..j-1].+-- Since rows are contiguous in row-major storage, this enables DoubleX4# SIMD.+rawCholColumnSIMD :: MutableByteArray s -> Int -> Int -> Int -> ST s ()+rawCholColumnSIMD (MutableByteArray mba) (I# off) (I# n) (I# j) = ST $ \s0 ->+  let jRowOff = off +# j *# n+      -- For each row i in [j..n-1], subtract dot(row[i][0..j-1], row[j][0..j-1])+      j4 = j -# (j `remInt#` 4#)  -- SIMD boundary for dot of length j++      goI i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let iRowOff = off +# i *# n+            in case mutRowDot iRowOff jRowOff 0# j4 j s of+                 (# s', dot #) ->+                   case readDoubleArray# mba (iRowOff +# j) s' of+                     (# s'', gij #) ->+                       case writeDoubleArray# mba (iRowOff +# j) (gij -## dot) s'' of+                         s''' -> goI (i +# 1#) s'''++      -- Dot product of two mutable row segments using SIMD+      mutRowDot r1 r2 k k4End kEnd s+        -- SIMD phase+        | isTrue# (k <# k4End) =+            case goSimd r1 r2 k k4End (broadcastDoubleX4# 0.0##) s of+              (# s', acc4 #) ->+                let !(# a, b, c, d #) = unpackDoubleX4# acc4+                    simdSum = a +## b +## c +## d+                in mutRowDotScalar r1 r2 k4End kEnd simdSum s'+        | otherwise = mutRowDotScalar r1 r2 k kEnd 0.0## s++      goSimd r1 r2 k k4End acc s+        | isTrue# (k >=# k4End) = (# s, acc #)+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (r1 +# k) s of+              (# s1, v1 #) ->+                case readDoubleArrayAsDoubleX4# mba (r2 +# k) s1 of+                  (# s2, v2 #) -> goSimd r1 r2 (k +# 4#) k4End (fmaddDoubleX4# v1 v2 acc) s2++      mutRowDotScalar r1 r2 k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            case readDoubleArray# mba (r1 +# k) s of+              (# s1, v1 #) ->+                case readDoubleArray# mba (r2 +# k) s1 of+                  (# s2, v2 #) -> mutRowDotScalar r1 r2 (k +# 1#) kEnd (acc +## v1 *## v2) s2++  in case goI j s0 of+       s1 ->+         -- Scale column: G[j,j] = sqrt(G[j,j]); G[i,j] /= G[j,j] for i > j+         case readDoubleArray# mba (jRowOff +# j) s1 of+           (# s2, gjj #) ->+             let sjj = sqrtDouble# gjj+             in case writeDoubleArray# mba (jRowOff +# j) sjj s2 of+                  s3 ->+                    let goScale i s+                          | isTrue# (i >=# n) = s+                          | otherwise =+                              case readDoubleArray# mba (off +# i *# n +# j) s of+                                (# s4, gij #) ->+                                  case writeDoubleArray# mba (off +# i *# n +# j) (gij /## sjj) s4 of+                                    s5 -> goScale (i +# 1#) s5+                    in (# goScale (j +# 1#) s3, () #)+{-# INLINE rawCholColumnSIMD #-}++-- | Like 'rawCholColumnSIMD' but the dot-product starts from column @fromCol@+-- instead of column 0.  Used by panel Cholesky: after applying the GEMM update+-- from previous panels, the within-panel factorisation only needs contributions+-- from columns @fromCol .. j-1@.+rawCholColumnSIMDFrom :: MutableByteArray s -> Int -> Int -> Int -> Int -> ST s ()+rawCholColumnSIMDFrom (MutableByteArray mba) (I# off) (I# n) (I# j) (I# fromCol) = ST $ \s0 ->+  let jRowOff = off +# j *# n+      dotLen  = j -# fromCol+      dotLen4 = dotLen -# (dotLen `remInt#` 4#)+      k4End   = fromCol +# dotLen4++      goI i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let iRowOff = off +# i *# n+            in case mutRowDot iRowOff jRowOff fromCol k4End j s of+                 (# s', dot #) ->+                   case readDoubleArray# mba (iRowOff +# j) s' of+                     (# s'', gij #) ->+                       case writeDoubleArray# mba (iRowOff +# j) (gij -## dot) s'' of+                         s''' -> goI (i +# 1#) s'''++      mutRowDot r1 r2 k kSimdEnd kEnd s+        | isTrue# (k <# kSimdEnd) =+            case goSimd r1 r2 k kSimdEnd (broadcastDoubleX4# 0.0##) s of+              (# s', acc4 #) ->+                let !(# a, b, c, d #) = unpackDoubleX4# acc4+                    simdSum = a +## b +## c +## d+                in mutRowDotScalar r1 r2 kSimdEnd kEnd simdSum s'+        | otherwise = mutRowDotScalar r1 r2 k kEnd 0.0## s++      goSimd r1 r2 k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            case readDoubleArrayAsDoubleX4# mba (r1 +# k) s of+              (# s1, v1 #) ->+                case readDoubleArrayAsDoubleX4# mba (r2 +# k) s1 of+                  (# s2, v2 #) -> goSimd r1 r2 (k +# 4#) kEnd (fmaddDoubleX4# v1 v2 acc) s2++      mutRowDotScalar r1 r2 k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            case readDoubleArray# mba (r1 +# k) s of+              (# s1, v1 #) ->+                case readDoubleArray# mba (r2 +# k) s1 of+                  (# s2, v2 #) -> mutRowDotScalar r1 r2 (k +# 1#) kEnd (acc +## v1 *## v2) s2++  in case goI j s0 of+       s1 ->+         case readDoubleArray# mba (jRowOff +# j) s1 of+           (# s2, gjj #) ->+             let sjj = sqrtDouble# gjj+             in case writeDoubleArray# mba (jRowOff +# j) sjj s2 of+                  s3 ->+                    let goScale i s+                          | isTrue# (i >=# n) = s+                          | otherwise =+                              case readDoubleArray# mba (off +# i *# n +# j) s of+                                (# s4, gij #) ->+                                  case writeDoubleArray# mba (off +# i *# n +# j) (gij /## sjj) s4 of+                                    s5 -> goScale (i +# 1#) s5+                    in (# goScale (j +# 1#) s3, () #)+{-# INLINE rawCholColumnSIMDFrom #-}++-- --------------------------------------------------------------------------+-- SIMD forward/back substitution kernels (dot-product formulation)+-- --------------------------------------------------------------------------++-- | SIMD forward substitution (unit lower triangular, dot-product formulation).+-- Solves Ly = b where L has unit diagonal; b is already in mba_x.+-- For each row i: x[i] -= dot(L[i, 0..i-1], x[0..i-1]).+-- L row slices are contiguous in row-major storage → SIMD-friendly.+rawForwardSubUnitPackedSIMD :: ByteArray -> Int -> Int+                            -> MutableByteArray s -> Int -> ST s ()+rawForwardSubUnitPackedSIMD (ByteArray ba_lu) (I# off_lu) (I# n)+                            (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let rowOff = off_lu +# i *# n+                dotLen = i+                d8End = dotLen -# (dotLen `remInt#` 8#)+                d4End = dotLen -# (dotLen `remInt#` 4#)+            in case goSimd8 rowOff 0# d8End (broadcastDoubleX4# 0.0##) (broadcastDoubleX4# 0.0##) s of+                 (# s1, acc0, acc1 #) ->+                   case goSimd4 rowOff d8End d4End acc0 s1 of+                     (# s2, acc0' #) ->+                       let !combined = plusDoubleX4# acc0' acc1+                           !(# a, b, c, d #) = unpackDoubleX4# combined+                           simdSum = a +## b +## c +## d+                       in case goScalar rowOff d4End dotLen simdSum s2 of+                            (# s3, dotVal #) ->+                              case readDoubleArray# mba_x (off_x +# i) s3 of+                                (# s4, xi #) ->+                                  case writeDoubleArray# mba_x (off_x +# i) (xi -## dotVal) s4 of+                                    s5 -> goI (i +# 1#) s5++      goSimd8 rowOff k k8End acc0 acc1 s+        | isTrue# (k >=# k8End) = (# s, acc0, acc1 #)+        | otherwise =+            let lv0 = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k)+                lv1 = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k +# 4#)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv0 #) ->+                   case readDoubleArrayAsDoubleX4# mba_x (off_x +# k +# 4#) s' of+                     (# s'', xv1 #) -> goSimd8 rowOff (k +# 8#) k8End+                                          (fmaddDoubleX4# lv0 xv0 acc0) (fmaddDoubleX4# lv1 xv1 acc1) s''++      goSimd4 rowOff k k4End acc s+        | isTrue# (k >=# k4End) = (# s, acc #)+        | otherwise =+            let lv = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv #) -> goSimd4 rowOff (k +# 4#) k4End (fmaddDoubleX4# lv xv acc) s'++      goScalar rowOff k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            let lk = indexDoubleArray# ba_lu (rowOff +# k)+            in case readDoubleArray# mba_x (off_x +# k) s of+                 (# s', xk #) -> goScalar rowOff (k +# 1#) kEnd (acc +## lk *## xk) s'++  in (# goI 0# s0, () #)+{-# INLINE rawForwardSubUnitPackedSIMD #-}++-- | SIMD back substitution (upper triangular, dot-product formulation).+-- Solves Ux = y where y is in mba_x; overwrites with x.+-- For each row i (n-1 down to 0): x[i] = (x[i] - dot(U[i, i+1..n-1], x[i+1..n-1])) / U[i,i].+rawBackSubPackedSIMD :: ByteArray -> Int -> Int+                     -> MutableByteArray s -> Int -> ST s ()+rawBackSubPackedSIMD (ByteArray ba_lu) (I# off_lu) (I# n)+                     (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goI i s+        | isTrue# (i <# 0#) = s+        | otherwise =+            let rowOff = off_lu +# i *# n+                dotStart = i +# 1#+                dotLen = n -# i -# 1#+                d8End = dotStart +# (dotLen -# (dotLen `remInt#` 8#))+                d4End = dotStart +# (dotLen -# (dotLen `remInt#` 4#))+            in case goSimd8 rowOff dotStart d8End (broadcastDoubleX4# 0.0##) (broadcastDoubleX4# 0.0##) s of+                 (# s1, acc0, acc1 #) ->+                   case goSimd4 rowOff d8End d4End acc0 s1 of+                     (# s2, acc0' #) ->+                       let !combined = plusDoubleX4# acc0' acc1+                           !(# a, b, c, d #) = unpackDoubleX4# combined+                           simdSum = a +## b +## c +## d+                       in case goScalar rowOff d4End n simdSum s2 of+                            (# s3, dotVal #) ->+                              let uii = indexDoubleArray# ba_lu (rowOff +# i)+                              in case readDoubleArray# mba_x (off_x +# i) s3 of+                                   (# s4, xi #) ->+                                     case writeDoubleArray# mba_x (off_x +# i) ((xi -## dotVal) /## uii) s4 of+                                       s5 -> goI (i -# 1#) s5++      goSimd8 rowOff k k8End acc0 acc1 s+        | isTrue# (k >=# k8End) = (# s, acc0, acc1 #)+        | otherwise =+            let uv0 = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k)+                uv1 = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k +# 4#)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv0 #) ->+                   case readDoubleArrayAsDoubleX4# mba_x (off_x +# k +# 4#) s' of+                     (# s'', xv1 #) -> goSimd8 rowOff (k +# 8#) k8End+                                          (fmaddDoubleX4# uv0 xv0 acc0) (fmaddDoubleX4# uv1 xv1 acc1) s''++      goSimd4 rowOff k k4End acc s+        | isTrue# (k >=# k4End) = (# s, acc #)+        | otherwise =+            let uv = indexDoubleArrayAsDoubleX4# ba_lu (rowOff +# k)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv #) -> goSimd4 rowOff (k +# 4#) k4End (fmaddDoubleX4# uv xv acc) s'++      goScalar rowOff k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            let uk = indexDoubleArray# ba_lu (rowOff +# k)+            in case readDoubleArray# mba_x (off_x +# k) s of+                 (# s', xk #) -> goScalar rowOff (k +# 1#) kEnd (acc +## uk *## xk) s'++  in (# goI (n -# 1#) s0, () #)+{-# INLINE rawBackSubPackedSIMD #-}++-- | SIMD Cholesky forward substitution (non-unit diagonal, dot-product formulation).+-- Solves Gy = b; b is in mba_x, overwrites with y.+-- For each row i: x[i] = (x[i] - dot(G[i, 0..i-1], x[0..i-1])) / G[i,i].+rawForwardSubCholPackedSIMD :: ByteArray -> Int -> Int+                            -> MutableByteArray s -> Int -> ST s ()+rawForwardSubCholPackedSIMD (ByteArray ba_g) (I# off_g) (I# n)+                            (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let rowOff = off_g +# i *# n+                dotLen = i+                d8End = dotLen -# (dotLen `remInt#` 8#)+                d4End = dotLen -# (dotLen `remInt#` 4#)+            in case goSimd8 rowOff 0# d8End (broadcastDoubleX4# 0.0##) (broadcastDoubleX4# 0.0##) s of+                 (# s1, acc0, acc1 #) ->+                   case goSimd4 rowOff d8End d4End acc0 s1 of+                     (# s2, acc0' #) ->+                       let !combined = plusDoubleX4# acc0' acc1+                           !(# a, b, c, d #) = unpackDoubleX4# combined+                           simdSum = a +## b +## c +## d+                       in case goScalar rowOff d4End dotLen simdSum s2 of+                            (# s3, dotVal #) ->+                              let gii = indexDoubleArray# ba_g (rowOff +# i)+                              in case readDoubleArray# mba_x (off_x +# i) s3 of+                                   (# s4, xi #) ->+                                     case writeDoubleArray# mba_x (off_x +# i) ((xi -## dotVal) /## gii) s4 of+                                       s5 -> goI (i +# 1#) s5++      goSimd8 rowOff k k8End acc0 acc1 s+        | isTrue# (k >=# k8End) = (# s, acc0, acc1 #)+        | otherwise =+            let gv0 = indexDoubleArrayAsDoubleX4# ba_g (rowOff +# k)+                gv1 = indexDoubleArrayAsDoubleX4# ba_g (rowOff +# k +# 4#)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv0 #) ->+                   case readDoubleArrayAsDoubleX4# mba_x (off_x +# k +# 4#) s' of+                     (# s'', xv1 #) -> goSimd8 rowOff (k +# 8#) k8End+                                          (fmaddDoubleX4# gv0 xv0 acc0) (fmaddDoubleX4# gv1 xv1 acc1) s''++      goSimd4 rowOff k k4End acc s+        | isTrue# (k >=# k4End) = (# s, acc #)+        | otherwise =+            let gv = indexDoubleArrayAsDoubleX4# ba_g (rowOff +# k)+            in case readDoubleArrayAsDoubleX4# mba_x (off_x +# k) s of+                 (# s', xv #) -> goSimd4 rowOff (k +# 4#) k4End (fmaddDoubleX4# gv xv acc) s'++      goScalar rowOff k kEnd acc s+        | isTrue# (k >=# kEnd) = (# s, acc #)+        | otherwise =+            let gk = indexDoubleArray# ba_g (rowOff +# k)+            in case readDoubleArray# mba_x (off_x +# k) s of+                 (# s', xk #) -> goScalar rowOff (k +# 1#) kEnd (acc +## gk *## xk) s'++  in (# goI 0# s0, () #)+{-# INLINE rawForwardSubCholPackedSIMD #-}++-- | SIMD Cholesky G^T back substitution (SAXPY formulation with broadcast).+-- Solves G^T x = y; y is in mba_x, overwrites with x.+-- For each j (n-1 down to 0): x[j] /= G[j,j], then for i=0..j-1:+-- x[i] -= G[j,i] * x[j] (SAXPY with broadcast x[j]).+-- G[j, 0..j-1] is contiguous in row-major → SIMD-friendly.+rawBackSubCholTPackedSIMD :: ByteArray -> Int -> Int+                          -> MutableByteArray s -> Int -> ST s ()+rawBackSubCholTPackedSIMD (ByteArray ba_g) (I# off_g) (I# n)+                          (MutableByteArray mba_x) (I# off_x) = ST $ \s0 ->+  let goJ j s+        | isTrue# (j <# 0#) = s+        | otherwise =+            let gjj = indexDoubleArray# ba_g (off_g +# j *# n +# j)+            in case readDoubleArray# mba_x (off_x +# j) s of+                 (# s', xj_ #) ->+                   let xj = xj_ /## gjj+                   in case writeDoubleArray# mba_x (off_x +# j) xj s' of+                        s'' ->+                          let jRowOff = off_g +# j *# n+                              negXj4 = broadcastDoubleX4# (negateDouble# xj)+                              updateLen = j+                              u8End = updateLen -# (updateLen `remInt#` 8#)+                              u4End = updateLen -# (updateLen `remInt#` 4#)++                              goSimd8 i s_+                                | isTrue# (i >=# u8End) = s_+                                | otherwise =+                                    let gv0 = indexDoubleArrayAsDoubleX4# ba_g (jRowOff +# i)+                                        gv1 = indexDoubleArrayAsDoubleX4# ba_g (jRowOff +# i +# 4#)+                                    in case readDoubleArrayAsDoubleX4# mba_x (off_x +# i) s_ of+                                         (# s1, xv0 #) ->+                                           case readDoubleArrayAsDoubleX4# mba_x (off_x +# i +# 4#) s1 of+                                             (# s2, xv1 #) ->+                                               case writeDoubleArrayAsDoubleX4# mba_x (off_x +# i) (fmaddDoubleX4# negXj4 gv0 xv0) s2 of+                                                 s3 -> case writeDoubleArrayAsDoubleX4# mba_x (off_x +# i +# 4#) (fmaddDoubleX4# negXj4 gv1 xv1) s3 of+                                                         s4 -> goSimd8 (i +# 8#) s4++                              goSimd4 i s_+                                | isTrue# (i >=# u4End) = s_+                                | otherwise =+                                    let gv = indexDoubleArrayAsDoubleX4# ba_g (jRowOff +# i)+                                    in case readDoubleArrayAsDoubleX4# mba_x (off_x +# i) s_ of+                                         (# s1, xv #) ->+                                           case writeDoubleArrayAsDoubleX4# mba_x (off_x +# i) (fmaddDoubleX4# negXj4 gv xv) s1 of+                                             s2 -> goSimd4 (i +# 4#) s2++                              goScalar i s_+                                | isTrue# (i >=# j) = s_+                                | otherwise =+                                    let gji = indexDoubleArray# ba_g (jRowOff +# i)+                                    in case readDoubleArray# mba_x (off_x +# i) s_ of+                                         (# s1, xi #) ->+                                           case writeDoubleArray# mba_x (off_x +# i) (xi -## gji *## xj) s1 of+                                             s2 -> goScalar (i +# 1#) s2++                          in goJ (j -# 1#) (goScalar u4End (goSimd4 u8End (goSimd8 0# s'')))+  in (# goJ (n -# 1#) s0, () #)+{-# INLINE rawBackSubCholTPackedSIMD #-}++-- --------------------------------------------------------------------------+-- Utilities+-- --------------------------------------------------------------------------++minI :: Int# -> Int# -> Int#+minI a b = if isTrue# (a <=# b) then a else b+{-# INLINE minI #-}
+ src/Numeric/LinearAlgebra/Massiv/Linear.hs view
@@ -0,0 +1,122 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Linear+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = Integration with the @linear@ library+--+-- This module provides conversion functions between the+-- <https://hackage.haskell.org/package/linear linear> library's types+-- (@'Linear.V.V'@, @'Linear.V2.V2'@, @'Linear.V3.V3'@, @'Linear.V4.V4'@)+-- and our dimensioned @'Vector'@ \/ @'Matrix'@ types.+--+-- == Why not typeclass instances?+--+-- The @linear@ library's typeclasses ('Linear.Additive.Additive',+-- 'Linear.Metric.Metric', 'Linear.Trace.Trace') expect types of kind+-- @* -> *@ (i.e., functors over the element type). Our @Vector n r e@ and+-- @Matrix m n r e@ carry additional type parameters (@n@, @r@) before @e@,+-- making direct functor-based instances impractical without additional+-- newtype wrappers.+--+-- Instead, equivalent operations are provided as standalone functions:+--+-- * "Numeric.LinearAlgebra.Massiv.BLAS.Level1" — 'dot', 'axpy', 'scal', 'nrm2'+-- * "Numeric.LinearAlgebra.Massiv.BLAS.Level3" — 'mAdd', 'mSub', 'mScale', 'matMul', 'transpose'+-- * "Numeric.LinearAlgebra.Massiv.Norms" — 'normFrob', 'norm1', 'normInf'+--+-- == Conversion semantics+--+-- 'fromLinearV' converts to the @'Data.Massiv.Array.B'@ (boxed) representation+-- because @linear@'s @V@ stores elements in a boxed @Data.Vector.Vector@.+-- For small fixed-size types (@V2@, @V3@, @V4@), 'fromV2' etc. produce+-- vectors in any representation @r@ satisfying @Manifest r e@.+module Numeric.LinearAlgebra.Massiv.Linear+  ( -- * Conversions with @linear@'s @V n a@+    fromLinearV+  , toLinearV+    -- * Small fixed-size vector conversions+  , fromV2+  , fromV3+  , fromV4+    -- * List-based matrix I\/O+  , toListMatrix+  , fromListMatrix+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..))+import qualified Data.Vector as BV+import GHC.TypeNats (KnownNat, natVal)+import Data.Proxy (Proxy(..))++import qualified Linear.V as L+import Linear.V2 (V2(..))+import Linear.V3 (V3(..))+import Linear.V4 (V4(..))++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Convert a @linear@ @'Linear.V.V' n a@ to our @'Vector' n 'Data.Massiv.Array.B' a@.+--+-- The result uses the boxed (@B@) representation since @linear@'s @V@ is+-- backed by a boxed @Data.Vector.Vector@.+fromLinearV :: forall n a. KnownNat n => L.V n a -> Vector n M.B a+fromLinearV lv =+  let bv = L.toVector lv+      nn = fromIntegral (natVal (Proxy @n))+      arr = M.compute @M.B $ M.makeArray @M.D M.Seq (M.Sz1 nn) (bv BV.!)+  in MkVector arr++-- | Convert our @'Vector' n 'Data.Massiv.Array.B' a@ to a @linear@ @'Linear.V.V' n a@.+--+-- This is the inverse of 'fromLinearV'. The dimension is checked by @linear@'s+-- 'Linear.V.fromVector' (which returns 'Maybe'); the 'error' case is unreachable+-- given correct type-level dimensions.+toLinearV :: forall n a. KnownNat n => Vector n M.B a -> L.V n a+toLinearV (MkVector arr) =+  let nn = fromIntegral (natVal (Proxy @n))+      bv = BV.generate nn (\i -> M.index' arr i)+  in case L.fromVector bv of+    Just v  -> v+    Nothing -> error "toLinearV: impossible dimension mismatch"++-- | Convert a @linear@ @'Linear.V2.V2'@ to a 2-element 'Vector'.+fromV2 :: M.Manifest r e => V2 e -> Vector 2 r e+fromV2 (V2 x y) = makeVector @2 $ \i -> case i of { 0 -> x; _ -> y }++-- | Convert a @linear@ @'Linear.V3.V3'@ to a 3-element 'Vector'.+fromV3 :: M.Manifest r e => V3 e -> Vector 3 r e+fromV3 (V3 x y z) = makeVector @3 $ \i -> case i of { 0 -> x; 1 -> y; _ -> z }++-- | Convert a @linear@ @'Linear.V4.V4'@ to a 4-element 'Vector'.+fromV4 :: M.Manifest r e => V4 e -> Vector 4 r e+fromV4 (V4 x y z w) = makeVector @4 $ \i -> case i of { 0 -> x; 1 -> y; 2 -> z; _ -> w }++-- | Convert a matrix to a list of lists (row-major order).+--+-- @+-- toListMatrix mat  ==  [[mat '!' (i,j) | j <- [0..n-1]] | i <- [0..m-1]]+-- @+toListMatrix :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+             => Matrix m n r e -> [[e]]+toListMatrix mat =+  let r = dimVal @m+      c = dimVal @n+  in [[mat ! (i, j) | j <- [0..c-1]] | i <- [0..r-1]]++-- | Create a matrix from a list of lists (row-major order).+--+-- Returns 'Nothing' if the list dimensions do not match the type-level+-- dimensions \(m\) and \(n\).+fromListMatrix :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e)+               => [[e]] -> Maybe (Matrix m n r e)+fromListMatrix rows_+  | length rows_ /= dimVal @m = Nothing+  | any (\row -> length row /= dimVal @n) rows_ = Nothing+  | otherwise = Just $ makeMatrix @m @n @r $ \i j -> (rows_ !! i) !! j
+ src/Numeric/LinearAlgebra/Massiv/Norms.hs view
@@ -0,0 +1,163 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Norms+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = Matrix and Vector Norms+--+-- Norms measure the "size" of vectors and matrices and are fundamental to+-- error analysis, convergence criteria, and conditioning estimates in+-- numerical linear algebra.+--+-- This module implements the norms described in:+--+-- * Golub, G. H. & Van Loan, C. F. (2013). /Matrix Computations/, 4th ed.+--   __Chapter 2: Matrix Analysis__, Sections 2.3–2.7, pp. 71–95.+--+-- == Vector norms (GVL4 Section 2.3, p. 71)+--+-- For a vector \(x \in \mathbb{R}^n\):+--+-- * 1-norm: \(\|x\|_1 = \sum_{i=1}^{n} |x_i|\) — see 'vnorm1'+-- * 2-norm (Euclidean): \(\|x\|_2 = \sqrt{\sum_{i=1}^{n} x_i^2}\) — see 'vnorm2'+-- * \(\infty\)-norm: \(\|x\|_\infty = \max_i |x_i|\) — see 'vnormInf'+--+-- == Matrix norms (GVL4 Section 2.3, pp. 71–78)+--+-- For a matrix \(A \in \mathbb{R}^{m \times n}\):+--+-- * Frobenius norm: \(\|A\|_F = \sqrt{\sum_{i,j} a_{ij}^2} = \sqrt{\text{trace}(A^T A)}\)+--   — see 'normFrob'+-- * 1-norm (max column sum): \(\|A\|_1 = \max_j \sum_i |a_{ij}|\) — see 'norm1'+-- * \(\infty\)-norm (max row sum): \(\|A\|_\infty = \max_i \sum_j |a_{ij}|\) — see 'normInf'+--+-- These satisfy the norm axioms: non-negativity, homogeneity, and the+-- triangle inequality \(\|A + B\| \leq \|A\| + \|B\|\).+--+-- == Condition numbers (GVL4 Section 2.7, pp. 87–95)+--+-- The condition number \(\kappa(A) = \|A\| \cdot \|A^{-1}\|\) measures+-- how sensitive the solution of \(Ax = b\) is to perturbations in \(A\)+-- and \(b\). See 'condFrob' for a placeholder using the Frobenius norm.+module Numeric.LinearAlgebra.Massiv.Norms+  ( -- * Vector norms (GVL4 Section 2.3, p. 71)+    vnorm1+  , vnorm2+  , vnormInf+    -- * Matrix norms (GVL4 Section 2.3, pp. 71–78)+  , normFrob+  , norm1+  , normInf+    -- * Condition number estimate (GVL4 Section 2.7, p. 87)+  , condFrob+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Vector 1-norm (GVL4 p. 71, eq. 2.3.1).+--+-- \[+--   \|x\|_1 = \sum_{i=1}^{n} |x_i|+-- \]+--+-- Complexity: \(O(n)\).+vnorm1 :: (KnownNat n, M.Manifest r e, Num e, Ord e)+        => Vector n r e -> e+vnorm1 (MkVector arr) = M.foldlS (\acc x -> acc + abs x) 0 arr++-- | Vector 2-norm, the Euclidean norm (GVL4 p. 71, eq. 2.3.2).+--+-- \[+--   \|x\|_2 = \sqrt{\sum_{i=1}^{n} x_i^2} = \sqrt{x^T x}+-- \]+--+-- Complexity: \(O(n)\).+vnorm2 :: (KnownNat n, M.Manifest r e, Floating e)+        => Vector n r e -> e+vnorm2 (MkVector arr) = sqrt $ M.foldlS (\acc x -> acc + x * x) 0 arr++-- | Vector \(\infty\)-norm (GVL4 p. 71, eq. 2.3.3).+--+-- \[+--   \|x\|_\infty = \max_{1 \leq i \leq n} |x_i|+-- \]+--+-- Complexity: \(O(n)\).+vnormInf :: (KnownNat n, M.Manifest r e, Num e, Ord e)+          => Vector n r e -> e+vnormInf (MkVector arr) = M.foldlS (\acc x -> max acc (abs x)) 0 arr++-- | Frobenius norm (GVL4 p. 72, eq. 2.3.7).+--+-- \[+--   \|A\|_F = \sqrt{\sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}^2}+--           = \sqrt{\text{trace}(A^T A)}+-- \]+--+-- The Frobenius norm is /not/ an operator norm (it is not subordinate+-- to any vector norm), but it is submultiplicative:+-- \(\|AB\|_F \leq \|A\|_F \|B\|_F\).+--+-- Complexity: \(O(mn)\).+normFrob :: (KnownNat m, KnownNat n, M.Manifest r e, Floating e)+          => Matrix m n r e -> e+normFrob (MkMatrix arr) = sqrt $ M.foldlS (\acc x -> acc + x * x) 0 arr++-- | Matrix 1-norm — maximum absolute column sum (GVL4 p. 72, eq. 2.3.10).+--+-- \[+--   \|A\|_1 = \max_{1 \leq j \leq n} \sum_{i=1}^{m} |a_{ij}|+-- \]+--+-- This is the operator norm subordinate to the vector 1-norm:+-- \(\|A\|_1 = \max_{\|x\|_1 = 1} \|Ax\|_1\).+--+-- Complexity: \(O(mn)\).+norm1 :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e, Ord e)+      => Matrix m n r e -> e+norm1 mat =+  let c = dimVal @n+      r = dimVal @m+      colSum j = foldl (\acc i -> acc + abs (mat ! (i, j))) 0 [0..r-1]+  in maximum $ map colSum [0..c-1]++-- | Matrix \(\infty\)-norm — maximum absolute row sum (GVL4 p. 72, eq. 2.3.11).+--+-- \[+--   \|A\|_\infty = \max_{1 \leq i \leq m} \sum_{j=1}^{n} |a_{ij}|+-- \]+--+-- This is the operator norm subordinate to the vector \(\infty\)-norm.+-- Note that \(\|A\|_\infty = \|A^T\|_1\).+--+-- Complexity: \(O(mn)\).+normInf :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e, Ord e)+        => Matrix m n r e -> e+normInf mat =+  let c = dimVal @n+      r = dimVal @m+      rowSum i = foldl (\acc j -> acc + abs (mat ! (i, j))) 0 [0..c-1]+  in maximum $ map rowSum [0..r-1]++-- | Estimate condition number using the Frobenius norm (GVL4 Section 2.7, p. 87).+--+-- The condition number is defined as+-- \(\kappa_F(A) = \|A\|_F \cdot \|A^{-1}\|_F\).+--+-- __Note__: This function currently returns only \(\|A\|_F\) as a placeholder.+-- Computing \(\|A^{-1}\|_F\) requires solving a linear system (e.g., via LU),+-- introducing a circular dependency. Users should compute the full condition+-- number by combining 'normFrob' with an explicit inverse or using SVD-based+-- estimates (\(\kappa_2 = \sigma_{\max} / \sigma_{\min}\)).+condFrob :: (KnownNat n, M.Manifest r e, Floating e)+         => Matrix n n r e -> e+condFrob = normFrob
+ src/Numeric/LinearAlgebra/Massiv/Orthogonal/Givens.hs view
@@ -0,0 +1,156 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Givens rotations for selective zeroing of matrix entries.+--+-- This module implements Givens (plane) rotations following Golub & Van+-- Loan, /Matrix Computations/, 4th edition (GVL4), Section 5.1.8,+-- pp. 240--243.+--+-- A Givens rotation is an orthogonal matrix that operates in a+-- two-dimensional subspace.  Given scalars \( a \) and \( b \), the+-- rotation matrix+--+-- \( G^T = \begin{bmatrix} c & -s \\ s & c \end{bmatrix} \)+--+-- is constructed so that+--+-- \( G^T \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} r \\ 0 \end{bmatrix} \)+--+-- where \( r = \sqrt{a^2 + b^2} \).  Our convention follows GVL4+-- Algorithm 5.1.3 (p. 240): \( c = a / r \), \( s = -b / r \).+--+-- Givens rotations are especially useful when only a small number of+-- sub-diagonal entries need to be zeroed (e.g., in Hessenberg or banded+-- matrices), whereas Householder reflections are preferred for zeroing+-- entire sub-columns at once.  Givens-based QR factorisation is the+-- method of choice for tridiagonal and Hessenberg eigenvalue problems+-- (GVL4 Section 5.2.8, p. 255).+--+-- __Complexity.__+--+-- * Computing a Givens rotation ('givensRotation'): \( O(1) \) flops+--   (one square root and a small number of divisions).+-- * Applying a Givens rotation to a row or column pair of an+--   \( m \times n \) matrix ('applyGivensLeft', 'applyGivensRight'):+--   \( O(n) \) or \( O(m) \) flops respectively (one pass over the+--   affected row or column pair).+module Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+  ( -- * Givens rotation+    givensRotation+    -- * Apply Givens rotation+  , applyGivensLeft+  , applyGivensRight+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Compute a Givens rotation (GVL4 Algorithm 5.1.3, p. 240).+--+-- Given scalars \( a \) and \( b \), compute cosine \( c \) and sine+-- \( s \) such that+--+-- \( \begin{bmatrix} c & s \\ -s & c \end{bmatrix}^T \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} r \\ 0 \end{bmatrix} \)+--+-- where \( r = \sqrt{a^2 + b^2} \).+--+-- The implementation avoids overflow and unnecessary computation by+-- distinguishing three cases:+--+-- * If \( b = 0 \), the rotation is the identity: \( c = 1, s = 0 \).+-- * If \( |b| > |a| \), the tangent \( \tau = -a/b \) is computed first,+--   then \( s = 1 / \sqrt{1 + \tau^2} \) and \( c = s \tau \).+-- * Otherwise, \( \tau = -b/a \), \( c = 1 / \sqrt{1 + \tau^2} \), and+--   \( s = c \tau \).+--+-- This avoids computing the potentially large quantity+-- \( r = \sqrt{a^2 + b^2} \) directly, which could overflow.+--+-- __Complexity:__ \( O(1) \) flops (one square root, a few multiplications+-- and divisions).+--+-- Returns @(c, s)@.+givensRotation :: (Floating e, Ord e) => e -> e -> (e, e)+givensRotation a b+  | b == 0    = (1, 0)+  | abs b > abs a =+      let tau = -a / b+          s = 1 / sqrt (1 + tau * tau)+          c = s * tau+      in (c, s)+  | otherwise =+      let tau = -b / a+          c = 1 / sqrt (1 + tau * tau)+          s = c * tau+      in (c, s)++-- | Apply a Givens rotation from the left to rows @i@ and @k@ of a matrix+-- (GVL4 Section 5.1.9, p. 241).+--+-- Performs the update+--+-- \( A([i,k], :) \leftarrow G^T \, A([i,k], :) \)+--+-- where \( G^T = \begin{bmatrix} c & -s \\ s & c \end{bmatrix} \).+-- Only rows @i@ and @k@ are modified; all other rows are untouched.+-- This is the standard operation used to zero out the \( (k, j) \)+-- entry of a matrix during Givens-based QR factorisation+-- (GVL4 Algorithm 5.2.3, p. 252).+--+-- __Complexity:__ \( O(n) \) flops, where \( n \) is the number of+-- columns.+applyGivensLeft :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+                => e    -- ^ c+                -> e    -- ^ s+                -> Int  -- ^ row i+                -> Int  -- ^ row k+                -> Matrix m n r e -> Matrix m n r e+applyGivensLeft c s ri rk a =+  makeMatrix @m @n @r $ \i j ->+    if i == ri then+      c * (a ! (ri, j)) - s * (a ! (rk, j))+    else if i == rk then+      s * (a ! (ri, j)) + c * (a ! (rk, j))+    else+      a ! (i, j)++-- | Apply a Givens rotation from the right to columns @i@ and @k@ of a+-- matrix (GVL4 Section 5.1.9, p. 242).+--+-- Performs the update+--+-- \( A(:, [i,k]) \leftarrow A(:, [i,k]) \, G \)+--+-- where \( G = \begin{bmatrix} c & s \\ -s & c \end{bmatrix} \).+-- Only columns @i@ and @k@ are modified; all other columns are+-- untouched.  Right-multiplication by a Givens rotation is typically+-- used to accumulate the orthogonal factor \( Q \) during QR+-- factorisation (GVL4 Section 5.1.9, p. 242).+--+-- __Complexity:__ \( O(m) \) flops, where \( m \) is the number of+-- rows.+applyGivensRight :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+                 => e    -- ^ c+                 -> e    -- ^ s+                 -> Int  -- ^ column i+                 -> Int  -- ^ column k+                 -> Matrix m n r e -> Matrix m n r e+applyGivensRight c s ci ck a =+  makeMatrix @m @n @r $ \i j ->+    if j == ci then+      c * (a ! (i, ci)) - s * (a ! (i, ck))+    else if j == ck then+      s * (a ! (i, ci)) + c * (a ! (i, ck))+    else+      a ! (i, j)
+ src/Numeric/LinearAlgebra/Massiv/Orthogonal/Householder.hs view
@@ -0,0 +1,176 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Householder reflections for orthogonal triangularisation.+--+-- This module implements the Householder reflection (also known as a+-- Householder transformation), following Golub & Van Loan, /Matrix+-- Computations/, 4th edition (GVL4), Section 5.1, pp. 236--243.+--+-- As GVL4 states (p. 236): \"The Householder reflection is the most+-- important tool in matrix computations.\"  A Householder reflector is a+-- matrix of the form+--+-- \( P = I - \beta v v^T \)+--+-- where \( v \) is the /Householder vector/ and \( \beta = 2 / (v^T v) \).+-- The key property of \( P \) is that it is both symmetric and orthogonal:+--+-- \( P = P^T = P^{-1} \)+--+-- Given an input vector \( x \), the Householder vector \( v \) and scalar+-- \( \beta \) are chosen so that+--+-- \( P x = (I - \beta v v^T) x = \| x \|_2 \, e_1 \)+--+-- where \( e_1 \) is the first standard basis vector.  This is the+-- fundamental operation behind Householder QR factorisation (GVL4+-- Algorithm 5.2.1) and many other matrix decompositions.+--+-- __Complexity.__+--+-- * Computing the Householder vector ('householderVector'): \( O(n) \) flops.+-- * Applying a Householder reflection to an \( m \times n \) matrix+--   ('applyHouseholderLeft', 'applyHouseholderRight'): \( O(mn) \) flops.+-- * Forming the explicit reflector matrix ('householderMatrix'): \( O(n^2) \)+--   flops, but this should be avoided in favour of implicit application+--   whenever possible.+module Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+  ( -- * Householder vector+    householderVector+    -- * Apply Householder reflection+  , applyHouseholderLeft+  , applyHouseholderRight+    -- * Construct explicit reflector+  , householderMatrix+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Compute a Householder vector (GVL4 Algorithm 5.1.1, p. 236).+--+-- Given a vector \( x \in \mathbb{R}^n \), compute the Householder vector+-- \( v \) and scalar \( \beta \) such that+--+-- \( (I - \beta \, v \, v^T) \, x = \| x \|_2 \, e_1 \)+--+-- where \( e_1 \) is the first standard basis vector.  By convention the+-- first component of \( v \) is normalised to \( v_1 = 1 \), which allows+-- it to be stored implicitly in the sub-diagonal part of a matrix during+-- QR factorisation.+--+-- The implementation follows GVL4 Algorithm 5.1.1 exactly, including the+-- careful treatment of the sign of \( x_1 \) to avoid catastrophic+-- cancellation.  When \( x \) is already a non-negative multiple of+-- \( e_1 \), the function returns \( \beta = 0 \) (i.e., the identity+-- transformation).+--+-- __Complexity:__ \( O(n) \) flops.+--+-- Returns @(v, beta)@.+householderVector :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+                  => Vector n r e -> (Vector n r e, e)+householderVector x =+  let nn = dimVal @n+      x0 = x !. 0+      -- σ = x(2:n)ᵀ · x(2:n)+      sigma = foldl' (\acc i -> acc + (x !. i) * (x !. i)) 0 [1..nn-1]+  in if sigma == 0 && x0 >= 0+    then -- x is already a positive multiple of e1+      ( makeVector @n @r $ \i -> if i == 0 then 1 else 0+      , 0+      )+    else if sigma == 0+    then -- x = -α·e₁+      ( makeVector @n @r $ \i -> if i == 0 then 1 else 0+      , 2+      )+    else+      let mu = sqrt (x0 * x0 + sigma)+          v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+          beta = 2 * v0 * v0 / (sigma + v0 * v0)+          v = makeVector @n @r $ \i ->+            if i == 0 then 1 else (x !. i) / v0+      in (v, beta)++-- | Apply a Householder reflection from the left (GVL4 Section 5.1, p. 236).+--+-- Given a Householder vector \( v \in \mathbb{R}^m \), scalar \( \beta \),+-- and matrix \( A \in \mathbb{R}^{m \times n} \), compute+--+-- \( A \leftarrow (I - \beta \, v \, v^T) \, A = A - \beta \, v \, (A^T v)^T \)+--+-- The computation is performed without forming \( P \) explicitly.+-- Instead, the intermediate vector \( w = \beta \, A^T v \) is computed+-- first, and then the rank-1 update \( A \leftarrow A - v \, w^T \) is+-- applied.  This is the standard technique described in GVL4+-- Section 5.1.4 (p. 238).+--+-- __Complexity:__ \( O(mn) \) flops.+applyHouseholderLeft :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+                     => Vector m r e -> e -> Matrix m n r e -> Matrix m n r e+applyHouseholderLeft v beta a =+  let mm = dimVal @m+      c  = dimVal @n+  in makeMatrix @m @n @r $ \i j ->+    let -- w = βAᵀv, w(j) = β · Σᵢ v(i)·A(i,j)+        wj = beta * foldl' (\acc k -> acc + (v !. k) * (a ! (k, j))) 0 [0..mm-1]+    in (a ! (i, j)) - (v !. i) * wj++-- | Apply a Householder reflection from the right (GVL4 Section 5.1, p. 236).+--+-- Given a matrix \( A \in \mathbb{R}^{m \times n} \), Householder vector+-- \( v \in \mathbb{R}^n \), and scalar \( \beta \), compute+--+-- \( A \leftarrow A \, (I - \beta \, v \, v^T) = A - \beta \, (A \, v) \, v^T \)+--+-- As with 'applyHouseholderLeft', the reflector is never formed+-- explicitly.  The intermediate vector \( w = \beta \, A \, v \) is+-- computed first, followed by the rank-1 update+-- \( A \leftarrow A - w \, v^T \).  See GVL4 Section 5.1.4 (p. 238).+--+-- __Complexity:__ \( O(mn) \) flops.+applyHouseholderRight :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Num e)+                      => Matrix m n r e -> Vector n r e -> e -> Matrix m n r e+applyHouseholderRight a v beta =+  let c = dimVal @n+  in makeMatrix @m @n @r $ \i j ->+    let -- w = β·A·v, w(i) = β · Σⱼ A(i,j)·v(j)+        wi = beta * foldl' (\acc k -> acc + (a ! (i, k)) * (v !. k)) 0 [0..c-1]+    in (a ! (i, j)) - wi * (v !. j)++-- | Construct the explicit Householder reflector matrix (GVL4 Section 5.1, p. 236).+--+-- Given a Householder vector \( v \in \mathbb{R}^n \) and scalar+-- \( \beta \), form the \( n \times n \) matrix+--+-- \( H = I - \beta \, v \, v^T \)+--+-- The resulting matrix is both symmetric and orthogonal:+-- \( H = H^T = H^{-1} \).+--+-- __Note:__ In most numerical algorithms it is preferable to apply the+-- Householder transformation implicitly via 'applyHouseholderLeft' or+-- 'applyHouseholderRight' rather than forming \( H \) explicitly.+-- Forming the explicit matrix costs \( O(n^2) \) flops and storage, and+-- subsequent multiplication with it costs \( O(n^3) \) rather than the+-- \( O(mn) \) achievable by implicit application.+--+-- __Complexity:__ \( O(n^2) \) flops.+householderMatrix :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+                  => Vector n r e -> e -> Matrix n n r e+householderMatrix v beta =+  makeMatrix @n @n @r $ \i j ->+    let ident = if i == j then 1 else 0+    in ident - beta * (v !. i) * (v !. j)
+ src/Numeric/LinearAlgebra/Massiv/Orthogonal/LeastSquares.hs view
@@ -0,0 +1,133 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Linear least squares solvers.+--+-- This module provides two methods for solving the linear least squares+-- problem+--+-- \( \min_x \| A x - b \|_2 \)+--+-- following Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4),+-- Section 5.3, pp. 260--270.+--+-- __Theorem 5.3.1 (Least squares existence, GVL4 p. 260).__  Let+-- \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \) and+-- \( \operatorname{rank}(A) = n \).  Then the least squares problem+-- \( \min_x \| A x - b \|_2 \) has a unique solution \( x^* \) given+-- by the /normal equations/+--+-- \( A^T A \, x = A^T b \)+--+-- Two solution methods are provided:+--+-- * __QR-based__ ('leastSquaresQR') -- GVL4 Algorithm 5.3.2 (p. 262).+--   Factor \( A = Q R \) via Householder QR, then solve+--   \( R_1 x = (Q^T b)_{1:n} \) by back-substitution, where \( R_1 \)+--   denotes the leading \( n \times n \) upper triangular block of+--   \( R \).  This is the recommended method: it is numerically stable+--   and does not square the condition number.+--+-- * __Normal equations__ ('leastSquaresNormal') -- GVL4 Section 5.3.2+--   (p. 261).  Form \( A^T A \) and \( A^T b \) explicitly, then solve+--   via Cholesky factorisation.  This is faster but squares the+--   condition number: \( \kappa_2(A^T A) = \kappa_2(A)^2 \), so it+--   should only be used when \( A \) is well-conditioned.+--+-- __Complexity.__+--+-- * QR-based: \( 2mn^2 \) flops (dominated by the QR factorisation).+-- * Normal equations: \( mn^2 + \tfrac{1}{3}n^3 \) flops+--   (\( mn^2 \) for forming \( A^T A \), \( \tfrac{1}{3}n^3 \) for+--   Cholesky).+module Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+  ( -- * Least squares via QR+    leastSquaresQR+    -- * Normal equations+  , leastSquaresNormal+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR (qr)+import Numeric.LinearAlgebra.Massiv.Solve.Triangular (backSub)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose)+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.Solve.Cholesky (choleskySolve)++-- | Solve the least squares problem \( \min_x \| A x - b \|_2 \) via QR+-- factorisation (GVL4 Algorithm 5.3.2, p. 262).+--+-- Given \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \) and full+-- column rank, and a right-hand side \( b \in \mathbb{R}^m \), compute+-- the unique least squares solution \( x^* \in \mathbb{R}^n \).+--+-- The algorithm proceeds in three steps:+--+-- 1. Factor \( A = Q R \) using Householder QR ('qr').+-- 2. Form the transformed right-hand side \( Q^T b \).+-- 3. Solve the \( n \times n \) upper triangular system+--    \( R_1 x = (Q^T b)_{1:n} \) by back-substitution, where \( R_1 \)+--    is the leading \( n \times n \) block of \( R \).+--+-- This method is numerically stable because orthogonal transformations+-- preserve the 2-norm and do not amplify rounding errors.  The condition+-- number relevant to the solution is \( \kappa_2(A) \), not+-- \( \kappa_2(A)^2 \) as with the normal equations.+--+-- __Complexity:__ \( 2mn^2 - \tfrac{2}{3}n^3 \) flops for the QR+-- factorisation plus \( 2mn \) flops for forming \( Q^T b \) and+-- \( n^2 \) flops for back-substitution, giving a total of+-- \( O(2mn^2) \) flops.+leastSquaresQR :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+               => Matrix m n r e -> Vector m r e -> Vector n r e+leastSquaresQR a b =+  let mm = dimVal @m+      nn = dimVal @n+      (q, r) = qr a+      qt = transpose q+      -- qtb = Qᵀ·b (dimension m)+      qtb = matvec qt b+      -- Extract top n×n of R (which is upper triangular)+      r1 = makeMatrix @n @n @r $ \i j -> r ! (i, j)+      -- Extract top n entries of Qᵀb+      qtb1 = makeVector @n @r $ \i -> qtb !. i+  in backSub r1 qtb1++-- | Solve the least squares problem \( \min_x \| A x - b \|_2 \) via the+-- normal equations (GVL4 Section 5.3.2, p. 261).+--+-- Given \( A \in \mathbb{R}^{m \times n} \) with full column rank and+-- \( b \in \mathbb{R}^m \), form the \( n \times n \) symmetric positive+-- definite system+--+-- \( A^T A \, x = A^T b \)+--+-- and solve it using Cholesky factorisation (\( A^T A = L L^T \)).+--+-- __Warning:__ The normal equations square the condition number of \( A \):+-- \( \kappa_2(A^T A) = \kappa_2(A)^2 \).  For ill-conditioned problems+-- this leads to a significant loss of accuracy compared to QR-based+-- methods.  Prefer 'leastSquaresQR' unless \( A \) is known to be+-- well-conditioned (GVL4 p. 261).+--+-- __Complexity:__ \( mn^2 \) flops for forming \( A^T A \), plus+-- \( \tfrac{1}{3}n^3 \) flops for the Cholesky factorisation, giving a+-- total of \( O(mn^2 + \tfrac{1}{3}n^3) \) flops.+leastSquaresNormal :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+                   => Matrix m n r e -> Vector m r e -> Vector n r e+leastSquaresNormal a b =+  let at = transpose a+      ata = matMul at a       -- n×n, symmetric positive definite+      atb = matvec at b       -- n×1+  in choleskySolve ata atb
+ src/Numeric/LinearAlgebra/Massiv/Orthogonal/QR.hs view
@@ -0,0 +1,481 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}+{-# LANGUAGE BangPatterns #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Orthogonal.QR+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- QR factorisation via Householder reflections and Givens rotations.+--+-- This module implements the QR factorisation of a matrix+-- \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \), following+-- Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4),+-- Section 5.2, pp. 246--260.+--+-- __Theorem 5.2.1 (QR existence, GVL4 p. 246).__  For any+-- \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \), there exists+-- an orthogonal matrix \( Q \in \mathbb{R}^{m \times m} \) and an upper+-- triangular matrix \( R \in \mathbb{R}^{m \times n} \) such that+--+-- \( A = Q \, R \)+--+-- If \( A \) has full column rank, the factorisation is unique up to+-- sign changes in the rows of \( R \) (equivalently, columns of \( Q \)).+--+-- Two algorithms are provided:+--+-- * __Householder QR__ ('qr', 'qrR') -- GVL4 Algorithm 5.2.1 (p. 249).+--   A sequence of Householder reflections \( H_1, H_2, \ldots, H_n \)+--   is applied to \( A \) from the left to produce+--   \( H_n \cdots H_2 \, H_1 \, A = R \), so that+--   \( Q = H_1 \, H_2 \cdots H_n \).+--+-- * __Givens QR__ ('qrGivens') -- GVL4 Section 5.2.4 (p. 252).+--   A sequence of Givens rotations zeroes out sub-diagonal entries one+--   at a time.  This variant is preferred for sparse or banded matrices,+--   particularly Hessenberg matrices, where the number of rotations is+--   proportional to the bandwidth rather than the matrix dimension.+--+-- __Complexity.__+--+-- * Householder QR: \( 2mn^2 - \tfrac{2}{3}n^3 \) flops (GVL4 p. 249).+-- * Givens QR: \( 3mn^2 - n^3 \) flops for a dense matrix, but+--   significantly fewer for structured (e.g., Hessenberg) matrices.+--+-- __Optimisation.__  The implementation uses in-place mutable arrays via+-- the 'ST' monad, storing Householder vectors implicitly in the+-- subdiagonal of the working matrix (the LAPACK convention).  The+-- orthogonal factor \( Q \) is formed via backward accumulation, and+-- 'qrR' avoids forming \( Q \) entirely.+module Numeric.LinearAlgebra.Massiv.Orthogonal.QR+  ( -- * QR factorisation (Householder)+    qr+  , qrP+  , qrR+    -- * QR factorisation (Givens)+  , qrGivens+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)+import Control.Monad (when)+import Control.Monad.ST (ST)+import GHC.Exts+import GHC.ST (ST(..))+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..), newByteArray, unsafeFreezeByteArray)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+  (givensRotation)+import Numeric.LinearAlgebra.Massiv.Internal.Kernel+  (rawMutSumSqColumn, rawMutSumProdColumns, rawMutHouseholderApply, rawMutQAccum,+   rawGemmKernel, rawZeroDoubles, rawNegateDoubles)++-- | Full QR factorisation via Householder reflections (GVL4 Algorithm 5.2.1, p. 249).+--+-- Given \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \), compute+-- the factorisation+--+-- \( A = Q \, R \)+--+-- where \( Q \in \mathbb{R}^{m \times m} \) is orthogonal and+-- \( R \in \mathbb{R}^{m \times n} \) is upper triangular.+--+-- The implementation uses in-place mutable arrays: the Householder+-- vectors are stored in the subdiagonal of the working copy of \( A \)+-- (LAPACK convention), and \( Q \) is formed via backward accumulation.+-- Total allocation: two matrices (one for \( R \), one for \( Q \)),+-- plus a small vector of \( \beta \) scalars.+--+-- __Complexity:__ \( 2mn^2 - \tfrac{2}{3}n^3 \) flops for the+-- triangularisation, plus \( 2m^2 n - \tfrac{2}{3}n^3 \) flops for+-- accumulating \( Q \) (GVL4 p. 249).+--+-- See also 'qrR' when only \( R \) is needed, and 'qrGivens' for a+-- Givens-based alternative.+qr :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+   => Matrix m n r e -> (Matrix m m r e, Matrix m n r e)+qr a =+  let mm = dimVal @m+      nn = dimVal @n+      steps = min mm nn++      -- Phase 1: In-place Householder triangularisation.+      -- After this, rArr holds R in the upper triangle and Householder+      -- vectors v_k in the subdiagonal of column k (with v_k(k) = 1 implicit).+      -- betaList holds the β scalars as a Haskell list.+      (betaList, rArr) = M.withMArrayST (unMatrix a) $ \mr -> do+        betas <- mapM (\k -> do+          -- Compute Householder vector for column k, rows k..m-1+          x0 <- M.readM mr (k :. k)+          sigma <- sumSqRange mr k mm k  -- σ = Σ R(i,k)² for i=k+1..m-1+          if sigma == 0 && x0 >= 0+            then pure 0     -- already in desired form+            else do+              let mu = sqrt (x0 * x0 + sigma)+                  v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                  beta = 2 * v0 * v0 / (sigma + v0 * v0)+              -- Store v_k in subdiagonal of column k (v_k(k) = 1 is implicit)+              -- Scale entries: v(i) = R(i,k) / v0 for i > k+              mapM_ (\i -> do+                rik <- M.readM mr (i :. k)+                M.write_ mr (i :. k) (rik / v0)+                ) [k+1..mm-1]+              -- Set diagonal: R(k,k) = μ (result of H_k applied to column k)+              M.write_ mr (k :. k) mu+              -- Apply H_k from the left to columns k+1..n-1 of R:+              -- (Column k is skipped: its diagonal is μ, subdiagonal stores v)+              -- For each column j: w_j = β(R(k,j) + Σ_{i>k} v(i)·R(i,j))+              --                    R(k,j) -= w_j; R(i,j) -= v(i)·w_j+              mapM_ (\j -> do+                rkj <- M.readM mr (k :. j)+                wj <- sumProdRange mr mr k mm k j+                let wj' = beta * (rkj + wj)+                M.write_ mr (k :. j) (rkj - wj')+                mapM_ (\i -> do+                  vi <- M.readM mr (i :. k)+                  rij <- M.readM mr (i :. j)+                  M.write_ mr (i :. j) (rij - vi * wj')+                  ) [k+1..mm-1]+                ) [k+1..nn-1]+              pure beta+          ) [0..steps-1]+        pure betas++      -- Phase 2: Backward accumulation of Q.+      -- Start with Q = I, then for k = steps-1 downto 0:+      --   Apply H_k from the right: Q <- Q·(I - β_k·v_k·v_k^T)+      qMat = createMatrix @m @m @r $ \mq -> do+        -- Initialize Q = I+        mapM_ (\i -> mapM_ (\j ->+          M.write_ mq (i :. j) (if i == j then 1 else 0)+          ) [0..mm-1]) [0..mm-1]+        -- Forward accumulation: Q <- Q·H_0·H_1·…·H_{n-1}+        mapM_ (\k -> do+          let beta_k = betaList !! k+          if beta_k == 0 then pure ()+          else+            -- Apply (I - β·v·v^T) from the right to Q+            -- For each row i: w_i = β·(Q(i,k) + Σ_{l>k} Q(i,l)·v(l))+            --                 Q(i,k) -= w_i; Q(i,l) -= w_i·v(l)+            mapM_ (\i -> do+              qik <- M.readM mq (i :. k)+              wi <- qvProd mq rArr i k mm+              let wi' = beta_k * (qik + wi)+              M.write_ mq (i :. k) (qik - wi')+              mapM_ (\l -> do+                let vl = M.index' rArr (l :. k)+                qil <- M.readM mq (i :. l)+                M.write_ mq (i :. l) (qil - wi' * vl)+                ) [k+1..mm-1]+              ) [0..mm-1]+          ) [0..steps-1]++      -- Extract clean R (zero out subdiagonal, which holds Householder vectors)+      rClean = makeMatrix @m @n @r $ \i j ->+        if i <= j then M.index' rArr (i :. j) else 0++  in (qMat, rClean)++-- | Specialised QR factorisation for @P Double@ using raw ByteArray# primops.+qrP :: forall m n. (KnownNat m, KnownNat n)+    => Matrix m n M.P Double -> (Matrix m m M.P Double, Matrix m n M.P Double)+qrP a =+  let mm = dimVal @m+      nn = dimVal @n+      steps = min mm nn++      -- Phase 1: In-place Householder triangularisation using raw kernels.+      (betaList, rArr) = M.withMArrayST (unMatrix a) $ \mr -> do+        let mbaR = unwrapMutableByteArray mr+            offR = unwrapMutableByteArrayOffset mr+        betas <- mapM (\k -> do+          -- Read x0 = R[k,k]+          x0 <- M.readM mr (k :. k)+          -- σ = Σ R[i,k]² for i = k+1..m-1 (using raw kernel)+          sigma <- rawMutSumSqColumn mbaR offR nn (k + 1) mm k+          if sigma == 0 && x0 >= 0+            then pure 0+            else do+              let mu = sqrt (x0 * x0 + sigma)+                  v0 = if x0 <= 0 then x0 - mu else -sigma / (x0 + mu)+                  beta = 2 * v0 * v0 / (sigma + v0 * v0)+              -- Scale v: v(i) = R(i,k) / v0 for i > k+              scaleColumn mbaR offR nn (k + 1) mm k (1.0 / v0)+              -- Set diagonal+              M.write_ mr (k :. k) mu+              -- Apply H_k to columns k+1..n-1 using raw kernel+              mapM_ (\j ->+                rawMutHouseholderApply mbaR offR nn beta k mm j+                ) [k+1..nn-1]+              pure beta+          ) [0..steps-1]+        pure betas++      -- Phase 2: Blocked WY Q accumulation using GEMM.+      -- Group Householder vectors into panels of size nb, compute T-factor+      -- via GEMM, and apply block reflectors via Level-3 GEMM.+      qMat = createMatrix @m @m @M.P $ \mq -> do+        let mbaQ = unwrapMutableByteArray mq+            offQ = unwrapMutableByteArrayOffset mq+            baR = unwrapByteArray rArr+            offFR = unwrapByteArrayOffset rArr+        -- Initialize Q = I via rawZeroDoubles + diagonal writes+        rawZeroDoubles mbaQ offQ (mm * mm)+        mapM_ (\i -> writeRawD mbaQ offQ (i * mm + i) 1) [0..mm-1]++        if steps <= 16+          then+            -- Small matrix: per-row accumulation (Level-2)+            mapM_ (\k -> do+              let beta_k = betaList !! k+              if beta_k == 0 then pure ()+              else+                mapM_ (\i ->+                  rawMutQAccum mbaQ offQ mm baR offFR nn beta_k k mm i+                  ) [0..mm-1]+              ) [0..steps-1]+          else do+            -- Blocked WY: batch nb Householder vectors at a time+            let !nb = min 32 steps+            mbaBetas <- newByteArray (steps * 8)+            mapM_ (\(i, b) -> writeRawD mbaBetas 0 i b) (zip [0..] betaList)+            mbaY  <- newByteArray (mm * nb * 8)+            mbaTf <- newByteArray (nb * nb * 8)+            mbaW1 <- newByteArray (mm * nb * 8)+            mbaW2 <- newByteArray (mm * nb * 8)+            mbaYT <- newByteArray (nb * mm * 8)+            mbaG  <- newByteArray (nb * nb * 8)++            let goBlock !k0+                  | k0 >= steps = pure ()+                  | otherwise = do+                      let !bs = min nb (steps - k0)+                      -- Pack Y (mm × bs): Y[:,j] = v_{k0+j}+                      -- v_k has implicit 1 at position k, stored values at k+1..mm-1+                      rawZeroDoubles mbaY 0 (mm * bs)+                      mapM_ (\j -> do+                        let !k = k0 + j+                        writeRawD mbaY 0 (k * bs + j) 1.0+                        mapM_ (\l ->+                          writeRawD mbaY 0 (l * bs + j) (indexRawD baR offFR (l * nn + k))+                          ) [k+1..mm-1]+                        ) [0..bs-1]++                      -- Transpose Y → Y^T (bs × mm) for GEMM reuse+                      rawZeroDoubles mbaYT 0 (bs * mm)+                      mapM_ (\j -> do+                        let !k = k0 + j+                        writeRawD mbaYT 0 (j * mm + k) 1.0+                        mapM_ (\l ->+                          writeRawD mbaYT 0 (j * mm + l) (indexRawD baR offFR (l * nn + k))+                          ) [k+1..mm-1]+                        ) [0..bs-1]++                      baY  <- unsafeFreezeByteArray mbaY+                      baYT <- unsafeFreezeByteArray mbaYT++                      -- Compute G = Y^T × Y (bs × bs) via GEMM+                      rawZeroDoubles mbaG 0 (bs * bs)+                      rawGemmKernel baYT 0 baY 0 mbaG 0 bs mm bs++                      -- Build T-factor (bs × bs upper-triangular)+                      rawZeroDoubles mbaTf 0 (bs * bs)+                      mapM_ (\j -> do+                        betaj <- readRawD mbaBetas 0 (k0 + j)+                        writeRawD mbaTf 0 (j * bs + j) betaj+                        when (j > 0 && betaj /= 0) $ do+                          -- T[0..j-1, j] = -betaj * T[0..j-1, 0..j-1] * G[0..j-1, j]+                          mapM_ (\i -> do+                            g_ij <- readRawD mbaG 0 (i * bs + j)+                            writeRawD mbaW1 0 i g_ij+                            ) [0..j-1]+                          mapM_ (\i -> do+                            let triLoop !l !acc+                                  | l >= j = pure acc+                                  | otherwise = do+                                      til <- readRawD mbaTf 0 (i * bs + l)+                                      dl  <- readRawD mbaW1 0 l+                                      triLoop (l+1) (acc + til * dl)+                            z <- triLoop 0 0+                            writeRawD mbaTf 0 (i * bs + j) (negate betaj * z)+                            ) [0..j-1]+                        ) [0..bs-1]++                      -- W1 = Q · Y (mm×mm * mm×bs → mm×bs)+                      baQ <- unsafeFreezeByteArray mbaQ+                      rawZeroDoubles mbaW1 0 (mm * bs)+                      rawGemmKernel baQ offQ baY 0 mbaW1 0 mm mm bs++                      -- W2 = W1 · T (mm×bs * bs×bs → mm×bs)+                      baW1 <- unsafeFreezeByteArray mbaW1+                      baTf <- unsafeFreezeByteArray mbaTf+                      rawZeroDoubles mbaW2 0 (mm * bs)+                      rawGemmKernel baW1 0 baTf 0 mbaW2 0 mm bs bs++                      -- Negate W2 in-place+                      rawNegateDoubles mbaW2 0 (mm * bs)++                      -- Q += (-W2) · Y^T (mm×bs * bs×mm → mm×mm)+                      baNW2 <- unsafeFreezeByteArray mbaW2+                      rawGemmKernel baNW2 0 baYT 0 mbaQ offQ mm bs mm++                      goBlock (k0 + bs)+            goBlock 0++      -- Extract clean R+      rClean = makeMatrix @m @n @M.P $ \i j ->+        if i <= j then M.index' rArr (i :. j) else 0++  in (qMat, rClean)+{-# NOINLINE qrP #-}++-- | Scale elements in a column of a mutable matrix: A[i,col] *= scale for i in [start..end-1].+scaleColumn :: MutableByteArray s -> Int -> Int -> Int -> Int -> Int -> Double -> ST s ()+scaleColumn (MutableByteArray mba) (I# off) (I# ncols) (I# start) (I# end) (I# col) (D# scale) = ST $ \s0 ->+  let go i s+        | isTrue# (i >=# end) = s+        | otherwise =+            case readDoubleArray# mba (off +# i *# ncols +# col) s of+              (# s', v #) ->+                case writeDoubleArray# mba (off +# i *# ncols +# col) (v *## scale) s' of+                  s'' -> go (i +# 1#) s''+  in (# go start s0, () #)+{-# INLINE scaleColumn #-}++-- | Helper: Σ R(i,col)² for i=start+1..end-1  (sum of squares below diagonal)+sumSqRange :: (M.Manifest r e, Num e) => M.MArray s r Ix2 e -> Int -> Int -> Int -> ST s e+sumSqRange mr start end col = go (start + 1) 0+  where+    go i !acc+      | i >= end = pure acc+      | otherwise = do+          v <- M.readM mr (i :. col)+          go (i + 1) (acc + v * v)++-- | Helper: Σ mr1(i,c1)·mr2(i,c2) for i=start+1..end-1+sumProdRange :: (M.Manifest r e, Num e)+             => M.MArray s r Ix2 e -> M.MArray s r Ix2 e+             -> Int -> Int -> Int -> Int -> ST s e+sumProdRange mr1 mr2 start end c1 c2 = go (start + 1) 0+  where+    go i !acc+      | i >= end = pure acc+      | otherwise = do+          v1 <- M.readM mr1 (i :. c1)+          v2 <- M.readM mr2 (i :. c2)+          go (i + 1) (acc + v1 * v2)++-- | Helper: Σ Q(row,l)·v(l) for l=start+1..end-1 where v is stored in rArr subdiag of col start+qvProd :: (M.Manifest r1 e, M.Manifest r2 e, Num e)+       => M.MArray s r1 Ix2 e -> M.Array r2 Ix2 e -> Int -> Int -> Int -> ST s e+qvProd mq rArr row start end = go (start + 1) 0+  where+    go l !acc+      | l >= end = pure acc+      | otherwise = do+          qrl <- M.readM mq (row :. l)+          let vl = M.index' rArr (l :. start)+          go (l + 1) (acc + qrl * vl)++-- | Compute only the upper triangular factor \( R \) from the QR+-- factorisation, without explicitly forming \( Q \)+-- (GVL4 Algorithm 5.2.1, p. 249).+--+-- Unlike 'qr', this function never forms the orthogonal factor,+-- saving \( O(m^2 n) \) flops.  The Householder vectors are computed+-- and applied in-place but discarded.+--+-- __Complexity:__ \( 2mn^2 - \tfrac{2}{3}n^3 \) flops.+qrR :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+    => Matrix m n r e -> Matrix m n r e+qrR a = snd (qr a)++-- | QR factorisation via Givens rotations (GVL4 Section 5.2.4, p. 252).+--+-- Given \( A \in \mathbb{R}^{m \times n} \) with \( m \ge n \), compute+-- the factorisation \( A = Q \, R \) by applying a sequence of Givens+-- rotations to zero out sub-diagonal entries one at a time.+--+-- The implementation computes \( R \) in-place via the 'ST' monad,+-- records the rotation parameters, then applies them to form \( Q \)+-- in a second in-place pass.+--+-- __Complexity:__ \( 3mn^2 - n^3 \) flops for a dense \( m \times n \)+-- matrix; \( O(mn) \) flops for an upper Hessenberg matrix+-- (GVL4 p. 253).+qrGivens :: forall m n r e. (KnownNat m, KnownNat n, M.Manifest r e, Floating e, Ord e)+          => Matrix m n r e -> (Matrix m m r e, Matrix m n r e)+qrGivens a =+  let mm = dimVal @m+      nn = dimVal @n+      steps = min mm nn++      -- Pass 1: compute R in-place, recording Givens rotations+      (rots, rArr) = M.withMArrayST (unMatrix a) $ \mr -> do+        let go j !acc+              | j >= steps = pure acc+              | otherwise = do+                  acc' <- goRows j (j+1) acc mr+                  go (j+1) acc'+            goRows j i !acc mr_+              | i >= mm = pure acc+              | otherwise = do+                  aij <- M.readM mr_ (i :. j)+                  if aij == 0 then goRows j (i+1) acc mr_+                  else do+                    ajj <- M.readM mr_ (j :. j)+                    let (c, s) = givensRotation ajj aij+                    -- Apply G^T to rows j and i+                    mapM_ (\col -> do+                      rjc <- M.readM mr_ (j :. col)+                      ric <- M.readM mr_ (i :. col)+                      M.write_ mr_ (j :. col) (c * rjc - s * ric)+                      M.write_ mr_ (i :. col) (s * rjc + c * ric)+                      ) [0..nn-1]+                    goRows j (i+1) (acc ++ [(c, s, j, i)]) mr_+        go 0 []++      -- Pass 2: form Q by applying recorded rotations to I+      qMat = createMatrix @m @m @r $ \mq -> do+        -- Initialize Q = I+        mapM_ (\i -> mapM_ (\j ->+          M.write_ mq (i :. j) (if i == j then 1 else 0)+          ) [0..mm-1]) [0..mm-1]+        -- Apply each rotation from the right: Q <- Q·G+        mapM_ (\(c, s, ci, ck) ->+          mapM_ (\row -> do+            qrc <- M.readM mq (row :. ci)+            qrk <- M.readM mq (row :. ck)+            M.write_ mq (row :. ci) (c * qrc - s * qrk)+            M.write_ mq (row :. ck) (s * qrc + c * qrk)+            ) [0..mm-1]+          ) rots++  in (qMat, MkMatrix rArr)++-- Raw ByteArray# helpers for blocked WY Q accumulation+readRawD :: MutableByteArray s -> Int -> Int -> ST s Double+readRawD (MutableByteArray mba) (I# off) (I# i) = ST $ \s ->+  case readDoubleArray# mba (off +# i) s of+    (# s', v #) -> (# s', D# v #)+{-# INLINE readRawD #-}++writeRawD :: MutableByteArray s -> Int -> Int -> Double -> ST s ()+writeRawD (MutableByteArray mba) (I# off) (I# i) (D# v) = ST $ \s ->+  case writeDoubleArray# mba (off +# i) v s of+    s' -> (# s', () #)+{-# INLINE writeRawD #-}++indexRawD :: ByteArray -> Int -> Int -> Double+indexRawD (ByteArray ba) (I# off) (I# i) =+  D# (indexDoubleArray# ba (off +# i))+{-# INLINE indexRawD #-}
+ src/Numeric/LinearAlgebra/Massiv/Solve/Banded.hs view
@@ -0,0 +1,342 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Solve.Banded+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = Banded and Tridiagonal System Solvers+--+-- Specialised LU and Cholesky factorizations for banded matrices, plus a+-- dedicated tridiagonal solver, following Golub & Van Loan,+-- /Matrix Computations/, 4th edition (GVL4), Section 4.3, pp. 174--182.+--+-- A matrix \(A \in \mathbb{R}^{n \times n}\) has /lower bandwidth/ \(p\)+-- and /upper bandwidth/ \(q\) when \(a_{ij} = 0\) for \(i > j + p\) or+-- \(j > i + q\). Exploiting this structure reduces the factorization cost+-- from \(O(n^3)\) to \(O(npq)\) (GVL4 p. 176), and the triangular-solve+-- cost from \(O(n^2)\) to \(O(np)\) or \(O(nq)\). The important special+-- case of a tridiagonal system (\(p = q = 1\)) is solvable in \(O(n)\)+-- flops.+--+-- +-------------------+-----------------------------------+----------------------------------++-- | Function          | Algorithm                         | Reference                        |+-- +===================+===================================+==================================++-- | 'bandLU'          | Band Gaussian elimination          | GVL4 Algorithm 4.3.1, p. 175    |+-- +-------------------+-----------------------------------+----------------------------------++-- | 'bandForwardSub'  | Band forward substitution          | GVL4 Algorithm 4.3.2, p. 176    |+-- +-------------------+-----------------------------------+----------------------------------++-- | 'bandBackSub'     | Band back substitution             | GVL4 Algorithm 4.3.3, p. 176    |+-- +-------------------+-----------------------------------+----------------------------------++-- | 'bandCholesky'    | Band Cholesky factorization        | GVL4 Algorithm 4.3.5, p. 178    |+-- +-------------------+-----------------------------------+----------------------------------++-- | 'tridiagSolve'    | SPD tridiagonal solver             | GVL4 Algorithm 4.3.6, p. 179    |+-- +-------------------+-----------------------------------+----------------------------------++--+-- == Complexity+--+-- * Band LU: \(O(npq)\) flops (GVL4 p. 176).+-- * Band triangular solve: \(O(np)\) or \(O(nq)\) flops (GVL4 p. 176).+-- * Band Cholesky: \(O(np^2)\) flops (GVL4 p. 178).+-- * Tridiagonal solver: \(O(n)\) flops (GVL4 p. 179).+--+-- == Type Safety+--+-- Matrix dimensions are tracked at the type level via 'KnownNat'. The+-- bandwidths \(p\) and \(q\) are passed as run-time 'Int' values because+-- they are often data-dependent. The matrix is stored in standard dense+-- format; only the band is accessed.+module Numeric.LinearAlgebra.Massiv.Solve.Banded+  ( -- * Band LU factorization+    bandLU+    -- * Band triangular solve+  , bandForwardSub+  , bandBackSub+    -- * Band Cholesky (\(A = GG^T\))+  , bandCholesky+    -- * Tridiagonal solver (\(Ax = b\), \(p = q = 1\))+  , tridiagSolve+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Band Gaussian elimination without pivoting (GVL4 Algorithm 4.3.1,+-- p. 175).+--+-- Given an \(n \times n\) matrix \(A\) with lower bandwidth \(p\) and upper+-- bandwidth \(q\), computes an in-place \(LU\) factorization where \(L\)+-- has lower bandwidth \(p\) and \(U\) has upper bandwidth \(q\).+--+-- The matrix is stored in standard dense format; only the entries within+-- the band are accessed or modified.+--+-- __Precondition.__ All leading principal submatrices of \(A\) must be+-- nonsingular (analogous to the dense case, GVL4 Theorem 3.2.1, p. 116).+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) statically ensures the matrix is square. The bandwidth+-- parameters \(p\) and \(q\) are run-time values.+--+-- ==== Complexity+--+-- \(O(npq)\) flops (GVL4 p. 176).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.3.1+-- (Band Gaussian Elimination), p. 175.+bandLU :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+       => Int  -- ^ Lower bandwidth \(p\)+       -> Int  -- ^ Upper bandwidth \(q\)+       -> Matrix n n r e -> Matrix n n r e+bandLU p q (MkMatrix a) =+  let nn = dimVal @n+  in MkMatrix $ snd $ M.withMArrayST a $ \ma ->+    mapM_ (\k -> do+      akk <- M.readM ma (k :. k)+      let imax = min (k + p) (nn - 1)+      -- Compute multipliers+      mapM_ (\i -> do+        aik <- M.readM ma (i :. k)+        M.write_ ma (i :. k) (aik / akk)+        ) [k+1..imax]+      -- Update+      let jmax = min (k + q) (nn - 1)+      mapM_ (\j ->+        mapM_ (\i -> do+          aij <- M.readM ma (i :. j)+          aik <- M.readM ma (i :. k)+          akj <- M.readM ma (k :. j)+          M.write_ ma (i :. j) (aij - aik * akj)+          ) [k+1..imax]+        ) [k+1..jmax]+      ) [0..nn-2]++-- | Band forward substitution (GVL4 Algorithm 4.3.2, p. 176).+--+-- Solves \(Lx = b\) where \(L\) is /unit/ lower triangular with lower+-- bandwidth \(p\). Only the \(p\) subdiagonals are accessed; the unit+-- diagonal is implicit, so no division is performed and the constraint+-- relaxes to 'Num'.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) ensures the dimensions of \(L\) and \(b\) agree at+-- compile time.+--+-- ==== Complexity+--+-- \(O(np)\) flops (GVL4 p. 176).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.3.2+-- (Band Forward Substitution), p. 176.+bandForwardSub :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+               => Int  -- ^ Lower bandwidth \(p\)+               -> Matrix n n r e -> Vector n r e -> Vector n r e+bandForwardSub p l b = createVector @n $ \mx -> do+  let nn = dimVal @n+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  mapM_ (\j -> do+    xj <- M.readM mx j+    let imax = min (j + p) (nn - 1)+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (l ! (i, j)) * xj)+      ) [j+1..imax]+    ) [0..nn-1]++-- | Band back substitution (GVL4 Algorithm 4.3.3, p. 176).+--+-- Solves \(Ux = b\) where \(U\) is upper triangular with upper bandwidth+-- \(q\). Only the diagonal and the \(q\) superdiagonals are accessed.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) ensures the dimensions of \(U\) and \(b\) agree at+-- compile time.+--+-- ==== Complexity+--+-- \(O(nq)\) flops (GVL4 p. 176).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.3.3+-- (Band Back Substitution), p. 176.+bandBackSub :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+            => Int  -- ^ Upper bandwidth \(q\)+            -> Matrix n n r e -> Vector n r e -> Vector n r e+bandBackSub q u b = createVector @n $ \mx -> do+  let nn = dimVal @n+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  mapM_ (\j -> do+    xj <- M.readM mx j+    let ujj = u ! (j, j)+        xj' = xj / ujj+    M.write_ mx j xj'+    let imin = max 0 (j - q)+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (u ! (i, j)) * xj')+      ) [imin..j-1]+    ) [nn-1, nn-2..0]++-- | Band Cholesky factorization (GVL4 Algorithm 4.3.5, p. 178).+--+-- Given a symmetric positive definite \(n \times n\) matrix \(A\) with+-- lower bandwidth \(p\), computes the lower triangular factor \(G\) (also+-- with bandwidth \(p\)) such that+--+-- \[+--   A = G G^T+-- \]+--+-- Only the lower band of \(A\) (entries \(a_{ij}\) with+-- \(0 \le i - j \le p\)) is read.+--+-- __Precondition.__ \(A\) must be symmetric positive definite. If this+-- condition is violated, the algorithm may encounter a negative value under+-- a square root and produce NaN.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) statically ensures the matrix is square. The bandwidth+-- \(p\) is a run-time value.+--+-- ==== Complexity+--+-- \(O(np^2)\) flops (GVL4 p. 178).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.3.5+-- (Band Cholesky), p. 178.+bandCholesky :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+             => Int  -- ^ Bandwidth \(p\)+             -> Matrix n n r e -> Matrix n n r e+bandCholesky p (MkMatrix a) =+  let nn = dimVal @n+  in MkMatrix $ M.createArrayST_ (M.Sz2 nn nn) $ \mg -> do+    -- Initialize to zero+    mapM_ (\i -> mapM_ (\j -> M.write_ mg (i :. j) 0) [0..nn-1]) [0..nn-1]+    -- Copy lower band of A+    mapM_ (\j ->+      let imax = min (j + p) (nn - 1)+      in mapM_ (\i -> M.write_ mg (i :. j) (M.index' a (i :. j))) [j..imax]+      ) [0..nn-1]+    -- Band Cholesky+    mapM_ (\j -> do+      -- Subtract contributions+      let kmin = max 0 (j - p)+      mapM_ (\k -> do+        gjk <- M.readM mg (j :. k)+        let lam = min (j + p) (nn - 1)+        mapM_ (\i -> do+          gij <- M.readM mg (i :. j)+          gik <- M.readM mg (i :. k)+          M.write_ mg (i :. j) (gij - gik * gjk)+          ) [j..lam]+        ) [kmin..j-1]+      -- Scale+      gjj <- M.readM mg (j :. j)+      let sjj = sqrt gjj+          lam = min (j + p) (nn - 1)+      mapM_ (\i -> do+        gij <- M.readM mg (i :. j)+        M.write_ mg (i :. j) (gij / sjj)+        ) [j..lam]+      ) [0..nn-1]++-- | Symmetric positive definite tridiagonal system solver (GVL4+-- Algorithm 4.3.6, p. 179).+--+-- Solves \(Ax = b\) where \(A\) is symmetric, tridiagonal, and positive+-- definite. The matrix \(A\) is specified compactly by its diagonal+-- \(\alpha_{1:n}\) and its superdiagonal \(\beta_{1:n-1}\).+--+-- The algorithm computes the \(LDL^T\) factorization of \(A\) and folds it+-- together with forward and back substitution in a single \(O(n)\) pass:+--+-- 1. __Factor:__ Compute \(A = LDL^T\) where \(L\) is unit lower+--    bidiagonal and \(D\) is diagonal.+-- 2. __Forward substitution:__ Solve \(Lz = b\).+-- 3. __Diagonal solve:__ Solve \(Dy = z\).+-- 4. __Back substitution:__ Solve \(L^T x = y\).+--+-- __Precondition.__ \(A\) must be symmetric positive definite. This is not+-- checked at run time.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) enforces that the diagonal, superdiagonal, right-hand+-- side, and solution vectors all have length \(n\) at compile time.+--+-- ==== Complexity+--+-- \(O(n)\) flops (GVL4 p. 179). This is optimal for tridiagonal systems.+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.3.6+-- (SPD Tridiagonal System Solver), p. 179.+tridiagSolve :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+             => Vector n r e     -- ^ Diagonal entries \(\alpha_{1:n}\)+             -> Vector n r e     -- ^ Superdiagonal entries \(\beta_{1:n-1}\) (length \(n\), only indices @0..n-2@ used)+             -> Vector n r e     -- ^ Right-hand side \(b\)+             -> Vector n r e     -- ^ Solution \(x\)+tridiagSolve diag supdiag b = createVector @n $ \mx -> do+  let nn = dimVal @n+  -- Working arrays for modified diagonal and superdiagonal+  alpha <- M.newMArray @r (M.Sz1 nn) (0 :: e)+  beta  <- M.newMArray @r (M.Sz1 nn) (0 :: e)+  -- Initialize+  mapM_ (\i -> do+    M.write_ alpha i (diag !. i)+    M.write_ beta i (if i < nn - 1 then supdiag !. i else 0)+    ) [0..nn-1]+  -- Copy b into result+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]++  -- LDLᵀ factorization and forward substitution combined+  -- for k = 2:n+  --   t = β(k-1), β(k-1) = t/α(k-1), α(k) = α(k) - t·β(k-1)+  mapM_ (\k -> do+    t <- M.readM beta (k - 1)+    ak1 <- M.readM alpha (k - 1)+    let bk1 = t / ak1+    M.write_ beta (k - 1) bk1+    ak <- M.readM alpha k+    M.write_ alpha k (ak - t * bk1)+    ) [1..nn-1]++  -- Forward substitution: b(k) = b(k) - β(k-1)·b(k-1)+  mapM_ (\k -> do+    bk <- M.readM mx k+    bk1 <- M.readM mx (k - 1)+    betaK1 <- M.readM beta (k - 1)+    M.write_ mx k (bk - betaK1 * bk1)+    ) [1..nn-1]++  -- Diagonal solve: b(n) = b(n)/α(n)+  bn <- M.readM mx (nn - 1)+  an <- M.readM alpha (nn - 1)+  M.write_ mx (nn - 1) (bn / an)++  -- Back substitution: b(k) = b(k)/α(k) - β(k)·b(k+1)+  mapM_ (\k -> do+    bk <- M.readM mx k+    ak <- M.readM alpha k+    betaK <- M.readM beta k+    bk1 <- M.readM mx (k + 1)+    M.write_ mx k (bk / ak - betaK * bk1)+    ) [nn-2, nn-3..0]
+ src/Numeric/LinearAlgebra/Massiv/Solve/Cholesky.hs view
@@ -0,0 +1,449 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}+{-# LANGUAGE BangPatterns #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Solve.Cholesky+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = Cholesky Factorization+--+-- Cholesky decomposition for symmetric positive definite (SPD) matrices,+-- following Golub & Van Loan, /Matrix Computations/, 4th edition (GVL4),+-- Section 4.2, pp. 163--169.+--+-- For any SPD matrix \(A \in \mathbb{R}^{n \times n}\), there exists a+-- unique lower triangular matrix \(G\) with positive diagonal entries such+-- that+--+-- \[+--   A = G G^T+-- \]+--+-- (GVL4 Theorem 4.2.1, p. 163). This is the /Cholesky factorization/.+-- Because it exploits symmetry, the Cholesky factorization requires only+-- half the work of a general LU factorization: \(O(n^3/3)\) flops vs.+-- \(O(2n^3/3)\) flops (GVL4 p. 165).+--+-- +-------------------+-------------------------------+---------------------------------++-- | Function          | Algorithm                     | Reference                       |+-- +===================+===============================+=================================++-- | 'cholesky'        | Outer-product Cholesky        | GVL4 Algorithm 4.2.1, p. 164    |+-- +-------------------+-------------------------------+---------------------------------++-- | 'choleskyGaxpy'   | Gaxpy (column-oriented)       | GVL4 Algorithm 4.2.2, p. 165    |+-- +-------------------+-------------------------------+---------------------------------++-- | 'choleskySolve'   | Solve via \(A = GG^T\)        | GVL4 Section 4.2, p. 166        |+-- +-------------------+-------------------------------+---------------------------------++--+-- == Complexity+--+-- The factorization costs \(O(n^3/3)\) flops -- exactly half of LU+-- (GVL4 p. 165). The subsequent pair of triangular solves adds \(O(n^2)\)+-- flops.+--+-- == Type Safety+--+-- Matrix dimensions are tracked at the type level via 'KnownNat', so the+-- compiler statically ensures the coefficient matrix is square and the+-- right-hand side vector has a conforming length. Note that positive+-- definiteness is /not/ checked at the type level; if the input matrix is+-- not SPD, the algorithm may produce NaN values from taking the square root+-- of a negative number.+module Numeric.LinearAlgebra.Massiv.Solve.Cholesky+  ( -- * Cholesky factorization (\(A = GG^T\))+    cholesky+  , choleskyGaxpy+    -- * Solving with Cholesky (\(Ax = b\))+  , choleskySolve+  , choleskySolveP+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)+import Control.Monad (when)+import GHC.Exts+import GHC.Prim+import GHC.ST (ST(..))+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..), newByteArray, unsafeFreezeByteArray)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Solve.Triangular (forwardSub, backSub)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (transpose)+import Numeric.LinearAlgebra.Massiv.Internal.Kernel+  (rawCholColumnSIMD, rawCholColumnSIMDFrom,+   rawForwardSubCholPackedSIMD, rawBackSubCholTPackedSIMD,+   rawGemmKernel, rawZeroDoubles)++-- | Outer-product Cholesky factorization (GVL4 Algorithm 4.2.1, p. 164).+--+-- Given a symmetric positive definite \(n \times n\) matrix \(A\), computes+-- the unique lower triangular matrix \(G\) with positive diagonal entries+-- such that+--+-- \[+--   A = G G^T+-- \]+--+-- Only the lower triangle of \(A\) is accessed; the upper triangle is+-- ignored.+--+-- The algorithm processes one column at a time using an /outer-product/+-- update of the trailing submatrix.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) statically ensures \(A\) is square. The 'Floating'+-- constraint provides 'sqrt'. Positive definiteness is a run-time+-- precondition; violation may produce NaN from \(\sqrt{g_{jj}}\) when+-- \(g_{jj} < 0\).+--+-- ==== Complexity+--+-- \(O(n^3/3)\) flops (GVL4 p. 165).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.2.1+-- (Outer Product Cholesky), p. 164. Existence: Theorem 4.2.1, p. 163.+cholesky :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+         => Matrix n n r e -> Matrix n n r e+cholesky (MkMatrix a) =+  let nn = dimVal @n+  in MkMatrix $ M.createArrayST_ (M.Sz2 nn nn) $ \mg -> do+    -- Initialize to zero+    mapM_ (\i -> mapM_ (\j -> M.write_ mg (i :. j) 0) [0..nn-1]) [0..nn-1]+    -- Copy lower triangle of A into working storage+    mapM_ (\j -> mapM_ (\i -> do+      let aij = M.index' a (i :. j)+      M.write_ mg (i :. j) aij+      ) [j..nn-1]) [0..nn-1]++    -- Outer product Cholesky+    mapM_ (\j -> do+      -- Subtract contributions from previous columns+      mapM_ (\k -> do+        gjk <- M.readM mg (j :. k)+        mapM_ (\i -> do+          gij <- M.readM mg (i :. j)+          gik <- M.readM mg (i :. k)+          M.write_ mg (i :. j) (gij - gik * gjk)+          ) [j..nn-1]+        ) [0..j-1]++      -- Scale column+      gjj <- M.readM mg (j :. j)+      let sjj = sqrt gjj+      M.write_ mg (j :. j) sjj+      mapM_ (\i -> do+        gij <- M.readM mg (i :. j)+        M.write_ mg (i :. j) (gij / sjj)+        ) [j+1..nn-1]+      ) [0..nn-1]++-- | Gaxpy (column-oriented) Cholesky factorization (GVL4 Algorithm 4.2.2,+-- p. 165).+--+-- Functionally equivalent to 'cholesky', but uses a /gaxpy/ (generalised+-- @y <- y - Gx@) inner loop that accumulates updates into each column+-- before normalising. This access pattern is advantageous for column-major+-- storage because it streams through contiguous memory.+--+-- ==== Mathematical definition+--+-- Computes the same \(G\) such that \(A = GG^T\) as 'cholesky'.+--+-- ==== Type-safety guarantees+--+-- Identical to 'cholesky'.+--+-- ==== Complexity+--+-- \(O(n^3/3)\) flops (GVL4 p. 165).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 4.2.2+-- (Gaxpy Cholesky), p. 165.+choleskyGaxpy :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+              => Matrix n n r e -> Matrix n n r e+choleskyGaxpy (MkMatrix a) =+  let nn = dimVal @n+  in MkMatrix $ M.createArrayST_ (M.Sz2 nn nn) $ \mg -> do+    -- Initialize: copy lower triangle of A+    mapM_ (\i -> mapM_ (\j ->+      if i >= j+        then M.write_ mg (i :. j) (M.index' a (i :. j))+        else M.write_ mg (i :. j) 0+      ) [0..nn-1]) [0..nn-1]++    -- Column by column+    mapM_ (\j -> do+      -- Update column j using previous columns (gaxpy)+      mapM_ (\k -> do+        gjk <- M.readM mg (j :. k)+        mapM_ (\i -> do+          gij <- M.readM mg (i :. j)+          gik <- M.readM mg (i :. k)+          M.write_ mg (i :. j) (gij - gik * gjk)+          ) [j..nn-1]+        ) [0..j-1]++      -- Scale by 1/sqrt(g(j,j))+      gjj <- M.readM mg (j :. j)+      let sjj = sqrt gjj+      mapM_ (\i -> do+        gij <- M.readM mg (i :. j)+        M.write_ mg (i :. j) (gij / sjj)+        ) [j..nn-1]+      ) [0..nn-1]++-- | Solve \(Ax = b\) where \(A\) is symmetric positive definite, using+-- Cholesky factorization (GVL4 Section 4.2, p. 166).+--+-- The algorithm proceeds in three stages:+--+-- 1. Factor \(A = GG^T\) via 'cholesky' (Algorithm 4.2.1).+-- 2. Solve \(Gy = b\) by forward substitution ('forwardSub').+-- 3. Solve \(G^T x = y\) by back substitution ('backSub').+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) enforces that \(A\) is \(n \times n\) and \(b\) has+-- length \(n\) at compile time.+--+-- ==== Complexity+--+-- \(O(n^3/3)\) flops for the factorization plus \(O(n^2)\) flops for the+-- two triangular solves (GVL4 p. 166).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Section 4.2,+-- pp. 163--169.+choleskySolve :: forall n r e. (KnownNat n, M.Manifest r e, Floating e, Ord e)+              => Matrix n n r e -> Vector n r e -> Vector n r e+choleskySolve a b =+  let g = cholesky a+      gt = transpose g+      y = forwardSub g b+  in backSub gt y++-- | Specialised Cholesky solve for @P Double@.+-- Does Cholesky factorisation + solve entirely using raw ByteArray# primops,+-- avoiding separate G/G^T matrix construction.+-- For n >= 64, uses panel (blocked) Cholesky factorisation with GEMM trailing update.+choleskySolveP :: forall n. KnownNat n+               => Matrix n n M.P Double -> Vector n M.P Double -> Vector n M.P Double+choleskySolveP (MkMatrix a) (MkVector b) =+  let nn = dimVal @n+  in createVector @n @M.P $ \mx -> do+    -- Allocate n×n working storage for G, copy lower triangle of A+    mg <- M.newMArray @M.P (M.Sz2 nn nn) (0 :: Double)+    let mbaG = unwrapMutableByteArray mg+        offG = unwrapMutableByteArrayOffset mg+    -- Copy lower triangle using raw primops+    copyLowerTriangle a mbaG offG nn++    -- Phase 1: Cholesky factorisation+    if nn >= 64+      then panelCholFactor mbaG offG nn 32+      else mapM_ (rawCholColumnSIMD mbaG offG nn) [0..nn-1]++    -- Phase 2: Freeze G and prepare RHS+    frozenG <- M.freezeS mg+    let baG = unwrapByteArray frozenG+        offFG = unwrapByteArrayOffset frozenG++    -- Copy b into output vector+    let mbaX = unwrapMutableByteArray mx+        offX = unwrapMutableByteArrayOffset mx+    copyVectorRaw b mbaX offX nn++    -- Phase 3: Forward substitution (Gy = b, SIMD dot-product)+    rawForwardSubCholPackedSIMD baG offFG nn mbaX offX++    -- Phase 4: Back substitution (G^T x = y, SIMD SAXPY)+    rawBackSubCholTPackedSIMD baG offFG nn mbaX offX+{-# NOINLINE choleskySolveP #-}++-- | Panel (blocked) Cholesky factorisation with GEMM trailing update.+-- For each panel of width @nb@:+--   1. Apply GEMM from previous panels: G[j:n, j:j+jb] -= L_prev × L_prev_panel^T+--   2. Factor the panel using within-panel Cholesky (dot from panel start)+panelCholFactor :: MutableByteArray s -> Int -> Int -> Int -> ST s ()+panelCholFactor mbaG offG nn nb = go 0+  where+    go !j+      | j >= nn = pure ()+      | otherwise = do+          let !jb = min nb (nn - j)+              !nBelow = nn - j  -- rows in [j..n-1]++          -- Step 1: update current panel with contributions from previous panels+          when (j > 0) $ do+            -- L_below = G[j:n, 0:j], shape nBelow × j+            -- L_panel = G[j:j+jb, 0:j], shape jb × j+            -- Update: G[j:n, j:j+jb] -= L_below × L_panel^T+            -- We compute this as: GEMM(L_below, L_panelT), where L_panelT = transpose(L_panel)++            -- Copy L_below to dense buffer (nBelow × j)+            bufA <- newByteArray (nBelow * j * 8)+            rawCopyCholSubmatrix mbaG offG nn j 0 nBelow j bufA 0++            -- Copy L_panel transposed to dense buffer (j × jb)+            -- L_panel is jb × j at rows [j..j+jb-1], cols [0..j-1]+            -- Transposed: j × jb+            bufBT <- newByteArray (j * jb * 8)+            rawCopyCholTranspose mbaG offG nn j 0 jb j bufBT 0++            -- Freeze for GEMM+            baA <- unsafeFreezeByteArray bufA+            baBT <- unsafeFreezeByteArray bufBT++            -- GEMM: C = L_below × L_panelT  (nBelow × jb)+            bufC <- newByteArray (nBelow * jb * 8)+            rawZeroDoubles bufC 0 (nBelow * jb)+            rawGemmKernel baA 0 baBT 0 bufC 0 nBelow j jb++            -- Subtract C from G[j:n, j:j+jb]+            baC <- unsafeFreezeByteArray bufC+            rawCholSubtractPanel baC 0 jb mbaG offG nn j j nBelow jb++          -- Step 2: factor panel using within-panel dependencies only+          mapM_ (\c -> rawCholColumnSIMDFrom mbaG offG nn c j) [j..j+jb-1]++          go (j + jb)++-- | Copy submatrix G[rowStart..rowStart+m-1, colStart..colStart+k-1] (stride n)+-- into dense buffer (stride k).+rawCopyCholSubmatrix :: MutableByteArray s -> Int -> Int+                     -> Int -> Int -> Int -> Int+                     -> MutableByteArray s -> Int+                     -> ST s ()+rawCopyCholSubmatrix (MutableByteArray mba_src) (I# off_src) (I# n)+                     (I# rowStart) (I# colStart) (I# m) (I# k)+                     (MutableByteArray mba_dst) (I# off_dst) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# m) = s+        | otherwise =+            let srcRow = off_src +# (rowStart +# i) *# n +# colStart+                dstRow = off_dst +# i *# k+                span_ = k -# (k `remInt#` 4#)+                goSimd j s_+                  | isTrue# (j >=# span_) = s_+                  | otherwise =+                      case readDoubleArrayAsDoubleX4# mba_src (srcRow +# j) s_ of+                        (# s1, v #) ->+                          case writeDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) v s1 of+                            s2 -> goSimd (j +# 4#) s2+                goScalar j s_+                  | isTrue# (j >=# k) = s_+                  | otherwise =+                      case readDoubleArray# mba_src (srcRow +# j) s_ of+                        (# s1, v #) ->+                          case writeDoubleArray# mba_dst (dstRow +# j) v s1 of+                            s2 -> goScalar (j +# 1#) s2+            in goI (i +# 1#) (goScalar span_ (goSimd 0# s))+  in (# goI 0# s0, () #)+{-# INLINE rawCopyCholSubmatrix #-}++-- | Copy and transpose: src[rowStart..rowStart+m-1, colStart..colStart+k-1] (stride n)+-- into dense buffer of shape k × m (stride m).+-- i.e., dst[j, i] = src[rowStart+i, colStart+j]+rawCopyCholTranspose :: MutableByteArray s -> Int -> Int+                     -> Int -> Int -> Int -> Int+                     -> MutableByteArray s -> Int+                     -> ST s ()+rawCopyCholTranspose (MutableByteArray mba_src) (I# off_src) (I# n)+                     (I# rowStart) (I# colStart) (I# m) (I# k)+                     (MutableByteArray mba_dst) (I# off_dst) = ST $ \s0 ->+  -- For each source row i, read k elements, write them as column i of dst+  let goI i s+        | isTrue# (i >=# m) = s+        | otherwise =+            let srcRow = off_src +# (rowStart +# i) *# n +# colStart+                goJ j s_+                  | isTrue# (j >=# k) = s_+                  | otherwise =+                      case readDoubleArray# mba_src (srcRow +# j) s_ of+                        (# s1, v #) ->+                          -- dst[j, i] at offset j * m + i+                          case writeDoubleArray# mba_dst (off_dst +# j *# m +# i) v s1 of+                            s2 -> goJ (j +# 1#) s2+            in goI (i +# 1#) (goJ 0# s)+  in (# goI 0# s0, () #)+{-# INLINE rawCopyCholTranspose #-}++-- | Subtract dense buffer C (m × k, stride srcStride) from+-- G[rowStart..rowStart+m-1, colStart..colStart+k-1] (stride n).+rawCholSubtractPanel :: ByteArray -> Int -> Int+                     -> MutableByteArray s -> Int -> Int+                     -> Int -> Int -> Int -> Int+                     -> ST s ()+rawCholSubtractPanel (ByteArray ba_src) (I# off_src) (I# srcStride)+                     (MutableByteArray mba_dst) (I# off_dst) (I# n)+                     (I# rowStart) (I# colStart) (I# m) (I# k) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# m) = s+        | otherwise =+            let srcRow = off_src +# i *# srcStride+                dstRow = off_dst +# (rowStart +# i) *# n +# colStart+                span_ = k -# (k `remInt#` 4#)+                goSimd j s_+                  | isTrue# (j >=# span_) = s_+                  | otherwise =+                      case readDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) s_ of+                        (# s1, aij #) ->+                          let cij = indexDoubleArrayAsDoubleX4# ba_src (srcRow +# j)+                              aij' = plusDoubleX4# aij (negateDoubleX4# cij)+                          in case writeDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) aij' s1 of+                               s2 -> goSimd (j +# 4#) s2+                goScalar j s_+                  | isTrue# (j >=# k) = s_+                  | otherwise =+                      case readDoubleArray# mba_dst (dstRow +# j) s_ of+                        (# s1, aij #) ->+                          let cij = indexDoubleArray# ba_src (srcRow +# j)+                          in case writeDoubleArray# mba_dst (dstRow +# j) (aij -## cij) s1 of+                               s2 -> goScalar (j +# 1#) s2+            in goI (i +# 1#) (goScalar span_ (goSimd 0# s))+  in (# goI 0# s0, () #)+{-# INLINE rawCholSubtractPanel #-}++-- | Copy lower triangle of an immutable 2D P array into a mutable byte array.+copyLowerTriangle :: M.Array M.P Ix2 Double -> MutableByteArray s -> Int -> Int -> ST s ()+copyLowerTriangle src (MutableByteArray mba_dst) (I# off_dst) (I# n) = ST $ \s0 ->+  let ba_src = case unwrapByteArray src of ByteArray ba -> ba+      off_src = case unwrapByteArrayOffset src of I# o -> o+      goI i s+        | isTrue# (i >=# n) = s+        | otherwise = goI (i +# 1#) (goJ i 0# s)+      goJ i j s+        | isTrue# (j ># i) = s+        | otherwise =+            let v = indexDoubleArray# ba_src (off_src +# i *# n +# j)+            in case writeDoubleArray# mba_dst (off_dst +# i *# n +# j) v s of+                 s' -> goJ i (j +# 1#) s'+  in (# goI 0# s0, () #)+{-# INLINE copyLowerTriangle #-}++-- | Copy an immutable P vector into a mutable byte array.+copyVectorRaw :: M.Array M.P Ix1 Double -> MutableByteArray s -> Int -> Int -> ST s ()+copyVectorRaw src (MutableByteArray mba_dst) (I# off_dst) (I# n) = ST $ \s0 ->+  let ba_src = case unwrapByteArray src of ByteArray ba -> ba+      off_src = case unwrapByteArrayOffset src of I# o -> o+      go i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let v = indexDoubleArray# ba_src (off_src +# i)+            in case writeDoubleArray# mba_dst (off_dst +# i) v s of+                 s' -> go (i +# 1#) s'+  in (# go 0# s0, () #)+{-# INLINE copyVectorRaw #-}
+ src/Numeric/LinearAlgebra/Massiv/Solve/LU.hs view
@@ -0,0 +1,562 @@+{-# LANGUAGE AllowAmbiguousTypes #-}+{-# LANGUAGE MagicHash #-}+{-# LANGUAGE UnboxedTuples #-}+{-# LANGUAGE BangPatterns #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Solve.LU+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = LU Factorization+--+-- LU decomposition with and without partial pivoting, plus derived+-- operations (linear solve and determinant), following Golub & Van Loan,+-- /Matrix Computations/, 4th edition (GVL4), Sections 3.2 and 3.4,+-- pp. 114--131.+--+-- Given an \(n \times n\) matrix \(A\), the factorization produces+--+-- \[+--   PA = LU+-- \]+--+-- where \(P\) is a permutation matrix, \(L\) is unit lower triangular, and+-- \(U\) is upper triangular (GVL4 Theorem 3.4.1, p. 125).+--+-- Without pivoting (\(P = I\)) the factorization exists if and only if all+-- leading principal submatrices of \(A\) are nonsingular+-- (GVL4 Theorem 3.2.1, p. 116). Partial pivoting guarantees existence for+-- any nonsingular \(A\) and improves numerical stability by bounding the+-- growth factor (GVL4 Section 3.4.6).+--+-- +-------------------+----------------------------------+-------------------------------++-- | Function          | Algorithm                        | Reference                     |+-- +===================+==================================+===============================++-- | 'lu'              | LU with partial pivoting         | GVL4 Algorithm 3.4.1, p. 126  |+-- +-------------------+----------------------------------+-------------------------------++-- | 'luNoPivot'       | Outer-product LU (no pivoting)   | GVL4 Algorithm 3.2.1, p. 115  |+-- +-------------------+----------------------------------+-------------------------------++-- | 'luSolve'         | Solve via \(PA = LU\)            | GVL4 Section 3.2, p. 118      |+-- +-------------------+----------------------------------+-------------------------------++-- | 'det'             | Determinant via \(PA = LU\)      | GVL4 Section 3.2, p. 120      |+-- +-------------------+----------------------------------+-------------------------------++--+-- == Complexity+--+-- The factorization requires \(O(2n^3/3)\) flops (GVL4 p. 118). Each+-- subsequent triangular solve adds \(O(n^2)\) flops.+--+-- == Type Safety+--+-- Matrix dimensions are tracked at the type level via 'KnownNat', so the+-- compiler statically enforces that the coefficient matrix is square and+-- that right-hand side vectors have conforming length.+module Numeric.LinearAlgebra.Massiv.Solve.LU+  ( -- * LU factorization+    lu+  , luNoPivot+    -- * Solving with LU (\(Ax = b\))+  , luSolve+  , luSolveP+    -- * Determinant+  , det+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..), unwrapByteArray, unwrapByteArrayOffset,+                          unwrapMutableByteArray, unwrapMutableByteArrayOffset)+import GHC.TypeNats (KnownNat)+import Data.Ord (comparing)+import Data.List (maximumBy)+import Control.Monad (when, forM)+import GHC.Exts+import GHC.Prim+import GHC.ST (ST(..))+import Data.Primitive.ByteArray (ByteArray(..), MutableByteArray(..), newByteArray, unsafeFreezeByteArray)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Solve.Triangular (forwardSubUnit, backSub)+import Numeric.LinearAlgebra.Massiv.Internal.Kernel+  (rawLUEliminateColumn, rawLUEliminateColumnTo, rawSwapRows, rawPivotSearch,+   rawForwardSubUnitPackedSIMD, rawBackSubPackedSIMD,+   rawGemmKernel, rawZeroDoubles)++-- | LU factorization with partial pivoting (GVL4 Algorithm 3.4.1, p. 126).+--+-- Given an \(n \times n\) matrix \(A\), computes the factorization+--+-- \[+--   PA = LU+-- \]+--+-- where+--+-- * \(P\) is a permutation matrix (returned as a pivot-index vector of type+--   @Array P Ix1 Int@),+-- * \(L\) is unit lower triangular (stored /below/ the diagonal of the+--   returned packed matrix), and+-- * \(U\) is upper triangular (stored /on and above/ the diagonal).+--+-- Partial pivoting selects the entry of largest absolute value in the+-- current column as the pivot, guaranteeing existence for any nonsingular+-- \(A\) (GVL4 Theorem 3.4.1, p. 125).+--+-- ==== Type-safety guarantees+--+-- The 'KnownNat' constraint on \(n\) statically ensures the matrix is+-- square. The 'Ord' constraint is required for pivot selection.+--+-- ==== Complexity+--+-- \(O(2n^3/3)\) flops (GVL4 p. 118).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.4.1+-- (Outer Product LU with Partial Pivoting), p. 126.+lu :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e, Ord e)+   => Matrix n n r e -> (Matrix n n r e, M.Array M.P Ix1 Int)+lu (MkMatrix a) =+  let nn = dimVal @n+      (pivArr, luArr) = M.withMArrayST a $ \ma -> do+        piv <- M.newMArray @M.P (M.Sz1 nn) 0+        mapM_ (\i -> M.write_ piv i i) [0..nn-1]++        mapM_ (\k -> do+          -- Find pivot row+          vals <- mapM (\i -> do+            v <- M.readM ma (i :. k)+            pure (i, abs v)+            ) [k..nn-1]+          let (pivRow, _) = maximumBy (comparing snd) vals++          -- Swap rows k and pivRow+          condM (pivRow /= k) $ do+            pk <- M.readM piv k+            pp <- M.readM piv pivRow+            M.write_ piv k pp+            M.write_ piv pivRow pk+            mapM_ (\j -> do+              akj <- M.readM ma (k :. j)+              apj <- M.readM ma (pivRow :. j)+              M.write_ ma (k :. j) apj+              M.write_ ma (pivRow :. j) akj+              ) [0..nn-1]++          -- Compute multipliers and update submatrix+          akk <- M.readM ma (k :. k)+          condM (akk /= 0) $+            mapM_ (\i -> do+              aik <- M.readM ma (i :. k)+              let mult = aik / akk+              M.write_ ma (i :. k) mult+              mapM_ (\j -> do+                aij <- M.readM ma (i :. j)+                akj <- M.readM ma (k :. j)+                M.write_ ma (i :. j) (aij - mult * akj)+                ) [k+1..nn-1]+              ) [k+1..nn-1]+          ) [0..nn-2]++        M.freezeS piv++  in (MkMatrix luArr, pivArr)++-- | LU factorization without pivoting (GVL4 Algorithm 3.2.1, p. 115).+--+-- Given an \(n \times n\) matrix \(A\) whose leading principal submatrices+-- are all nonsingular, computes the factorization \(A = LU\) where \(L\) is+-- unit lower triangular and \(U\) is upper triangular. Both factors are+-- packed into a single returned matrix: \(L\) occupies the strictly lower+-- triangle (the unit diagonal is implicit) and \(U\) occupies the upper+-- triangle including the diagonal.+--+-- __Precondition.__ All leading principal submatrices+-- \(A(1{:}k, 1{:}k)\), \(k = 1, \ldots, n\), must be nonsingular+-- (GVL4 Theorem 3.2.1, p. 116). Violating this precondition results in+-- division by zero.+--+-- ==== Type-safety guarantees+--+-- The 'KnownNat' constraint on \(n\) statically ensures the input is a+-- square matrix.+--+-- ==== Complexity+--+-- \(O(2n^3/3)\) flops (GVL4 p. 118).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.2.1+-- (Outer Product LU Factorization), p. 115.+luNoPivot :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+          => Matrix n n r e -> Matrix n n r e+luNoPivot (MkMatrix a) =+  let nn = dimVal @n+  in MkMatrix $ snd $ M.withMArrayST a $ \ma ->+    mapM_ (\k -> do+      akk <- M.readM ma (k :. k)+      mapM_ (\i -> do+        aik <- M.readM ma (i :. k)+        M.write_ ma (i :. k) (aik / akk)+        ) [k+1..nn-1]+      mapM_ (\j ->+        mapM_ (\i -> do+          aij <- M.readM ma (i :. j)+          aik <- M.readM ma (i :. k)+          akj <- M.readM ma (k :. j)+          M.write_ ma (i :. j) (aij - aik * akj)+          ) [k+1..nn-1]+        ) [k+1..nn-1]+      ) [0..nn-2]++-- | Solve \(Ax = b\) using LU factorization with partial pivoting+-- (GVL4 Section 3.2, p. 118).+--+-- The algorithm proceeds in three stages:+--+-- 1. Factor \(PA = LU\) via 'lu' (Algorithm 3.4.1).+-- 2. Solve \(Ly = Pb\) by forward substitution ('forwardSubUnit').+-- 3. Solve \(Ux = y\) by back substitution ('backSub').+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) enforces that \(A\) is \(n \times n\) and \(b\) has+-- length \(n\) at compile time.+--+-- ==== Complexity+--+-- \(O(2n^3/3)\) flops for the factorization plus \(O(n^2)\) flops for each+-- of the two triangular solves (GVL4 p. 118).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Section 3.2,+-- pp. 114--120.+luSolve :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e, Ord e)+        => Matrix n n r e -> Vector n r e -> Vector n r e+luSolve a b =+  let (luMat, pivArr) = lu a+      -- Extract L (unit lower triangular)+      l = makeMatrix @n @n @r $ \i j ->+        if i == j then 1+        else if i > j then luMat ! (i, j)+        else 0+      -- Extract U (upper triangular)+      u = makeMatrix @n @n @r $ \i j ->+        if i <= j then luMat ! (i, j)+        else 0+      -- Apply permutation to b: pb = P·b+      pb = makeVector @n @r $ \i ->+        b !. M.index' pivArr i+      -- Solve Ly = Pb+      y = forwardSubUnit l pb+      -- Solve Ux = y+  in backSub u y++-- | Specialised LU solve for @P Double@.+-- Does LU factorisation + solve entirely using raw ByteArray# primops,+-- avoiding L/U matrix reconstruction and massiv's per-element overhead.+-- For n >= 64, uses panel (blocked) LU factorisation with GEMM trailing update.+luSolveP :: forall n. KnownNat n+         => Matrix n n M.P Double -> Vector n M.P Double -> Vector n M.P Double+luSolveP (MkMatrix a) (MkVector b) =+  let nn = dimVal @n+  in createVector @n @M.P $ \mx -> do+    -- Thaw matrix for in-place LU factorisation+    ma <- M.thawS a+    let mbaA = unwrapMutableByteArray ma+        offA = unwrapMutableByteArrayOffset ma++    -- Phase 1: LU factorisation with partial pivoting+    pivots <- if nn >= 64+      then panelLUFactor mbaA offA nn 32+      else mapM (\k -> do+        pivRow <- rawPivotSearch mbaA offA nn k k+        condM (pivRow /= k) $+          rawSwapRows mbaA offA nn k pivRow 0+        rawLUEliminateColumn mbaA offA nn k+        pure (k, pivRow)+        ) [0..nn-2]++    -- Phase 2: Freeze LU and prepare RHS+    frozenLU <- M.freezeS ma+    let baLU = unwrapByteArray frozenLU+        offLU = unwrapByteArrayOffset frozenLU++    -- Copy b into the output vector mx+    let mbaX = unwrapMutableByteArray mx+        offX = unwrapMutableByteArrayOffset mx+    copyVector b mbaX offX nn++    -- Apply pivot permutation to x+    applyPivotsForward mbaX offX pivots++    -- Phase 3: Forward substitution (unit lower triangular, SIMD dot-product)+    rawForwardSubUnitPackedSIMD baLU offLU nn mbaX offX++    -- Phase 4: Back substitution (upper triangular, SIMD dot-product)+    rawBackSubPackedSIMD baLU offLU nn mbaX offX+{-# NOINLINE luSolveP #-}++-- | Panel (blocked) LU factorisation with GEMM trailing update.+-- Processes columns in panels of width @nb@.  Within each panel, elimination+-- is restricted to the panel columns; the trailing submatrix is updated via a+-- single GEMM call, converting O(n) column-by-column Level-2 updates into one+-- cache-friendly Level-3 GEMM.+panelLUFactor :: MutableByteArray s -> Int -> Int -> Int -> ST s [(Int, Int)]+panelLUFactor mbaA offA nn nb = go 0 []+  where+    go !k0 !pivAcc+      | k0 >= nn - 1 = pure (reverse pivAcc)+      | otherwise = do+          let !panelEnd = min (k0 + nb) nn+              !actualNb = panelEnd - k0++          -- Factor panel columns k0..panelEnd-1 with restricted trailing update+          panelPivs <- forM [k0..panelEnd-1] $ \k -> do+            pivRow <- rawPivotSearch mbaA offA nn k k+            condM (pivRow /= k) $+              rawSwapRows mbaA offA nn k pivRow 0+            rawLUEliminateColumnTo mbaA offA nn k panelEnd+            pure (k, pivRow)++          -- Apply panel's L to trailing columns: triangular solve for U12+          when (panelEnd < nn) $ do+            rawTriSolvePanelTrail mbaA offA nn k0 panelEnd++            -- GEMM update: A22 -= L21 × U12+            let !mTrail = nn - panelEnd+                !nTrail = nn - panelEnd++            -- Copy L21 to dense buffer (mTrail × actualNb)+            bufL <- newByteArray (mTrail * actualNb * 8)+            rawCopySubmatrixToDense mbaA offA nn panelEnd k0 mTrail actualNb bufL 0++            -- Copy U12 to dense buffer (actualNb × nTrail)+            bufU <- newByteArray (actualNb * nTrail * 8)+            rawCopySubmatrixToDense mbaA offA nn k0 panelEnd actualNb nTrail bufU 0++            -- Freeze for immutable GEMM inputs+            baL <- unsafeFreezeByteArray bufL+            baU <- unsafeFreezeByteArray bufU++            -- GEMM: C = L21 × U12+            bufC <- newByteArray (mTrail * nTrail * 8)+            rawZeroDoubles bufC 0 (mTrail * nTrail)+            rawGemmKernel baL 0 baU 0 bufC 0 mTrail actualNb nTrail++            -- Subtract C from A22+            baC <- unsafeFreezeByteArray bufC+            rawSubtractFromStrided baC 0 nTrail mbaA offA nn panelEnd panelEnd mTrail nTrail++          go panelEnd (reverse panelPivs ++ pivAcc)++-- | Apply unit lower triangular solve from the panel to trailing columns.+-- After factoring panel columns [k0..panelEnd-1] with restricted updates,+-- the trailing columns [panelEnd..n-1] need: for each k in the panel, apply+-- the multipliers to rows k+1..panelEnd-1 of the trailing columns.+-- i.e.  A[i,j] -= A[i,k] * A[k,j]  for  k0 <= k < panelEnd, k < i < panelEnd, panelEnd <= j < n+rawTriSolvePanelTrail :: MutableByteArray s -> Int -> Int -> Int -> Int -> ST s ()+rawTriSolvePanelTrail (MutableByteArray mba) (I# off) (I# n) (I# k0) (I# panelEnd) = ST $ \s0 ->+  let -- For each column k in the panel+      goK k s+        | isTrue# (k >=# panelEnd) = s+        | otherwise =+            let kRowOff = off +# k *# n+            in goI k (k +# 1#) kRowOff s+        where+          -- For each row i in [k+1..panelEnd-1]+          goI k_ i kRowOff s_+            | isTrue# (i >=# panelEnd) = goK (k_ +# 1#) s_+            | otherwise =+                let iRowOff = off +# i *# n+                in case readDoubleArray# mba (iRowOff +# k_) s_ of+                     (# s1, lik #) ->+                       let negLik = negateDouble# lik+                           negLikV = broadcastDoubleX4# negLik+                           jSpan = n -# panelEnd+                           j4End = panelEnd +# (jSpan -# (jSpan `remInt#` 4#))+                           -- SIMD j-loop+                           goJSimd j s__+                             | isTrue# (j >=# j4End) = s__+                             | otherwise =+                                 case readDoubleArrayAsDoubleX4# mba (iRowOff +# j) s__ of+                                   (# s2, aij #) ->+                                     case readDoubleArrayAsDoubleX4# mba (kRowOff +# j) s2 of+                                       (# s3, akj #) ->+                                         let aij' = fmaddDoubleX4# negLikV akj aij+                                         in case writeDoubleArrayAsDoubleX4# mba (iRowOff +# j) aij' s3 of+                                              s4 -> goJSimd (j +# 4#) s4+                           -- Scalar cleanup+                           goJScalar j s__+                             | isTrue# (j >=# n) = s__+                             | otherwise =+                                 case readDoubleArray# mba (iRowOff +# j) s__ of+                                   (# s2, aij #) ->+                                     case readDoubleArray# mba (kRowOff +# j) s2 of+                                       (# s3, akj #) ->+                                         case writeDoubleArray# mba (iRowOff +# j) (aij +## negLik *## akj) s3 of+                                           s4 -> goJScalar (j +# 1#) s4+                       in goI k_ (i +# 1#) kRowOff (goJScalar j4End (goJSimd panelEnd s1))+  in (# goK k0 s0, () #)+{-# INLINE rawTriSolvePanelTrail #-}++-- | Copy a submatrix A[rowStart..rowStart+m-1, colStart..colStart+k-1] (stride n)+-- into a dense buffer (stride k).+rawCopySubmatrixToDense :: MutableByteArray s -> Int -> Int  -- src, offset, n+                        -> Int -> Int -> Int -> Int          -- rowStart, colStart, m, k+                        -> MutableByteArray s -> Int          -- dst, dstOffset+                        -> ST s ()+rawCopySubmatrixToDense (MutableByteArray mba_src) (I# off_src) (I# n)+                        (I# rowStart) (I# colStart) (I# m) (I# k)+                        (MutableByteArray mba_dst) (I# off_dst) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# m) = s+        | otherwise =+            let srcRow = off_src +# (rowStart +# i) *# n +# colStart+                dstRow = off_dst +# i *# k+                span_ = k -# (k `remInt#` 4#)+                goSimd j s_+                  | isTrue# (j >=# span_) = s_+                  | otherwise =+                      case readDoubleArrayAsDoubleX4# mba_src (srcRow +# j) s_ of+                        (# s1, v #) ->+                          case writeDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) v s1 of+                            s2 -> goSimd (j +# 4#) s2+                goScalar j s_+                  | isTrue# (j >=# k) = s_+                  | otherwise =+                      case readDoubleArray# mba_src (srcRow +# j) s_ of+                        (# s1, v #) ->+                          case writeDoubleArray# mba_dst (dstRow +# j) v s1 of+                            s2 -> goScalar (j +# 1#) s2+            in goI (i +# 1#) (goScalar span_ (goSimd 0# s))+  in (# goI 0# s0, () #)+{-# INLINE rawCopySubmatrixToDense #-}++-- | Subtract a dense matrix C (m × k, stride srcStride) from a strided submatrix+-- A[rowStart..rowStart+m-1, colStart..colStart+k-1] (stride n).+rawSubtractFromStrided :: ByteArray -> Int -> Int              -- src, srcOffset, srcStride+                       -> MutableByteArray s -> Int -> Int     -- dst, dstOffset, n+                       -> Int -> Int -> Int -> Int             -- rowStart, colStart, m, k+                       -> ST s ()+rawSubtractFromStrided (ByteArray ba_src) (I# off_src) (I# srcStride)+                       (MutableByteArray mba_dst) (I# off_dst) (I# n)+                       (I# rowStart) (I# colStart) (I# m) (I# k) = ST $ \s0 ->+  let goI i s+        | isTrue# (i >=# m) = s+        | otherwise =+            let srcRow = off_src +# i *# srcStride+                dstRow = off_dst +# (rowStart +# i) *# n +# colStart+                span_ = k -# (k `remInt#` 4#)+                goSimd j s_+                  | isTrue# (j >=# span_) = s_+                  | otherwise =+                      case readDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) s_ of+                        (# s1, aij #) ->+                          let cij = indexDoubleArrayAsDoubleX4# ba_src (srcRow +# j)+                              aij' = plusDoubleX4# aij (negateDoubleX4# cij)+                          in case writeDoubleArrayAsDoubleX4# mba_dst (dstRow +# j) aij' s1 of+                               s2 -> goSimd (j +# 4#) s2+                goScalar j s_+                  | isTrue# (j >=# k) = s_+                  | otherwise =+                      case readDoubleArray# mba_dst (dstRow +# j) s_ of+                        (# s1, aij #) ->+                          let cij = indexDoubleArray# ba_src (srcRow +# j)+                          in case writeDoubleArray# mba_dst (dstRow +# j) (aij -## cij) s1 of+                               s2 -> goScalar (j +# 1#) s2+            in goI (i +# 1#) (goScalar span_ (goSimd 0# s))+  in (# goI 0# s0, () #)+{-# INLINE rawSubtractFromStrided #-}++-- | Copy an immutable P vector into a mutable byte array.+copyVector :: M.Array M.P Ix1 Double -> MutableByteArray s -> Int -> Int -> ST s ()+copyVector src (MutableByteArray mba_dst) (I# off_dst) (I# n) = ST $ \s0 ->+  let ba_src = case unwrapByteArray src of ByteArray ba -> ba+      off_src = case unwrapByteArrayOffset src of I# o -> o+      go i s+        | isTrue# (i >=# n) = s+        | otherwise =+            let v = indexDoubleArray# ba_src (off_src +# i)+            in case writeDoubleArray# mba_dst (off_dst +# i) v s of+                 s' -> go (i +# 1#) s'+  in (# go 0# s0, () #)+{-# INLINE copyVector #-}++-- | Apply pivot permutation to a vector (forward direction).+applyPivotsForward :: MutableByteArray s -> Int -> [(Int, Int)] -> ST s ()+applyPivotsForward (MutableByteArray mba) (I# off) pivots = ST $ \s0 ->+  let go [] s = s+      go ((I# k, I# pivRow) : rest) s+        | isTrue# (k ==# pivRow) = go rest s+        | otherwise =+            case readDoubleArray# mba (off +# k) s of+              (# s1, vk #) ->+                case readDoubleArray# mba (off +# pivRow) s1 of+                  (# s2, vp #) ->+                    case writeDoubleArray# mba (off +# k) vp s2 of+                      s3 -> case writeDoubleArray# mba (off +# pivRow) vk s3 of+                              s4 -> go rest s4+  in (# go pivots s0, () #)+{-# INLINE applyPivotsForward #-}++-- | Compute the determinant of an \(n \times n\) matrix via LU factorization+-- (GVL4 Section 3.2, p. 120).+--+-- ==== Mathematical definition+--+-- From \(PA = LU\) it follows that+--+-- \[+--   \det(A) = (-1)^s \prod_{i=1}^{n} u_{ii}+-- \]+--+-- where \(s\) is the number of row transpositions performed during partial+-- pivoting.+--+-- ==== Type-safety guarantees+--+-- 'KnownNat' \(n\) ensures \(A\) is square at compile time.+--+-- ==== Complexity+--+-- \(O(2n^3/3)\) flops, dominated by the LU factorization.+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Section 3.2, p. 120.+det :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e, Ord e)+    => Matrix n n r e -> e+det a =+  let nn = dimVal @n+      (luMat, pivArr) = lu a+      -- Product of U diagonal+      diagProd = foldl' (\acc i -> acc * (luMat ! (i, i))) 1 [0..nn-1]+      -- Count transpositions: number of i where piv[i] /= i+      pivList = map (M.index' pivArr) [0..nn-1]+      nswaps = countSwaps pivList+      sign = if even nswaps then 1 else -1+  in sign * diagProd++-- | Count the number of swaps in a permutation.+countSwaps :: [Int] -> Int+countSwaps perm = go (zip [0..] perm) 0+  where+    go [] n = n+    go ((i,p):rest) n+      | i == p = go rest n+      | otherwise =+          -- Swap p into position i by finding where i is+          let rest' = map (\(idx, v) -> if v == i then (idx, p) else (idx, v)) rest+          in go rest' (n + 1)++-- | Conditional monadic action.+condM :: Applicative m => Bool -> m () -> m ()+condM True act = act+condM False _ = pure ()
+ src/Numeric/LinearAlgebra/Massiv/Solve/Triangular.hs view
@@ -0,0 +1,250 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Solve.Triangular+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- = Triangular System Solvers+--+-- Forward and back substitution for lower- and upper-triangular linear+-- systems, following Golub & Van Loan, /Matrix Computations/, 4th edition+-- (GVL4), Section 3.1, pp. 106--113.+--+-- The section presents four algorithmic variants organised along two axes:+--+-- * __Row-oriented vs. column-oriented__ inner loops, affecting data-access+--   patterns and cache behaviour.+-- * __General vs. unit-triangular__ coefficient matrices, where the unit+--   variants avoid division by the (known-to-be-one) diagonal.+--+-- This module exposes the following mapping:+--+-- +-------------------+---------------------------+---------------------------------++-- | Function          | Algorithm                 | Reference                       |+-- +===================+===========================+=================================++-- | 'forwardSub'      | Row-oriented forward sub  | GVL4 Algorithm 3.1.1, p. 106    |+-- +-------------------+---------------------------+---------------------------------++-- | 'backSub'         | Row-oriented back sub     | GVL4 Algorithm 3.1.2, p. 107    |+-- +-------------------+---------------------------+---------------------------------++-- | 'forwardSubUnit'  | Column-oriented fwd sub   | GVL4 Algorithm 3.1.3, p. 108    |+-- +-------------------+---------------------------+---------------------------------++-- | 'backSubUnit'     | Column-oriented back sub  | GVL4 Algorithm 3.1.4, p. 109    |+-- +-------------------+---------------------------+---------------------------------++--+-- == Complexity+--+-- Every solver performs \(O(n^2 / 2)\) floating-point operations (flops) for+-- an \(n \times n\) triangular system (GVL4 p. 109).+--+-- == Type Safety+--+-- The matrix dimension \(n\) is tracked at the type level via 'KnownNat',+-- so the compiler statically guarantees that the coefficient matrix is+-- square and that the right-hand side vector has a conforming length.+module Numeric.LinearAlgebra.Massiv.Solve.Triangular+  ( -- * Forward substitution (\(Lx = b\))+    forwardSub+    -- * Back substitution (\(Ux = b\))+  , backSub+    -- * Unit-triangular variants+  , forwardSubUnit+  , backSubUnit+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Sz(..))+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal++-- | Row-oriented forward substitution (GVL4 Algorithm 3.1.1, p. 106).+--+-- Solves the lower-triangular system \(Lx = b\) where \(L \in \mathbb{R}^{n \times n}\)+-- is lower triangular with nonzero diagonal entries.+--+-- ==== Mathematical definition+--+-- For \(j = 1, \ldots, n\):+--+-- \[+--   x_j = \frac{1}{\ell_{jj}} \left( b_j - \sum_{k=1}^{j-1} \ell_{jk}\, x_k \right)+-- \]+--+-- ==== Type-safety guarantees+--+-- The type-level natural \(n\) ('KnownNat') ensures that \(L\) is square+-- and that \(b\) has exactly \(n\) entries. A dimension mismatch is a+-- compile-time error.+--+-- ==== Complexity+--+-- \(O(n^2 / 2)\) flops (GVL4 p. 109).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.1.1+-- (Row-Oriented Forward Substitution), p. 106.+forwardSub :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+           => Matrix n n r e -> Vector n r e -> Vector n r e+forwardSub l b = createVector @n $ \mx -> do+  let nn = dimVal @n+  -- Copy b into the mutable result+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  -- Forward elimination+  mapM_ (\j -> do+    xj <- M.readM mx j+    let ldiag = l ! (j, j)+        xj' = xj / ldiag+    M.write_ mx j xj'+    -- Update remaining entries+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (l ! (i, j)) * xj')+      ) [j+1..nn-1]+    ) [0..nn-1]++-- | Row-oriented back substitution (GVL4 Algorithm 3.1.2, p. 107).+--+-- Solves the upper-triangular system \(Ux = b\) where \(U \in \mathbb{R}^{n \times n}\)+-- is upper triangular with nonzero diagonal entries.+--+-- ==== Mathematical definition+--+-- For \(j = n, n-1, \ldots, 1\):+--+-- \[+--   x_j = \frac{1}{u_{jj}} \left( b_j - \sum_{k=j+1}^{n} u_{jk}\, x_k \right)+-- \]+--+-- ==== Type-safety guarantees+--+-- The type-level natural \(n\) ('KnownNat') ensures that \(U\) is square+-- and that \(b\) has exactly \(n\) entries. A dimension mismatch is a+-- compile-time error.+--+-- ==== Complexity+--+-- \(O(n^2 / 2)\) flops (GVL4 p. 109).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.1.2+-- (Row-Oriented Back Substitution), p. 107.+backSub :: forall n r e. (KnownNat n, M.Manifest r e, Fractional e)+        => Matrix n n r e -> Vector n r e -> Vector n r e+backSub u b = createVector @n $ \mx -> do+  let nn = dimVal @n+  -- Copy b into the mutable result+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  -- Back elimination+  mapM_ (\j -> do+    xj <- M.readM mx j+    let udiag = u ! (j, j)+        xj' = xj / udiag+    M.write_ mx j xj'+    -- Update remaining entries+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (u ! (i, j)) * xj')+      ) [0..j-1]+    ) [nn-1, nn-2..0]++-- | Column-oriented forward substitution for unit lower triangular systems+-- (GVL4 Algorithm 3.1.3, p. 108).+--+-- Solves \(Lx = b\) where \(L \in \mathbb{R}^{n \times n}\) is /unit/ lower+-- triangular, i.e. \(\ell_{jj} = 1\) for all \(j\). Because the diagonal is+-- implicitly one, no division is needed and the constraint relaxes from+-- 'Fractional' to 'Num'.+--+-- ==== Mathematical definition+--+-- For \(j = 1, \ldots, n\):+--+-- \[+--   x_j = b_j - \sum_{k=1}^{j-1} \ell_{jk}\, x_k+-- \]+--+-- The implementation uses a /column-oriented/ (saxpy) loop: once \(x_j\) is+-- determined, rows \(i > j\) are updated by subtracting \(\ell_{ij}\, x_j\).+--+-- ==== Type-safety guarantees+--+-- Identical to 'forwardSub': the dimensions are enforced at compile time+-- via 'KnownNat'.+--+-- ==== Complexity+--+-- \(O(n^2 / 2)\) flops (GVL4 p. 109).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.1.3+-- (Column-Oriented Forward Substitution), p. 108.+forwardSubUnit :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+               => Matrix n n r e -> Vector n r e -> Vector n r e+forwardSubUnit l b = createVector @n $ \mx -> do+  let nn = dimVal @n+  -- Copy b into the mutable result+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  -- Column-oriented forward substitution+  mapM_ (\j -> do+    xj <- M.readM mx j+    -- Subtract l(i,j)*xj from x(i) for i > j+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (l ! (i, j)) * xj)+      ) [j+1..nn-1]+    ) [0..nn-1]++-- | Column-oriented back substitution for unit upper triangular systems+-- (GVL4 Algorithm 3.1.4, p. 109).+--+-- Solves \(Ux = b\) where \(U \in \mathbb{R}^{n \times n}\) is /unit/ upper+-- triangular, i.e. \(u_{jj} = 1\) for all \(j\). Because the diagonal is+-- implicitly one, no division is needed and the constraint relaxes from+-- 'Fractional' to 'Num'.+--+-- ==== Mathematical definition+--+-- For \(j = n, n-1, \ldots, 1\):+--+-- \[+--   x_j = b_j - \sum_{k=j+1}^{n} u_{jk}\, x_k+-- \]+--+-- The implementation uses a /column-oriented/ loop: once \(x_j\) is known,+-- rows \(i < j\) are updated by subtracting \(u_{ij}\, x_j\).+--+-- ==== Type-safety guarantees+--+-- Identical to 'backSub': the dimensions are enforced at compile time via+-- 'KnownNat'.+--+-- ==== Complexity+--+-- \(O(n^2 / 2)\) flops (GVL4 p. 109).+--+-- ==== Reference+--+-- Golub & Van Loan, /Matrix Computations/, 4th ed., Algorithm 3.1.4+-- (Column-Oriented Back Substitution), p. 109.+backSubUnit :: forall n r e. (KnownNat n, M.Manifest r e, Num e)+            => Matrix n n r e -> Vector n r e -> Vector n r e+backSubUnit u b = createVector @n $ \mx -> do+  let nn = dimVal @n+  -- Copy b into the mutable result+  mapM_ (\i -> M.write_ mx i (b !. i)) [0..nn-1]+  -- Column-oriented back substitution+  mapM_ (\j -> do+    xj <- M.readM mx j+    -- Subtract u(i,j)*xj from x(i) for i < j+    mapM_ (\i -> do+      xi <- M.readM mx i+      M.write_ mx i (xi - (u ! (i, j)) * xj)+      ) [0..j-1]+    ) [nn-1, nn-2..0]
+ src/Numeric/LinearAlgebra/Massiv/Types.hs view
@@ -0,0 +1,185 @@+{-# LANGUAGE UndecidableInstances #-}+{-# LANGUAGE AllowAmbiguousTypes #-}++-- |+-- Module      : Numeric.LinearAlgebra.Massiv.Types+-- Copyright   : (c) Nadia Chambers 2026+-- License     : BSD-3-Clause+-- Maintainer  : nadia.chambers@iohk.io+-- Stability   : experimental+--+-- Core type definitions for type-safe dimensioned matrices and vectors+-- backed by <https://hackage.haskell.org/package/massiv massiv> arrays.+--+-- The central types are 'Matrix' and 'Vector', which wrap massiv arrays+-- with phantom type-level natural number parameters encoding their dimensions.+-- This ensures that dimensionally incorrect operations (e.g., adding matrices+-- of different sizes, or multiplying matrices with incompatible inner dimensions)+-- are caught at compile time by GHC's type checker.+--+-- = Type-level dimension encoding+--+-- Dimensions are encoded as GHC @DataKinds@ promoted @'GHC.TypeNats.Nat'@+-- values. The constraint @'GHC.TypeNats.KnownNat' n@ provides access to the+-- runtime value via @'GHC.TypeNats.natVal'@.+--+-- @+-- -- A 3x4 matrix of Doubles using Primitive representation+-- type MyMatrix = Matrix 3 4 P Double+--+-- -- A 5-element vector of Doubles using Unboxed representation+-- type MyVector = Vector 5 U Double+-- @+--+-- = Existential wrappers+--+-- For situations where dimensions are not known until runtime (e.g., reading+-- a matrix from a file), use 'SomeMatrix' and 'SomeVector'. These existentially+-- quantify the dimension parameters while retaining 'KnownNat' evidence.+--+-- See "Numeric.LinearAlgebra.Massiv.Internal" for construction helpers.+module Numeric.LinearAlgebra.Massiv.Types+  ( -- * Dimensioned matrix type+    Matrix(..)+    -- * Dimensioned vector type+  , Vector(..)+    -- * Smart constructors+  , matrix+  , vector+    -- * Existential wrappers+  , SomeMatrix(..)+  , SomeVector(..)+  , someMatrix+  , someVector+    -- * Dimension queries+  , rows+  , cols+  , size+    -- * Type-level helpers+  , type KnownDims+  ) where++import Data.Massiv.Array (Array, Ix2(..), Sz(..), Ix1, Comp(..))+import qualified Data.Massiv.Array as M+import GHC.TypeNats (Nat, KnownNat, natVal, SomeNat(..), someNatVal)+import Data.Proxy (Proxy(..))+import Control.DeepSeq (NFData(..))++-- | Constraint synonym for two known dimensions.+--+-- @KnownDims m n@ is equivalent to @(KnownNat m, KnownNat n)@.+type KnownDims m n = (KnownNat m, KnownNat n)++-- | A matrix with compile-time known dimensions \(m\) (rows) \(\times\) \(n\) (cols).+--+-- Wraps a massiv @'Data.Massiv.Array.Array' r 'Data.Massiv.Array.Ix2' e@.+-- The phantom type parameters \(m\) and \(n\) enforce dimensional conformance+-- at compile time. For example, matrix multiplication via 'matMul' requires+-- the inner dimensions to unify:+--+-- @+-- 'matMul' :: Matrix m __k__ r e -> Matrix __k__ n r e -> Matrix m n r e+-- @+--+-- The representation parameter @r@ selects the massiv array backend:+--+-- * @'Data.Massiv.Array.P'@ — Primitive (best for 'Double', 'Int'; pinned memory)+-- * @'Data.Massiv.Array.U'@ — Unboxed (via @Data.Vector.Unboxed@)+-- * @'Data.Massiv.Array.S'@ — Storable (via @Foreign.ForeignPtr@; useful for FFI)+-- * @'Data.Massiv.Array.B'@ — Boxed (polymorphic but slower; GC overhead)+newtype Matrix (m :: Nat) (n :: Nat) r e = MkMatrix { unMatrix :: Array r Ix2 e }++deriving instance Show (Array r Ix2 e) => Show (Matrix m n r e)+deriving instance Eq (Array r Ix2 e) => Eq (Matrix m n r e)++instance NFData (Array r Ix2 e) => NFData (Matrix m n r e) where+  rnf (MkMatrix arr) = rnf arr++-- | A vector with compile-time known dimension \(n\).+--+-- Wraps a massiv @'Data.Massiv.Array.Array' r 'Data.Massiv.Array.Ix1' e@.+-- The phantom parameter \(n\) ensures that vector operations (e.g., 'dot',+-- 'axpy') are only applied to vectors of matching dimension.+newtype Vector (n :: Nat) r e = MkVector { unVector :: Array r Ix1 e }++deriving instance Show (Array r Ix1 e) => Show (Vector n r e)+deriving instance Eq (Array r Ix1 e) => Eq (Vector n r e)++instance NFData (Array r Ix1 e) => NFData (Vector n r e) where+  rnf (MkVector arr) = rnf arr++-- | Smart constructor for matrices. Checks at runtime that the array+-- dimensions match the type-level dimensions \(m\) and \(n\).+--+-- Returns 'Nothing' if the dimensions do not match.+--+-- @+-- let arr = M.makeArray Seq (Sz2 3 4) (\\(i :. j) -> fromIntegral (i + j))+-- matrix \@3 \@4 arr  -- Just (MkMatrix arr)+-- matrix \@2 \@4 arr  -- Nothing+-- @+matrix :: forall m n r e. (KnownDims m n, M.Size r)+       => Array r Ix2 e -> Maybe (Matrix m n r e)+matrix arr+  | M.Sz2 r c <- M.size arr+  , r == fromIntegral (natVal (Proxy @m))+  , c == fromIntegral (natVal (Proxy @n))+  = Just (MkMatrix arr)+  | otherwise = Nothing++-- | Smart constructor for vectors. Checks at runtime that the array+-- size matches the type-level dimension \(n\).+--+-- Returns 'Nothing' if the size does not match.+vector :: forall n r e. (KnownNat n, M.Size r)+       => Array r Ix1 e -> Maybe (Vector n r e)+vector arr+  | M.Sz1 n <- M.size arr+  , n == fromIntegral (natVal (Proxy @n))+  = Just (MkVector arr)+  | otherwise = Nothing++-- | Get the number of rows at the value level. \(O(1)\).+rows :: forall m n r e. KnownNat m => Matrix m n r e -> Int+rows _ = fromIntegral (natVal (Proxy @m))++-- | Get the number of columns at the value level. \(O(1)\).+cols :: forall m n r e. KnownNat n => Matrix m n r e -> Int+cols _ = fromIntegral (natVal (Proxy @n))++-- | Get the size of a vector at the value level. \(O(1)\).+size :: forall n r e. KnownNat n => Vector n r e -> Int+size _ = fromIntegral (natVal (Proxy @n))++-- | Existentially quantified matrix with runtime-determined dimensions.+--+-- Use 'someMatrix' to wrap a massiv array whose dimensions are not known+-- at compile time. Pattern matching on 'SomeMatrix' brings 'KnownNat'+-- evidence into scope:+--+-- @+-- case someMatrix arr of+--   SomeMatrix (mat :: Matrix m n r e) -> ...+--   -- m and n are now in scope as KnownNat+-- @+data SomeMatrix r e where+  SomeMatrix :: (KnownNat m, KnownNat n) => Matrix m n r e -> SomeMatrix r e++-- | Existentially quantified vector with runtime-determined dimensions.+data SomeVector r e where+  SomeVector :: KnownNat n => Vector n r e -> SomeVector r e++-- | Wrap a massiv 2D array into an existentially typed matrix.+someMatrix :: M.Size r => Array r Ix2 e -> SomeMatrix r e+someMatrix arr =+  let M.Sz2 r c = M.size arr+  in case someNatVal (fromIntegral r) of+    SomeNat (_ :: Proxy m) -> case someNatVal (fromIntegral c) of+      SomeNat (_ :: Proxy n) -> SomeMatrix @m @n (MkMatrix arr)++-- | Wrap a massiv 1D array into an existentially typed vector.+someVector :: M.Size r => Array r Ix1 e -> SomeVector r e+someVector arr =+  let M.Sz1 n = M.size arr+  in case someNatVal (fromIntegral n) of+    SomeNat (_ :: Proxy n) -> SomeVector @n (MkVector arr)
+ test/Spec.hs view
@@ -0,0 +1,18 @@+module Main (main) where++import Test.Tasty++import Test.BLAS (blasTests)+import Test.Solve (solveTests)+import Test.Orthogonal (orthogonalTests)+import Test.Eigen (eigenTests)+import Test.Norms (normTests)++main :: IO ()+main = defaultMain $ testGroup "linear-massiv"+  [ blasTests+  , solveTests+  , orthogonalTests+  , eigenTests+  , normTests+  ]
+ test/Test/BLAS.hs view
@@ -0,0 +1,172 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Test.BLAS (blasTests) where++import Test.Tasty+import Test.Tasty.QuickCheck+import Test.Tasty.HUnit+import qualified Data.Massiv.Array as M++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1+import Numeric.LinearAlgebra.Massiv.BLAS.Level2+import Numeric.LinearAlgebra.Massiv.BLAS.Level3+import Numeric.LinearAlgebra.Massiv.Norms (normFrob)+import Test.Types (genMatrix, genVector, hilbertMatrix, (~=), matApproxEq, vecApproxEq)+import Test.Residuals (machineEps)++blasTests :: TestTree+blasTests = testGroup "BLAS"+  [ testGroup "Level 1"+    [ testProperty "dot product commutative" prop_dotCommutative+    , testProperty "dot with zero vector" prop_dotZero+    , testCase "dot Hilbert columns" test_dotHilbertColumns+    , testProperty "axpy identity" prop_axpyIdentity+    , testProperty "scal by 1" prop_scalIdentity+    , testProperty "nrm2 non-negative" prop_nrm2NonNeg+    ]+  , testGroup "Level 2"+    [ testProperty "matvec with identity" prop_matvecIdentity+    , testProperty "gemv alpha=1 beta=0" prop_gemvSimple+    ]+  , testGroup "Level 3"+    [ testProperty "matMul with identity (left)" prop_matMulIdentityLeft+    , testProperty "matMul with identity (right)" prop_matMulIdentityRight+    , testProperty "transpose involution" prop_transposeInvolution+    , testProperty "mAdd commutative" prop_mAddCommutative+    , testProperty "matMul associative 5x5" prop_matMulAssociative5+    , testProperty "matMul with identity 10x10" prop_matMulIdentity10+    , testCase "3x3 matmul known" test_matMulKnown+    , testCase "gemm larger 3x3" test_gemmLarger+    ]+  ]++-- Level 1 properties++prop_dotCommutative :: Property+prop_dotCommutative = forAll ((,) <$> genVector @4 <*> genVector @4) $ \(x, y) ->+  dot x y ~= dot y x++prop_dotZero :: Property+prop_dotZero = forAll (genVector @4) $ \x ->+  let z = zeroVector @4 @M.P :: Vector 4 M.P Double+  in dot x z ~= 0++prop_axpyIdentity :: Property+prop_axpyIdentity = forAll (genVector @4) $ \x ->+  let z = zeroVector @4 @M.P :: Vector 4 M.P Double+  in vecApproxEq @4 (axpy 1 z x) x++prop_scalIdentity :: Property+prop_scalIdentity = forAll (genVector @4) $ \x ->+  vecApproxEq @4 (scal 1 x) x++prop_nrm2NonNeg :: Property+prop_nrm2NonNeg = forAll (genVector @4) $ \x ->+  nrm2 x >= (0 :: Double)++-- Level 2 properties++prop_matvecIdentity :: Property+prop_matvecIdentity = forAll (genVector @3) $ \x ->+  let eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in vecApproxEq @3 (matvec eye x) x++prop_gemvSimple :: Property+prop_gemvSimple = forAll ((,) <$> genMatrix @3 @3 <*> genVector @3) $ \(a, x) ->+  let z = zeroVector @3 @M.P :: Vector 3 M.P Double+      result = gemv 1.0 a x 0.0 z+      expected = matvec a x+  in vecApproxEq @3 result expected++-- Level 3 properties++prop_matMulIdentityLeft :: Property+prop_matMulIdentityLeft = forAll (genMatrix @3 @3) $ \a ->+  let eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in matApproxEq @3 @3 (matMul eye a) a++prop_matMulIdentityRight :: Property+prop_matMulIdentityRight = forAll (genMatrix @3 @3) $ \a ->+  let eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in matApproxEq @3 @3 (matMul a eye) a++prop_transposeInvolution :: Property+prop_transposeInvolution = forAll (genMatrix @3 @4) $ \a ->+  matApproxEq @3 @4 (transpose (transpose a)) a++prop_mAddCommutative :: Property+prop_mAddCommutative = forAll ((,) <$> genMatrix @3 @3 <*> genMatrix @3 @3) $ \(a, b) ->+  matApproxEq @3 @3 (mAdd a b) (mAdd b a)++-- | (AB)C ≈ A(BC) for random 5×5 matrices.+prop_matMulAssociative5 :: Property+prop_matMulAssociative5 =+  forAll ((,,) <$> genMatrix @5 @5 <*> genMatrix @5 @5 <*> genMatrix @5 @5) $ \(a, b, c) ->+    let lhs = matMul (matMul a b) c+        rhs = matMul a (matMul b c)+    in normFrob (mSub lhs rhs) / (normFrob lhs + 1e-15) < 1e-6++-- | I·A = A for 10×10 matrices.+prop_matMulIdentity10 :: Property+prop_matMulIdentity10 = forAll (genMatrix @10 @10) $ \a ->+  let eye = identityMatrix @10 @M.P :: Matrix 10 10 M.P Double+  in matApproxEq @10 @10 (matMul eye a) a++-- | Dot product of columns 0 and 1 of hilbertMatrix @5.+-- col0 = [1, 1/2, 1/3, 1/4, 1/5]+-- col1 = [1/2, 1/3, 1/4, 1/5, 1/6]+-- dot  = sum_{k=0}^{4} 1/((k+1)*(k+2)) = 1/2 + 1/6 + 1/12 + 1/20 + 1/30 = 50/60 = 5/6+test_dotHilbertColumns :: Assertion+test_dotHilbertColumns = do+  let h = hilbertMatrix @5 :: Matrix 5 5 M.P Double+      col0 = makeVector @5 @M.P $ \k -> h ! (k, 0)+      col1 = makeVector @5 @M.P $ \k -> h ! (k, 1)+      result = dot col0 col1+      expected = 5 / 6 :: Double+  assertBool ("dot of Hilbert cols 0,1 = 5/6, got " ++ show result)+    $ abs (result - expected) < 1e-12++-- Known-value test for 3×3 matrix multiplication+test_matMulKnown :: Assertion+test_matMulKnown = do+  -- A = [[1,2],[3,4]], B = [[5,6],[7,8]]+  -- AB = [[19,22],[43,50]]+  let a = makeMatrix @2 @2 @M.P $ \i j -> case (i,j) of+            (0,0) -> 1; (0,1) -> 2; (1,0) -> 3; (1,1) -> 4; _ -> 0 :: Double+      b = makeMatrix @2 @2 @M.P $ \i j -> case (i,j) of+            (0,0) -> 5; (0,1) -> 6; (1,0) -> 7; (1,1) -> 8; _ -> 0 :: Double+      c = matMul a b+  assertBool "c(0,0) = 19" $ (c ! (0,0)) ~= 19+  assertBool "c(0,1) = 22" $ (c ! (0,1)) ~= 22+  assertBool "c(1,0) = 43" $ (c ! (1,0)) ~= 43+  assertBool "c(1,1) = 50" $ (c ! (1,1)) ~= 50++-- | GEMM with α=2.0, β=0.5 on known 3×3 matrices.+-- A = [[1,2,3],[4,5,6],[7,8,9]]+-- B = [[9,8,7],[6,5,4],[3,2,1]]+-- C = [[1,0,0],[0,1,0],[0,0,1]]+-- Result = α*A*B + β*C+--+-- A*B = [[30,24,18],[84,69,54],[138,114,90]]+-- α*A*B = [[60,48,36],[168,138,108],[276,228,180]]+-- β*C   = [[0.5,0,0],[0,0.5,0],[0,0,0.5]]+-- Final = [[60.5,48,36],[168,138.5,108],[276,228,180.5]]+test_gemmLarger :: Assertion+test_gemmLarger = do+  let a = makeMatrix @3 @3 @M.P $ \i j ->+            fromIntegral (i * 3 + j + 1) :: Double+      b = makeMatrix @3 @3 @M.P $ \i j ->+            fromIntegral (9 - (i * 3 + j)) :: Double+      c = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+      result = gemm 2.0 a b 0.5 c+  assertBool "gemm(0,0) = 60.5" $ (result ! (0,0)) ~= 60.5+  assertBool "gemm(0,1) = 48"   $ (result ! (0,1)) ~= 48+  assertBool "gemm(0,2) = 36"   $ (result ! (0,2)) ~= 36+  assertBool "gemm(1,0) = 168"  $ (result ! (1,0)) ~= 168+  assertBool "gemm(1,1) = 138.5"$ (result ! (1,1)) ~= 138.5+  assertBool "gemm(1,2) = 108"  $ (result ! (1,2)) ~= 108+  assertBool "gemm(2,0) = 276"  $ (result ! (2,0)) ~= 276+  assertBool "gemm(2,1) = 228"  $ (result ! (2,1)) ~= 228+  assertBool "gemm(2,2) = 180.5"$ (result ! (2,2)) ~= 180.5
+ test/Test/Eigen.hs view
@@ -0,0 +1,587 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Test.Eigen (eigenTests) where++import Test.Tasty+import Test.Tasty.QuickCheck+import Test.Tasty.HUnit+import qualified Data.Massiv.Array as M+import Data.List (sort)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, matMulP, transpose, mSub)+import Numeric.LinearAlgebra.Massiv.BLAS.Level1 (nrm2, scal)+import Numeric.LinearAlgebra.Massiv.Eigen.Power+import Numeric.LinearAlgebra.Massiv.Eigen.Hessenberg+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric+import Numeric.LinearAlgebra.Massiv.Eigen.SVD (svd, svdP, svdGKP)+import Data.Proxy (Proxy(..))+import Numeric.LinearAlgebra.Massiv.Eigen.Schur (schur, eigenvalues)+import Numeric.LinearAlgebra.Massiv.Norms (normFrob, vnorm2)+import Test.Types+import Test.Residuals++eigenTests :: TestTree+eigenTests = testGroup "Eigenvalue"+  [ testGroup "Power Method"+    [ testCase "dominant eigenvalue of diagonal" test_powerDiagonal+    ]+  , testGroup "Hessenberg"+    [ testProperty "A = QHQᵀ reconstruction" prop_hessenbergReconstruction+    , testProperty "H is upper Hessenberg" prop_hessenbergForm+    ]+  , testGroup "Symmetric"+    [ testProperty "A = QΛQᵀ reconstruction" prop_symmetricEigenReconstruction+    , testProperty "Q orthogonal" prop_symmetricQOrthogonal+    , testCase "eigenvalues of diagonal" test_symmetricDiagonal+    ]+  , testGroup "Jacobi"+    [ testCase "jacobi eigenvalues of known matrix" test_jacobiKnown+    ]+  , testGroup "SVD"+    [ testProperty "A ≈ UΣVᵀ reconstruction" prop_svdReconstruction+    , testCase "singular values of diagonal" test_svdDiagonal+    , testCase "svdGKP reconstruction 10x10" test_svdGKReconstruction+    ]+  , testGroup "Standard test matrices"+    [ testCase "Wilkinson eigenvalues" test_wilkinsonEigen+    , testCase "Hilbert eigenvalues positive" test_hilbertEigen+    , testCase "Frank eigenvalues positive" test_frankEigen+    , testProperty "clustered eigenvalues" prop_clusteredEigen+    ]+  , testGroup "Eigen residuals"+    [ testProperty "eigenpair scaled residuals 3x3" prop_eigenResiduals+    ]+  , testGroup "SVD residuals"+    [ testProperty "SVD scaled residual 3x3" prop_svdScaledResidual+    , testProperty "SVD orthogonality U and V 3x3" prop_svdOrthogonality+    , testCase "SVD diagonal 5x5 sorted" test_svdDiagonalLarger+    ]+  , testGroup "Eigenvalue ordering"+    [ testProperty "symmetric eigenvalues sorted 4x4" prop_symmetricEigenvaluesSorted+    ]+  , testGroup "D&C eigensolver"+    [ testCase "D&C eigenvalues of diagonal 10x10" test_dcEigenDiagonal+    , testCase "D&C reconstruction 50x50" test_dcEigenReconstruction50+    , testCase "D&C orthogonality 50x50" test_dcEigenOrthogonal50+    , testCase "D&C matches QR at 30x30" test_dcMatchesQR+    , testCase "D&C orthogonality 30x30" test_dcOrtho30+    , testCase "D&C orthogonality 52x52" test_dcOrtho52+    , testCase "D&C orthogonality 60x60" test_dcOrtho60+    , testCase "D&C orthogonality 80x80" test_dcOrtho80+    , testCase "D&C orthogonality 90x90" test_dcOrtho90+    , testCase "D&C orthogonality 95x95" test_dcOrtho95+    , testCase "D&C ortho diagonal 100x100" test_dcOrthoDiag100+    , testCase "D&C ortho alt-matrix 100x100" test_dcOrthoAlt100+    , testCase "D&C reconstruction 100x100" test_dcEigenReconstruction100+    , testCase "D&C reconstruction 128x128" test_dcEigenReconstruction128+    ]+  , testGroup "Panel tridiag (n >= 256)"+    [ testCase "tridiag match 128x128" test_panelTridiagReconstruction128+    , testCase "eigenreconstruction 200x200" test_panelTridiagReconstruction200+    , testCase "orthogonality 200x200" test_panelTridiagOrthogonal200+    , testCase "eigenreconstruction 300x300" test_panelTridiagReconstruction300+    ]+  ]++-- Power method++test_powerDiagonal :: Assertion+test_powerDiagonal = do+  -- A = diag(3, 2, 1) → dominant eigenvalue = 3+  let a = makeMatrix @3 @3 @M.P $ \i j ->+            if i == j then case i of { 0 -> 3; 1 -> 2; _ -> 1 } else 0 :: Double+      q0 = makeVector @3 @M.P $ \_ -> 1 / sqrt 3 :: Double+      (lam, _) = powerMethod a q0 100 1e-10+  assertBool "eigenvalue ~ 3" $ abs (lam - 3) < 0.01++-- Hessenberg++prop_hessenbergReconstruction :: Property+prop_hessenbergReconstruction = forAll (genMatrix @4 @4) $ \a ->+  let (q, h) = hessenberg a+      qt = transpose q+      qhqt = matMul q (matMul h qt)+  in matApproxEq @4 @4 a qhqt++prop_hessenbergForm :: Property+prop_hessenbergForm = forAll (genMatrix @4 @4) $ \a ->+  let (_, h) = hessenberg a+  in all (\(i, j) -> abs (h ! (i, j)) < 1e-8)+     [(i, j) | i <- [0..3], j <- [0..3], i > j + 1]++-- Symmetric eigenvalue++prop_symmetricEigenReconstruction :: Property+prop_symmetricEigenReconstruction = forAll (genSPDMatrix @3) $ \a ->+  let (eigvals, q) = symmetricEigen a 500 1e-12+      qt = transpose q+      -- Reconstruct: A ≈ Q diag(λ) Qᵀ+      lambda = makeMatrix @3 @3 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      -- Use relaxed tolerance for iterative eigenvalue decomposition+  in all (\(i, j) -> abs (a ! (i,j) - qlqt ! (i,j)) < 1e-4 * (1 + abs (a ! (i,j))))+     [(i, j) | i <- [0..2], j <- [0..2]]++prop_symmetricQOrthogonal :: Property+prop_symmetricQOrthogonal = forAll (genSPDMatrix @3) $ \a ->+  let (_, q) = symmetricEigen a 500 1e-12+      qt = transpose q+      qtq = matMul qt q+      eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in matApproxEq @3 @3 qtq eye++test_symmetricDiagonal :: Assertion+test_symmetricDiagonal = do+  -- Eigenvalues of diag(5, 3, 1) should be {1, 3, 5}+  let a = makeMatrix @3 @3 @M.P $ \i j ->+            if i == j then case i of { 0 -> 5; 1 -> 3; _ -> 1 } else 0 :: Double+      (eigvals, _) = symmetricEigen a 100 1e-12+      evs = sort [eigvals !. 0, eigvals !. 1, eigvals !. 2]+  assertBool "eigenvalue 1" $ abs (evs !! 0 - 1) < 0.01+  assertBool "eigenvalue 3" $ abs (evs !! 1 - 3) < 0.01+  assertBool "eigenvalue 5" $ abs (evs !! 2 - 5) < 0.01++-- Jacobi++test_jacobiKnown :: Assertion+test_jacobiKnown = do+  let a = makeMatrix @3 @3 @M.P $ \i j -> case (i,j) of+            (0,0) -> 4; (0,1) -> 1; (0,2) -> 0+            (1,0) -> 1; (1,1) -> 3; (1,2) -> 1+            (2,0) -> 0; (2,1) -> 1; (2,2) -> 2+            _ -> 0 :: Double+      (eigvals, q) = jacobiEigen a 100 1e-12+      qt = transpose q+      lambda = makeMatrix @3 @3 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+  assertBool "Jacobi reconstruction" $ matApproxEq @3 @3 a qlqt++-- SVD++prop_svdReconstruction :: Property+prop_svdReconstruction = forAll (genMatrix @3 @3) $ \a ->+  let (u, sigma, v) = svd a+      vt = transpose v+      -- Reconstruct: U * diag(σ) * Vᵀ+      sigMat = makeMatrix @3 @3 @M.P $ \i j ->+        if i == j then sigma !. i else 0+      usv = matMul u (matMul sigMat vt)+  in matApproxEq @3 @3 a usv++test_svdDiagonal :: Assertion+test_svdDiagonal = do+  -- SVD of diag(5, 3, 1) should give singular values {5, 3, 1}+  let a = makeMatrix @3 @3 @M.P $ \i j ->+            if i == j then case i of { 0 -> 5; 1 -> 3; _ -> 1 } else 0 :: Double+      (_, sigma, _) = svd a+      svs = sort [sigma !. 0, sigma !. 1, sigma !. 2]+  assertBool "sv 1" $ abs (svs !! 0 - 1) < 0.1+  assertBool "sv 3" $ abs (svs !! 1 - 3) < 0.1+  assertBool "sv 5" $ abs (svs !! 2 - 5) < 0.1++test_svdGKReconstruction :: Assertion+test_svdGKReconstruction = do+  -- Test 1: diagonal matrix (trivial bidiag, no QR needed)+  let diag5 = makeMatrix @5 @5 @M.P $ \i j ->+                if i == j then fromIntegral (5 - i) else 0 :: Double+      (_, sigDiag, _) = svdGKP diag5+      diagSorted = sort [sigDiag !. i | i <- [0..4]]+      diagExpected = [1,2,3,4,5] :: [Double]+      diagErr = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) diagSorted diagExpected+  assertBool ("svdGKP diagonal sigma " ++ show diagSorted ++ " err=" ++ show diagErr) $ diagErr < 0.1+  -- Test 2: already-bidiagonal matrix (tests QR iteration in isolation)+  -- B = [[3,1,0],[0,2,1],[0,0,1]] — bidiag with d=[3,2,1], e=[1,1]+  let bidiag3 = makeMatrix @3 @3 @M.P $ \i j -> case (i,j) of+                  (0,0) -> 3; (0,1) -> 1; (1,1) -> 2; (1,2) -> 1; (2,2) -> 1+                  _ -> 0 :: Double+      (_, sigBidiag, _) = svdGKP bidiag3+      (_, sigBidiagRef, _) = svdP bidiag3+      bidiagSorted = sort [sigBidiag !. i | i <- [0..2]]+      bidiagRefSorted = sort [sigBidiagRef !. i | i <- [0..2]]+      bidiagDiff = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) bidiagSorted bidiagRefSorted+  assertBool ("svdGKP bidiag diff " ++ show bidiagDiff+              ++ "\n  gk=" ++ show bidiagSorted+              ++ "\n  ref=" ++ show bidiagRefSorted) $ bidiagDiff < 1e-6+  -- Test 3: general matrix+  let a = makeMatrix @5 @5 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in 1.0 / d + if i == j then 5 else 0+      (_, sigmaGK, _) = svdGKP a+      (_, sigmaRef, _) = svdP a+      gkSorted = sort [sigmaGK !. i | i <- [0..4]]+      refSorted = sort [sigmaRef !. i | i <- [0..4]]+      svDiff = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) gkSorted refSorted+  assertBool ("svdGKP sigma diff " ++ show svDiff ++ "\n  gk=" ++ show gkSorted+              ++ "\n  ref=" ++ show refSorted) $ svDiff < 1e-6+  -- Test 4: singular values match for 10×10+  let a10 = makeMatrix @10 @10 @M.P $ \i j ->+              let d = fromIntegral (abs (i - j) + 1) :: Double+              in 1.0 / d + if i == j then 10 else 0+      (u10, sig10, v10) = svdGKP a10+      (_, sigRef10, _) = svdP a10+      gk10Sorted = sort [sig10 !. i | i <- [0..9]]+      ref10Sorted = sort [sigRef10 !. i | i <- [0..9]]+      svDiff10 = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) gk10Sorted ref10Sorted+  assertBool ("svdGKP 10x10 sigma diff " ++ show svDiff10+              ++ "\n  gk=" ++ show gk10Sorted+              ++ "\n  ref=" ++ show ref10Sorted) $ svDiff10 < 1e-6+  -- Test 5: reconstruction A ≈ U Σ V^T for 10×10+  let sigMat10 = makeMatrix @10 @10 @M.P $ \i j ->+        if i == j then sig10 !. i else 0+      usv10 = matMulP u10 (matMulP sigMat10 (transpose v10))+      reconErr10 = maximum [abs (a10 ! (i,j) - usv10 ! (i,j))+                           | i <- [0..9], j <- [0..9]]+  assertBool ("svdGKP 10x10 reconstruction err " ++ show reconErr10) $ reconErr10 < 1e-10+  -- Test 6: orthogonality of U and V+  let utu = matMulP (transpose u10) u10+      vtv = matMulP (transpose v10) v10+      eye10 = identityMatrix @10 @M.P :: Matrix 10 10 M.P Double+      uErr = maximum [abs (utu ! (i,j) - eye10 ! (i,j)) | i <- [0..9], j <- [0..9]]+      vErr = maximum [abs (vtv ! (i,j) - eye10 ! (i,j)) | i <- [0..9], j <- [0..9]]+  assertBool ("svdGKP 10x10 U ortho err " ++ show uErr) $ uErr < 1e-10+  assertBool ("svdGKP 10x10 V ortho err " ++ show vErr) $ vErr < 1e-10++-- Standard test matrices++test_wilkinsonEigen :: Assertion+test_wilkinsonEigen = do+  let a = wilkinsonMatrix @7 :: Matrix 7 7 M.P Double+      (eigvals, q) = symmetricEigen a 500 1e-12+      nn = 7+      -- Verify we get 7 eigenvalues+      evList = map (\i -> eigvals !. i) [0..nn-1]+  assertBool "got 7 eigenvalues" $ length evList == 7+  -- Verify reconstruction: A ≈ Q diag(λ) Qᵀ+  let diag_lambda = makeMatrix @7 @7 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qt = transpose q+      qlqt = matMul q (matMul diag_lambda qt)+      residual = normFrob (mSub a qlqt) / (normFrob a + 1e-15)+  assertBool "Wilkinson reconstruction" $ residual < 1e-4++test_hilbertEigen :: Assertion+test_hilbertEigen = do+  let a = hilbertMatrix @5 :: Matrix 5 5 M.P Double+      (eigvals, _) = symmetricEigen a 500 1e-12+      evList = map (\i -> eigvals !. i) [0..4]+  -- Hilbert matrix is SPD, so all eigenvalues must be positive+  assertBool "all eigenvalues positive" $ all (> 0) evList++test_frankEigen :: Assertion+test_frankEigen = do+  let a = frankMatrix @5 :: Matrix 5 5 M.P Double+      (_, t) = schur a 200 1e-10+      evs = eigenvalues @5 t+  -- Frank matrix has all positive real eigenvalues+  assertBool "all eigenvalues positive" $ all (> 0) evs++prop_clusteredEigen :: Property+prop_clusteredEigen = withMaxSuccess 10 $ forAll (genClusteredEigenMatrix @4 5.0) $ \a ->+  let (eigvals, q) = symmetricEigen a 500 1e-12+      qt = transpose q+      diag_lambda = makeMatrix @4 @4 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul diag_lambda qt)+      residual = normFrob (mSub a qlqt) / (normFrob a + 1e-15)+      -- Relaxed tolerance since clustered eigenvalues are harder+  in residual < 1e-3++-- Eigen residuals++prop_eigenResiduals :: Property+prop_eigenResiduals = withMaxSuccess 20 $ forAll (genSPDMatrix @3) $ \a ->+  let (eigvals, q) = symmetricEigen a 500 1e-12+      nn = 3+      -- Check each eigenpair+      checks = map (\i ->+        let lambda_i = eigvals !. i+            v_i = makeVector @3 @M.P $ \k -> q ! (k, i)+        in scaledResidualEigen a lambda_i v_i < 1000+        ) [0..nn-1]+  in all id checks++-- SVD residuals++prop_svdScaledResidual :: Property+prop_svdScaledResidual = forAll (genMatrix @3 @3) $ \a ->+  let (u, sigma, v) = svd a+  in scaledResidualSVD a u sigma v < 1000++prop_svdOrthogonality :: Property+prop_svdOrthogonality = forAll (genMatrix @3 @3) $ \a ->+  let (u, _, v) = svd a+      -- U orthogonality can be looser because the SVD implementation+      -- constructs U columns as Av/sigma, which may accumulate error.+      -- V comes from eigendecomposition of A^T A so is typically tighter.+  in orthogonalityResidual @3 u < 500000 && orthogonalityResidual @3 v < 5000++test_svdDiagonalLarger :: Assertion+test_svdDiagonalLarger = do+  let a = makeMatrix @5 @5 @M.P $ \i j ->+            if i == j then case i of+              0 -> 7; 1 -> 5; 2 -> 3; 3 -> 2; _ -> 1+            else 0 :: Double+      (_, sigma, _) = svd a+      svs = sort [sigma !. 0, sigma !. 1, sigma !. 2, sigma !. 3, sigma !. 4]+      expected = [1, 2, 3, 5, 7] :: [Double]+  assertBool "sorted singular values match" $+    all (\(s, e) -> abs (s - e) < 0.1) (zip svs expected)++-- Eigenvalue ordering++prop_symmetricEigenvaluesSorted :: Property+prop_symmetricEigenvaluesSorted = forAll (genSPDMatrix @4) $ \a ->+  let (eigvals, _) = symmetricEigen a 500 1e-12+      evList = sort $ map (\i -> eigvals !. i) [0..3]+      -- Verify non-decreasing order after sorting+  in and $ zipWith (<=) evList (tail evList)++-- D&C eigensolver tests++test_dcEigenDiagonal :: Assertion+test_dcEigenDiagonal = do+  -- Eigenvalues of diag(10, 9, 8, ..., 1) should be {1..10}+  let a = makeMatrix @10 @10 @M.P $ \i j ->+            if i == j then fromIntegral (10 - i) else 0 :: Double+      (eigvals, _) = symmetricEigenPDC a 1e-12+      evs = sort [eigvals !. i | i <- [0..9]]+  mapM_ (\(i, expected) ->+    assertBool ("eigenvalue " ++ show expected) $+      abs (evs !! i - expected) < 0.01)+    (zip [0..] [1..10 :: Double])++test_dcEigenReconstruction50 :: Assertion+test_dcEigenReconstruction50 = do+  -- A = QΛQ^T reconstruction for a 50x50 SPD matrix+  let a = mkSPD50+      (eigvals, q) = symmetricEigenPDC a 1e-12+      qt = transpose q+      lambda = makeMatrix @50 @50 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..49], j <- [0..49]]+  assertBool ("reconstruction error " ++ show maxErr ++ " < 1e-8") $ maxErr < 1e-8++test_dcEigenOrthogonal50 :: Assertion+test_dcEigenOrthogonal50 = do+  let a = mkSPD50+      (_, q) = symmetricEigenPDC a 1e-12+      qt = transpose q+      qtq = matMul qt q+      eye = identityMatrix @50 @M.P :: Matrix 50 50 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..49], j <- [0..49]]+  assertBool ("orthogonality error " ++ show maxErr ++ " < 1e-8") $ maxErr < 1e-8++test_dcMatchesQR :: Assertion+test_dcMatchesQR = do+  -- D&C and QR should produce same eigenvalues for a 30x30 SPD matrix+  let a = makeMatrix @30 @30 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 30 + fromIntegral i else 1.0 / d+      (eigsDC, _) = symmetricEigenPDC a 1e-12+      (eigsQR, _) = symmetricEigenP a 3000 1e-12+      dcSorted = sort [eigsDC !. i | i <- [0..29]]+      qrSorted = sort [eigsQR !. i | i <- [0..29]]+      maxDiff = maximum $ zipWith (\a' b' -> abs (a' - b')) dcSorted qrSorted+  assertBool ("D&C vs QR diff " ++ show maxDiff ++ " < 1e-8") $ maxDiff < 1e-8++test_dcEigenReconstruction100 :: Assertion+test_dcEigenReconstruction100 = do+  let a = makeMatrix @100 @100 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 100 + fromIntegral i else 1.0 / d+      (eigvalsDC, qDC) = symmetricEigenPDC a 1e-12+      (eigvalsQR, _)   = symmetricEigenP a 10000 1e-12+      -- Compare eigenvalues+      dcSorted = sort [eigvalsDC !. i | i <- [0..99]]+      qrSorted = sort [eigvalsQR !. i | i <- [0..99]]+      evDiff = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) dcSorted qrSorted+  assertBool ("D&C 100 eigenvalue diff " ++ show evDiff) $ evDiff < 1e-6+  -- Check orthogonality of Q+  let qtq = matMulP (transpose qDC) qDC+      eye100 = identityMatrix @100 @M.P :: Matrix 100 100 M.P Double+      orthoErr = maximum [abs (qtq ! (i,j) - eye100 ! (i,j)) | i <- [0..99], j <- [0..99]]+  assertBool ("D&C 100 orthogonality error " ++ show orthoErr) $ orthoErr < 1e-6+  -- Full reconstruction+  let qt = transpose qDC+      lambda = makeMatrix @100 @100 @M.P $ \i j ->+        if i == j then eigvalsDC !. i else 0+      qlqt = matMul qDC (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..99], j <- [0..99]]+  assertBool ("D&C 100 reconstruction error " ++ show maxErr ++ " < 1e-7") $ maxErr < 1e-7++test_dcEigenReconstruction128 :: Assertion+test_dcEigenReconstruction128 = do+  let a = makeMatrix @128 @128 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 128 + fromIntegral i else 1.0 / d+      (eigvals, q) = symmetricEigenPDC a 1e-12+      qt = transpose q+      lambda = makeMatrix @128 @128 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..127], j <- [0..127]]+  assertBool ("D&C 128 reconstruction error " ++ show maxErr ++ " < 5e-7") $ maxErr < 5e-7++-- Panel tridiag tests (n >= 128 crossover)++test_panelTridiagReconstruction128 :: Assertion+test_panelTridiagReconstruction128 = do+  let nn = 128+      a = makeMatrix @128 @128 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 128 + fromIntegral i else 1.0 / d+      (eigvals, q) = symmetricEigenP a 10000 1e-12+      qt = transpose q+      lambda = makeMatrix @128 @128 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..nn-1], j <- [0..nn-1]]+  assertBool ("reconstruction error " ++ show maxErr ++ " < 1e-6") $ maxErr < 1e-6++test_panelTridiagReconstruction200 :: Assertion+test_panelTridiagReconstruction200 = do+  let nn = 200+      a = mkSPD200+      (eigvals, q) = symmetricEigenP a 10000 1e-12+      qt = transpose q+      lambda = makeMatrix @200 @200 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..nn-1], j <- [0..nn-1]]+  assertBool ("reconstruction error " ++ show maxErr ++ " < 1e-6") $ maxErr < 1e-6++test_panelTridiagOrthogonal200 :: Assertion+test_panelTridiagOrthogonal200 = do+  let nn = 200+      a = mkSPD200+      (_, q) = symmetricEigenP a 10000 1e-12+      qt = transpose q+      qtq = matMul qt q+      eye = identityMatrix @200 @M.P :: Matrix 200 200 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..nn-1], j <- [0..nn-1]]+  assertBool ("orthogonality error " ++ show maxErr ++ " < 1e-8") $ maxErr < 1e-8++test_panelTridiagReconstruction300 :: Assertion+test_panelTridiagReconstruction300 = do+  let nn = 300+      a = makeMatrix @300 @300 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 300 + fromIntegral i else 1.0 / d+      (eigvals, q) = symmetricEigenP a 15000 1e-12+      qt = transpose q+      lambda = makeMatrix @300 @300 @M.P $ \i j ->+        if i == j then eigvals !. i else 0+      qlqt = matMul q (matMul lambda qt)+      maxErr = maximum [abs (a ! (i,j) - qlqt ! (i,j)) | i <- [0..nn-1], j <- [0..nn-1]]+  assertBool ("reconstruction error " ++ show maxErr ++ " < 1e-6") $ maxErr < 1e-6++mkSPD200 :: Matrix 200 200 M.P Double+mkSPD200 = makeMatrix @200 @200 @M.P $ \i j ->+  let d = fromIntegral (abs (i - j) + 1) :: Double+  in if i == j then 200 + fromIntegral i else 1.0 / d++-- Helper: 50x50 SPD matrix for D&C tests+mkSPD50 :: Matrix 50 50 M.P Double+mkSPD50 = makeMatrix @50 @50 @M.P $ \i j ->+  let d = fromIntegral (abs (i - j) + 1) :: Double+  in if i == j then 50 + fromIntegral i else 1.0 / d++-- Granular D&C orthogonality tests at various sizes+test_dcOrtho30 :: Assertion+test_dcOrtho30 = do+  let a = makeMatrix @30 @30 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 30 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @30 @M.P :: Matrix 30 30 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..29], j <- [0..29]]+  assertBool ("D&C 30 ortho err " ++ show maxErr) $ maxErr < 1e-8++test_dcOrtho52 :: Assertion+test_dcOrtho52 = do+  let a = makeMatrix @52 @52 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 52 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @52 @M.P :: Matrix 52 52 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..51], j <- [0..51]]+  assertBool ("D&C 52 ortho err " ++ show maxErr) $ maxErr < 1e-8++test_dcOrtho60 :: Assertion+test_dcOrtho60 = do+  let a = makeMatrix @60 @60 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 60 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @60 @M.P :: Matrix 60 60 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..59], j <- [0..59]]+  assertBool ("D&C 60 ortho err " ++ show maxErr) $ maxErr < 1e-8++test_dcOrtho80 :: Assertion+test_dcOrtho80 = do+  let a = makeMatrix @80 @80 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 80 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @80 @M.P :: Matrix 80 80 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..79], j <- [0..79]]+  assertBool ("D&C 80 ortho err " ++ show maxErr) $ maxErr < 1e-8++test_dcOrtho90 :: Assertion+test_dcOrtho90 = do+  let a = makeMatrix @90 @90 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 90 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @90 @M.P :: Matrix 90 90 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..89], j <- [0..89]]+  assertBool ("D&C 90 ortho err " ++ show maxErr) $ maxErr < 1e-8++test_dcOrtho95 :: Assertion+test_dcOrtho95 = do+  let a = makeMatrix @95 @95 @M.P $ \i j ->+            let d = fromIntegral (abs (i - j) + 1) :: Double+            in if i == j then 95 + fromIntegral i else 1.0 / d+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @95 @M.P :: Matrix 95 95 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..94], j <- [0..94]]+  assertBool ("D&C 95 ortho err " ++ show maxErr) $ maxErr < 1e-8++-- Test D&C with a purely diagonal 100×100 matrix+test_dcOrthoDiag100 :: Assertion+test_dcOrthoDiag100 = do+  let a = makeMatrix @100 @100 @M.P $ \i j ->+            if i == j then fromIntegral (i + 1) else 0 :: Double+      (eigvals, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @100 @M.P :: Matrix 100 100 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..99], j <- [0..99]]+      evSorted = sort [eigvals !. i | i <- [0..99]]+      evDiff = maximum $ zipWith (\a_ b_ -> abs (a_ - b_)) evSorted [1..100]+  assertBool ("D&C diag100 ortho err " ++ show maxErr) $ maxErr < 1e-8+  assertBool ("D&C diag100 eigenvalue diff " ++ show evDiff) $ evDiff < 1e-8++-- Test D&C with a different matrix at 100×100 (sparser off-diagonal)+test_dcOrthoAlt100 :: Assertion+test_dcOrthoAlt100 = do+  let a = makeMatrix @100 @100 @M.P $ \i j ->+            if i == j then 500 + fromIntegral i+            else if abs (i - j) == 1 then 0.1+            else 0 :: Double+      (_, q) = symmetricEigenPDC a 1e-12+      qtq = matMulP (transpose q) q+      eye = identityMatrix @100 @M.P :: Matrix 100 100 M.P Double+      maxErr = maximum [abs (qtq ! (i,j) - eye ! (i,j)) | i <- [0..99], j <- [0..99]]+  assertBool ("D&C alt100 ortho err " ++ show maxErr) $ maxErr < 1e-8
+ test/Test/Norms.hs view
@@ -0,0 +1,112 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Test.Norms (normTests) where++import Test.Tasty+import Test.Tasty.QuickCheck+import Test.Tasty.HUnit+import qualified Data.Massiv.Array as M++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.Norms+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (transpose, matMul)+import Numeric.LinearAlgebra.Massiv.Eigen.SVD (singularValues)+import Test.Types++normTests :: TestTree+normTests = testGroup "Norms"+  [ testGroup "Vector norms"+    [ testProperty "vnorm2 non-negative" prop_vnorm2NonNeg+    , testProperty "vnorm1 non-negative" prop_vnorm1NonNeg+    , testProperty "vnormInf <= vnorm1" prop_vnormInfLeqVnorm1+    , testCase "known vector norm" test_knownVnorm+    ]+  , testGroup "Matrix norms"+    [ testProperty "normFrob non-negative" prop_normFrobNonNeg+    , testProperty "norm1 = normInf of transpose" prop_norm1TransposeNormInf+    , testProperty "triangle inequality (Frobenius)" prop_triangleInequality+    , testProperty "norm consistency with sigma_max" prop_normConsistency+    , testProperty "submultiplicativity" prop_submultiplicativity+    , testCase "known Frobenius norm" test_knownFrobNorm+    , testCase "Hilbert matrix norm relationships" test_hilbertNorms+    ]+  ]++-- Vector norms++prop_vnorm2NonNeg :: Property+prop_vnorm2NonNeg = forAll (genVector @4) $ \x ->+  vnorm2 x >= (0 :: Double)++prop_vnorm1NonNeg :: Property+prop_vnorm1NonNeg = forAll (genVector @4) $ \x ->+  vnorm1 x >= (0 :: Double)++prop_vnormInfLeqVnorm1 :: Property+prop_vnormInfLeqVnorm1 = forAll (genVector @4) $ \x ->+  vnormInf x <= vnorm1 x + 1e-12++test_knownVnorm :: Assertion+test_knownVnorm = do+  let v = makeVector @3 @M.P $ \i -> case i of { 0 -> 3; 1 -> 4; _ -> 0 } :: Double+  assertBool "vnorm2 of [3,4,0] = 5" $ abs (vnorm2 v - 5) < 1e-12+  assertBool "vnorm1 of [3,4,0] = 7" $ abs (vnorm1 v - 7) < 1e-12+  assertBool "vnormInf of [3,4,0] = 4" $ abs (vnormInf v - 4) < 1e-12++-- Matrix norms++prop_normFrobNonNeg :: Property+prop_normFrobNonNeg = forAll (genMatrix @3 @3) $ \a ->+  normFrob a >= (0 :: Double)++prop_norm1TransposeNormInf :: Property+prop_norm1TransposeNormInf = forAll (genMatrix @3 @4) $ \a ->+  let at = transpose a+  in abs (norm1 a - normInf at) < 1e-8++prop_triangleInequality :: Property+prop_triangleInequality = forAll ((,) <$> genMatrix @3 @3 <*> genMatrix @3 @3) $ \(a, b) ->+  let ab = makeMatrix @3 @3 @M.P $ \i j -> (a ! (i,j)) + (b ! (i,j))+  in normFrob ab <= normFrob a + normFrob b + 1e-10++-- | For 5×5 matrices: sigma_max ≤ normFrob A ≤ sqrt(5) * sigma_max.+--+-- The Frobenius norm satisfies ‖A‖_F = sqrt(sum sigma_i^2), so+-- sigma_max ≤ ‖A‖_F ≤ sqrt(n) * sigma_max.+prop_normConsistency :: Property+prop_normConsistency = forAll (genMatrix @5 @5) $ \a ->+  let sv = singularValues a+      sigmaMax = sv !. 0+      frobA = normFrob a+  in sigmaMax <= frobA + 1e-10+     && frobA <= sqrt 5 * sigmaMax + 1e-10++-- | Submultiplicativity: ‖A·B‖_F ≤ ‖A‖_F · ‖B‖_F for 3×3 matrices.+prop_submultiplicativity :: Property+prop_submultiplicativity =+  forAll ((,) <$> genMatrix @3 @3 <*> genMatrix @3 @3) $ \(a, b) ->+    normFrob (matMul a b) <= normFrob a * normFrob b + 1e-10++-- | Hilbert matrix norm relationships for hilbertMatrix @4:+-- norm1 ≤ normFrob * sqrt(n) and normFrob ≤ sqrt(n) * normInf.+test_hilbertNorms :: Assertion+test_hilbertNorms = do+  let h = hilbertMatrix @4 :: Matrix 4 4 M.P Double+      frobH = normFrob h+      n1H = norm1 h+      niH = normInf h+      sqrtN = sqrt 4 :: Double+  assertBool ("norm1 <= normFrob * sqrt(4): norm1=" ++ show n1H+              ++ " normFrob*2=" ++ show (frobH * sqrtN))+    $ n1H <= frobH * sqrtN + 1e-12+  assertBool ("normFrob <= sqrt(4) * normInf: normFrob=" ++ show frobH+              ++ " 2*normInf=" ++ show (sqrtN * niH))+    $ frobH <= sqrtN * niH + 1e-12++test_knownFrobNorm :: Assertion+test_knownFrobNorm = do+  -- ‖[[1,2],[3,4]]‖_F = sqrt(1+4+9+16) = sqrt(30)+  let a = makeMatrix @2 @2 @M.P $ \i j -> case (i,j) of+            (0,0) -> 1; (0,1) -> 2; (1,0) -> 3; (1,1) -> 4; _ -> 0 :: Double+  assertBool "Frobenius norm" $ abs (normFrob a - sqrt 30) < 1e-12
+ test/Test/Orthogonal.hs view
@@ -0,0 +1,198 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Test.Orthogonal (orthogonalTests) where++import Test.Tasty+import Test.Tasty.QuickCheck+import Test.Tasty.HUnit+import qualified Data.Massiv.Array as M++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose, mSub)+import Numeric.LinearAlgebra.Massiv.Orthogonal.Householder+import Numeric.LinearAlgebra.Massiv.Orthogonal.Givens+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR (qr, qrGivens)+import Numeric.LinearAlgebra.Massiv.Orthogonal.LeastSquares+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.Norms (normFrob, vnorm2)+import Test.Types+import Test.Residuals++orthogonalTests :: TestTree+orthogonalTests = testGroup "Orthogonal"+  [ testGroup "Householder"+    [ testProperty "Householder zeros subdiagonal" prop_householderZeros+    , testProperty "Householder matrix orthogonal" prop_householderOrthogonal+    ]+  , testGroup "Givens"+    [ testProperty "Givens zeros target" prop_givensZeros+    , testProperty "Givens preserves norm" prop_givensNorm+    ]+  , testGroup "QR"+    [ testProperty "A = QR reconstruction" prop_qrReconstruction+    , testProperty "Q orthogonal" prop_qOrthogonal+    , testProperty "R upper triangular" prop_rUpperTriangular+    , testCase "3x3 QR known" test_qrKnown+    ]+  , testGroup "Least Squares"+    [ testCase "overdetermined system" test_leastSquares+    ]+  , testGroup "QR scaled residuals"+    [ testProperty "QR scaled residual 5x5" prop_qrScaledResidual5+    , testProperty "Q orthogonality residual 5x5" prop_qOrthogonalityResidual5+    , testProperty "Q orthogonality residual 10x10" prop_qOrthogonalityResidual10+    ]+  , testGroup "QR reconstruction (larger)"+    [ testProperty "QR reconstruction 10x10" prop_qrReconstruction10x10+    , testProperty "QR reconstruction 10x5 rectangular" prop_qrReconstruction10x5+    ]+  , testGroup "Givens vs Householder"+    [ testProperty "Givens matches Householder 5x5" prop_givensMatchesHouseholder+    ]+  , testGroup "Least squares (property)"+    [ testProperty "normal equations 10x5" prop_leastSquares10x5+    ]+  ]++-- Householder tests++prop_householderZeros :: Property+prop_householderZeros = forAll (genVector @4) $ \x ->+  let (v, beta) = householderVector x+      h = householderMatrix @4 @M.P v beta+      hx = matvec h x+      -- All entries below first should be ~0+  in all (\i -> abs (hx !. i) < 1e-6) [1..3]++prop_householderOrthogonal :: Property+prop_householderOrthogonal = forAll (genVector @3) $ \x ->+  let (v, beta) = householderVector x+      h = householderMatrix @3 @M.P v beta+      ht = transpose h+      hht = matMul h ht+      eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in matApproxEq @3 @3 hht eye++-- Givens tests++prop_givensZeros :: Property+prop_givensZeros = forAll ((,) <$> choose (-10, 10) <*> choose (-10, 10)) $ \(a, b) ->+  let (c, s) = givensRotation a b+      -- Convention: [c, -s; s, c] * [a; b] = [r; 0]+      zero = s * a + c * b  -- should be ~0+  in abs zero < (1e-10 :: Double)++prop_givensNorm :: Property+prop_givensNorm = forAll ((,) <$> choose (-10, 10) <*> choose (-10, 10)) $ \(a, b) ->+  let (c, s) = givensRotation a (b :: Double)+      r = c * a - s * b  -- r = sqrt(a² + b²)+  in abs (r * r - (a * a + b * b)) < 1e-8++-- QR tests++prop_qrReconstruction :: Property+prop_qrReconstruction = forAll (genMatrix @4 @3) $ \a ->+  let (q, r) = qr a+      qr_ = matMul q r+  in matApproxEq @4 @3 a qr_++prop_qOrthogonal :: Property+prop_qOrthogonal = forAll (genMatrix @3 @3) $ \a ->+  let (q, _) = qr a+      qt = transpose q+      qtq = matMul qt q+      eye = identityMatrix @3 @M.P :: Matrix 3 3 M.P Double+  in matApproxEq @3 @3 qtq eye++prop_rUpperTriangular :: Property+prop_rUpperTriangular = forAll (genMatrix @3 @3) $ \a ->+  let (_, r) = qr a+  in all (\(i, j) -> abs (r ! (i, j)) < 1e-8)+     [(i, j) | i <- [0..2], j <- [0..2], i > j]++test_qrKnown :: Assertion+test_qrKnown = do+  let a = makeMatrix @3 @3 @M.P $ \i j -> case (i,j) of+            (0,0) -> 12; (0,1) -> -51; (0,2) -> 4+            (1,0) -> 6;  (1,1) -> 167; (1,2) -> -68+            (2,0) -> -4; (2,1) -> 24;  (2,2) -> -41+            _ -> 0 :: Double+      (q, r) = qr a+      qr_ = matMul q r+  assertBool "QR reconstruction" $ matApproxEq @3 @3 a qr_++-- Least squares test++test_leastSquares :: Assertion+test_leastSquares = do+  -- Overdetermined system: 4 equations, 2 unknowns+  -- A = [[1,1],[1,2],[1,3],[1,4]], b = [6,5,7,10]+  -- Least squares fit: y = a + bx+  let a = makeMatrix @4 @2 @M.P $ \i j -> case (i,j) of+            (0,0) -> 1; (0,1) -> 1+            (1,0) -> 1; (1,1) -> 2+            (2,0) -> 1; (2,1) -> 3+            (3,0) -> 1; (3,1) -> 4+            _ -> 0 :: Double+      b = makeVector @4 @M.P $ \i -> case i of+            0 -> 6; 1 -> 5; 2 -> 7; _ -> 10 :: Double+      x = leastSquaresQR a b+  -- The solution should minimize ‖Ax - b‖₂+  -- With these values: x ≈ [3.5, 1.4]+  assertBool "intercept reasonable" $ abs (x !. 0 - 3.5) < 0.5+  assertBool "slope reasonable" $ abs (x !. 1 - 1.4) < 0.5++-- QR scaled residual tests++prop_qrScaledResidual5 :: Property+prop_qrScaledResidual5 = forAll (genMatrix @5 @5) $ \a ->+  let (q, r) = qr a+  in scaledResidualQR a q r < 100++prop_qOrthogonalityResidual5 :: Property+prop_qOrthogonalityResidual5 = forAll (genMatrix @5 @5) $ \a ->+  let (q, _) = qr a+  in orthogonalityResidual @5 q < 100++prop_qOrthogonalityResidual10 :: Property+prop_qOrthogonalityResidual10 = withMaxSuccess 20 $ forAll (genMatrix @10 @10) $ \a ->+  let (q, _) = qr a+  in orthogonalityResidual @10 q < 100++-- QR reconstruction (larger)++prop_qrReconstruction10x10 :: Property+prop_qrReconstruction10x10 = forAll (genMatrix @10 @10) $ \a ->+  let (q, r) = qr a+  in normFrob (mSub a (matMul q r)) / (normFrob a + 1e-15) < 1e-6++prop_qrReconstruction10x5 :: Property+prop_qrReconstruction10x5 = withMaxSuccess 20 $ forAll (genMatrix @10 @5) $ \a ->+  let (q, r) = qr a+  in normFrob (mSub a (matMul q r)) / (normFrob a + 1e-15) < 1e-6++-- Givens vs Householder++prop_givensMatchesHouseholder :: Property+prop_givensMatchesHouseholder = forAll (genMatrix @5 @5) $ \a ->+  let (q1, r1) = qr a+      (q2, r2) = qrGivens a+      a1 = matMul q1 r1+      a2 = matMul q2 r2+  in normFrob (mSub a1 a2) / (normFrob a1 + 1e-15) < 1e-6++-- Least squares (property)++prop_leastSquares10x5 :: Property+prop_leastSquares10x5 = forAll ((,) <$> genMatrix @10 @5 <*> genVector @10) $ \(a, b) ->+  let x = leastSquaresQR a b+      -- Compute Ax as a 10-vector+      ax = matvec a x+      -- Compute residual r = Ax - b+      r_vec = makeVector @10 @M.P $ \i -> (ax !. i) - (b !. i)+      -- Compute A^T r as a 5-vector+      at = transpose a+      atr = matvec at r_vec+      -- Normal equations: A^T(Ax - b) should be approximately zero+  in vnorm2 atr / (vnorm2 b + 1e-15) < 1e-4
+ test/Test/Residuals.hs view
@@ -0,0 +1,233 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- | LAPACK-style scaled residual functions for NLA verification.+--+-- This module provides the standard residual metrics used by LAPACK+-- (Anderson et al., 1999) and recommended by Higham (2002) for+-- verifying numerical linear algebra routines. Rather than comparing+-- element-wise against a fixed tolerance, these functions compute+-- /scaled residuals/ that account for problem size, matrix norms,+-- and machine precision.+--+-- A scaled residual less than \(O(1)\) indicates backward stability;+-- values less than ~10--100 are considered acceptable.+--+-- __References:__+--+-- * Higham, N. J. (2002). /Accuracy and Stability of Numerical+--   Algorithms/, 2nd ed., SIAM. Chapter 1.+-- * Anderson, E. et al. (1999). /LAPACK Users' Guide/, 3rd ed., SIAM.+-- * LAPACK Working Note 41: Installation Guide.+-- * Golub, G. H. & Van Loan, C. F. (2013). /Matrix Computations/,+--   4th ed., Chapter 2.+module Test.Residuals+  ( -- * Machine epsilon+    machineEps+    -- * Scaled residual functions+  , scaledResidualLinear+  , scaledResidualEigen+  , scaledResidualQR+  , scaledResidualSVD+  , scaledResidualCholesky+  , scaledResidualLU+  , orthogonalityResidual+    -- * Condition-number-aware tolerance+  , conditionTolerance+  , safeCondition2+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix1, Ix2(..), Array)+import GHC.TypeNats (KnownNat)++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level1 (scal)+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose, mSub)+import Numeric.LinearAlgebra.Massiv.Norms (normFrob, normInf, vnorm2, vnormInf)+import Numeric.LinearAlgebra.Massiv.Eigen.SVD (singularValues)++-- | IEEE 754 double precision machine epsilon (unit roundoff).+--+-- \(\varepsilon = 2^{-52} \approx 2.22 \times 10^{-16}\)+--+-- This is the smallest value such that \(1 + \varepsilon > 1\) in+-- double precision floating-point arithmetic.+machineEps :: Double+machineEps = 2.220446049250313e-16++-- | Scaled residual for a linear system solve \(Ax = b\).+--+-- \[+-- \eta(\hat{x}) = \frac{\|A\hat{x} - b\|_\infty}+--                      {\|A\|_\infty \|\hat{x}\|_\infty + \|b\|_\infty}+-- \]+--+-- Per Higham (2002), Section 7.1, this is the normwise backward error.+-- A backward-stable solver should produce \(\eta \le O(n \varepsilon)\).+scaledResidualLinear :: forall n. KnownNat n+  => Matrix n n M.P Double -> Vector n M.P Double -> Vector n M.P Double -> Double+scaledResidualLinear a x b =+  let ax = matvec a x+      nn = dimVal @n+      -- residual r = Ax - b+      r = makeVector @n @M.P $ \i -> (ax !. i) - (b !. i)+      numr = vnormInf r+      denom = normInf a * vnormInf x + vnormInf b + fromIntegral nn * machineEps+  in numr / denom++-- | Scaled residual for an eigenpair \((\\lambda, v)\).+--+-- \[+-- \frac{\|Av - \lambda v\|_2}{\|A\|_F \|v\|_2}+-- \]+--+-- Per Higham (2002), Section 14.1. A well-computed eigenpair+-- should have residual \(O(n \varepsilon)\).+scaledResidualEigen :: forall n. KnownNat n+  => Matrix n n M.P Double -> Double -> Vector n M.P Double -> Double+scaledResidualEigen a lambda v =+  let av = matvec a v+      lambdaV = scal lambda v+      r = makeVector @n @M.P $ \i -> (av !. i) - (lambdaV !. i)+      numr = vnorm2 r+      denom = normFrob a * vnorm2 v + machineEps+  in numr / denom++-- | Scaled residual for QR factorization \(A = QR\).+--+-- \[+-- \frac{\|A - QR\|_F}{\|A\|_F \cdot n \cdot \varepsilon}+-- \]+--+-- Per LAPACK testing methodology (LAWN 41). A value less than+-- \(O(1)\) (typically < 10--100) indicates the factorization+-- is backward stable.+scaledResidualQR :: forall m n. (KnownNat m, KnownNat n)+  => Matrix m n M.P Double -> Matrix m m M.P Double -> Matrix m n M.P Double -> Double+scaledResidualQR a q r =+  let qr_prod = matMul q r+      diff = mSub a qr_prod+      nn = dimVal @n+      numr = normFrob diff+      denom = normFrob a * fromIntegral nn * machineEps + machineEps+  in numr / denom++-- | Scaled residual for SVD \(A = U \Sigma V^T\).+--+-- \[+-- \frac{\|A - U \Sigma V^T\|_F}{\|A\|_F \cdot \max(m,n) \cdot \varepsilon}+-- \]+--+-- Per LAPACK testing methodology (LAWN 41).+scaledResidualSVD :: forall m n. (KnownNat m, KnownNat n)+  => Matrix m n M.P Double -> Matrix m m M.P Double -> Vector n M.P Double+  -> Matrix n n M.P Double -> Double+scaledResidualSVD a u sigma v =+  let mm = dimVal @m+      nn = dimVal @n+      -- Build U * diag(sigma) by scaling columns of U+      -- Then multiply by V^T+      -- U is m x m, sigma has n entries, V is n x n+      -- We need U(:,0:n-1) * diag(sigma) * V^T+      -- Since our SVD returns m x m U, we take the first n columns conceptually+      uSigma = makeMatrix @m @n @M.P $ \i j ->+        (u ! (i, j)) * (sigma !. j)+      uSigmaVt = matMul uSigma (transpose v)+      diff = mSub a uSigmaVt+      numr = normFrob diff+      denom = normFrob a * fromIntegral (max mm nn) * machineEps + machineEps+  in numr / denom++-- | Scaled residual for Cholesky factorization \(A = GG^T\).+--+-- \[+-- \frac{\|A - GG^T\|_F}{\|A\|_F \cdot n \cdot \varepsilon}+-- \]+scaledResidualCholesky :: forall n. KnownNat n+  => Matrix n n M.P Double -> Matrix n n M.P Double -> Double+scaledResidualCholesky a g =+  let ggt = matMul g (transpose g)+      diff = mSub a ggt+      nn = dimVal @n+      numr = normFrob diff+      denom = normFrob a * fromIntegral nn * machineEps + machineEps+  in numr / denom++-- | Scaled residual for LU factorization \(PA = LU\).+--+-- \[+-- \frac{\|PA - LU\|_F}{\|A\|_F \cdot n \cdot \varepsilon}+-- \]+--+-- Takes the original matrix, the packed LU matrix, and the pivot array.+-- Extracts L (unit lower triangular) and U (upper triangular) from the+-- packed form.+scaledResidualLU :: forall n. KnownNat n+  => Matrix n n M.P Double -> Matrix n n M.P Double+  -> Array M.P Ix1 Int -> Double+scaledResidualLU a lu_packed pivots =+  let nn = dimVal @n+      -- Extract L (unit lower triangular)+      l = makeMatrix @n @n @M.P $ \i j ->+        if i == j then 1+        else if i > j then lu_packed ! (i, j)+        else 0+      -- Extract U (upper triangular)+      u_mat = makeMatrix @n @n @M.P $ \i j ->+        if i <= j then lu_packed ! (i, j)+        else 0+      -- Construct PA by applying the permutation+      pa = makeMatrix @n @n @M.P $ \i j ->+        let pi_i = M.index' pivots i+        in a ! (pi_i, j)+      lu_prod = matMul l u_mat+      diff = mSub pa lu_prod+      numr = normFrob diff+      denom = normFrob a * fromIntegral nn * machineEps + machineEps+  in numr / denom++-- | Orthogonality residual for a matrix \(Q\).+--+-- \[+-- \frac{\|Q^T Q - I\|_F}{n \cdot \varepsilon}+-- \]+--+-- Per LAPACK testing methodology. A value less than \(O(1)\)+-- means orthogonality is preserved to within rounding error.+orthogonalityResidual :: forall n. KnownNat n+  => Matrix n n M.P Double -> Double+orthogonalityResidual q =+  let nn = dimVal @n+      qtq = matMul (transpose q) q+      diff = mSub qtq (identityMatrix @n @M.P)+      numr = normFrob diff+      denom = fromIntegral nn * machineEps+  in numr / denom++-- | Condition-number-aware tolerance: \(\kappa \cdot n \cdot \varepsilon\).+--+-- For a linear system \(Ax = b\), the forward error satisfies+-- \(\|\hat{x} - x\| / \|x\| \le \kappa(A) \cdot \eta\) where+-- \(\eta\) is the backward error. A backward-stable algorithm+-- achieves \(\eta \approx n \varepsilon\), so the expected+-- forward error is \(O(\kappa \cdot n \cdot \varepsilon)\).+--+-- See Higham (2002), Theorem 7.2.+conditionTolerance :: Double -> Int -> Double+conditionTolerance kappa n = kappa * fromIntegral n * machineEps++-- | Estimate the 2-norm condition number \(\kappa_2(A) = \sigma_{\max} / \sigma_{\min}\).+--+-- Uses SVD to compute singular values. Returns \(10^{16}\) for+-- numerically singular matrices (where \(\sigma_{\min} < \varepsilon \cdot \sigma_{\max}\)).+safeCondition2 :: forall n. KnownNat n => Matrix n n M.P Double -> Double+safeCondition2 a =+  let sv = singularValues a+      nn = dimVal @n+      sigmaMax = sv !. 0+      sigmaMin = sv !. (nn - 1)+  in if abs sigmaMin < machineEps * abs sigmaMax+     then 1e16+     else abs sigmaMax / abs sigmaMin
+ test/Test/Solve.hs view
@@ -0,0 +1,278 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++module Test.Solve (solveTests) where++import Test.Tasty+import Test.Tasty.QuickCheck+import Test.Tasty.HUnit+import qualified Data.Massiv.Array as M++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level2 (matvec)+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose)+import Numeric.LinearAlgebra.Massiv.Solve.Triangular+import Numeric.LinearAlgebra.Massiv.Solve.LU+import Numeric.LinearAlgebra.Massiv.Solve.Cholesky+import Numeric.LinearAlgebra.Massiv.Solve.Banded (tridiagSolve)+import Numeric.LinearAlgebra.Massiv.Eigen.Symmetric (symmetricEigen)+import Test.Types+import Test.Residuals++solveTests :: TestTree+solveTests = testGroup "Solve"+  [ testGroup "Triangular"+    [ testProperty "forward sub: Lx = b roundtrip" prop_forwardSubRoundtrip+    , testProperty "back sub: Ux = b roundtrip" prop_backSubRoundtrip+    ]+  , testGroup "LU"+    [ testProperty "PA = LU reconstruction" prop_luReconstruction+    , testProperty "luSolve: Ax = b roundtrip" prop_luSolveRoundtrip+    , testCase "3x3 LU known" test_luKnown+    ]+  , testGroup "Cholesky"+    [ testProperty "A = GGᵀ reconstruction" prop_choleskyReconstruction+    , testProperty "choleskySolve roundtrip" prop_choleskySolveRoundtrip+    ]+  , testGroup "Tridiagonal"+    [ testCase "tridiag solve known" test_tridiagKnown+    ]+  , testGroup "Ill-conditioned"+    [ testCase "LU on Hilbert 5×5" test_luHilbert+    , testProperty "Cholesky near-singular SPD" prop_choleskyNearSingular+    ]+  , testGroup "Randsvd"+    [ testProperty "luSolve with randsvd matrix" prop_luSolveRandsvd+    , testProperty "choleskySolve with randsvd SPD" prop_choleskySolveRandsvd+    ]+  , testGroup "Scaled Residuals"+    [ testProperty "LU residual bound 3×3" prop_luResidualBound+    , testProperty "Cholesky residual bound 3×3" prop_choleskyResidualBound+    ]+  , testGroup "Cross-module"+    [ testCase "det ≈ product of eigenvalues" test_detEqualsEigenProduct+    ]+  , testGroup "Larger Sizes"+    [ testProperty "LU reconstruction 5×5" prop_luReconstruction5+    , testProperty "Cholesky reconstruction 5×5" prop_choleskyReconstruction5+    , testProperty "luSolve 10×10" prop_luSolve10+    ]+  ]++-- Triangular tests++prop_forwardSubRoundtrip :: Property+prop_forwardSubRoundtrip = forAll ((,) <$> genLowerTriangular @3 <*> genVector @3) $ \(l, b) ->+  let x = forwardSub l b+      b' = matvec l x+  in vecApproxEq @3 b b'++prop_backSubRoundtrip :: Property+prop_backSubRoundtrip = forAll ((,) <$> genUpperTriangular @3 <*> genVector @3) $ \(u, b) ->+  let x = backSub u b+      b' = matvec u x+  in vecApproxEq @3 b b'++-- LU tests++prop_luReconstruction :: Property+prop_luReconstruction = forAll (genMatrix @3 @3) $ \a ->+  let (luMat, pivArr) = lu a+      nn = 3 :: Int+      l = makeMatrix @3 @3 @M.P $ \i j ->+        if i == j then 1+        else if i > j then luMat ! (i, j)+        else 0 :: Double+      u = makeMatrix @3 @3 @M.P $ \i j ->+        if i <= j then luMat ! (i, j)+        else 0 :: Double+      lu_ = matMul l u+      -- PA+      pa = makeMatrix @3 @3 @M.P $ \i j ->+        a ! (M.index' pivArr i, j)+  in matApproxEq @3 @3 pa lu_++prop_luSolveRoundtrip :: Property+prop_luSolveRoundtrip = forAll ((,) <$> genMatrix @3 @3 <*> genVector @3) $ \(a, b) ->+  -- Only test for non-singular matrices (skip near-singular)+  let (_, pivArr) = lu a+      luMat = fst (lu a)+      -- Check diagonal of U+      diagOk = all (\i -> abs (luMat ! (i, i)) > 1e-6) [0..2]+  in diagOk ==>+    let x = luSolve a b+        b' = matvec a x+    in vecApproxEq @3 b b'++test_luKnown :: Assertion+test_luKnown = do+  -- A = [[2,1,1],[4,3,3],[8,7,9]]+  let a = makeMatrix @3 @3 @M.P $ \i j -> case (i,j) of+            (0,0) -> 2; (0,1) -> 1; (0,2) -> 1+            (1,0) -> 4; (1,1) -> 3; (1,2) -> 3+            (2,0) -> 8; (2,1) -> 7; (2,2) -> 9+            _ -> 0 :: Double+      b = makeVector @3 @M.P $ \i -> case i of+            0 -> 1; 1 -> 1; _ -> 1 :: Double+      x = luSolve a b+      b' = matvec a x+  assertBool "LU solve roundtrip" $ vecApproxEq @3 b b'++-- Cholesky tests++prop_choleskyReconstruction :: Property+prop_choleskyReconstruction = forAll (genSPDMatrix @3) $ \a ->+  let g = cholesky a+      gt = transpose g+      ggt = matMul g gt+  in matApproxEq @3 @3 a ggt++prop_choleskySolveRoundtrip :: Property+prop_choleskySolveRoundtrip = forAll ((,) <$> genSPDMatrix @3 <*> genVector @3) $ \(a, b) ->+  let x = choleskySolve a b+      b' = matvec a x+  in vecApproxEq @3 b b'++-- Tridiagonal test++test_tridiagKnown :: Assertion+test_tridiagKnown = do+  -- Tridiagonal SPD: A = [[2,-1,0],[-1,2,-1],[0,-1,2]]+  -- b = [1, 0, 1]+  -- x should be [1, 1, 1]+  let diag_ = makeVector @3 @M.P $ \_ -> 2 :: Double+      supdiag = makeVector @3 @M.P $ \i -> if i < 2 then -1 else 0 :: Double+      b = makeVector @3 @M.P $ \i -> case i of { 0 -> 1; 1 -> 0; _ -> 1 } :: Double+      x = tridiagSolve diag_ supdiag b+      -- Verify Ax = b manually: 2*1 + (-1)*1 = 1, (-1)*1 + 2*1 + (-1)*1 = 0, (-1)*1 + 2*1 = 1+  assertBool "x(0) ~ 1" $ (x !. 0) ~= 1+  assertBool "x(1) ~ 1" $ (x !. 1) ~= 1+  assertBool "x(2) ~ 1" $ (x !. 2) ~= 1++------------------------------------------------------------------------+-- Ill-conditioned tests+------------------------------------------------------------------------++-- | Test 1: LU solve on the 5×5 Hilbert matrix (very ill-conditioned).+test_luHilbert :: Assertion+test_luHilbert = do+  let h = hilbertMatrix @5+      x = makeVector @5 @M.P $ \i -> fromIntegral (i + 1)+      b = matvec h x+      x' = luSolve h b+      residual = scaledResidualLinear @5 h x' b+  assertBool ("LU Hilbert residual too large: " ++ show residual)+    (residual < 1e-4)++-- | Test 2: Cholesky on near-singular SPD matrix (cond = 1e4).+prop_choleskyNearSingular :: Property+prop_choleskyNearSingular = withMaxSuccess 20 $+  forAll (genSPDMatrixWithCond @5 1e4) $ \a ->+    let g = cholesky a+        residual = scaledResidualCholesky @5 a g+    in counterexample ("Cholesky residual = " ++ show residual) $+       residual < 1000++------------------------------------------------------------------------+-- Randsvd tests+------------------------------------------------------------------------++-- | Test 3: LU solve with a randsvd matrix (cond = 100).+prop_luSolveRandsvd :: Property+prop_luSolveRandsvd = withMaxSuccess 30 $+  forAll ((,) <$> genMatrixWithCond @5 @5 100.0 <*> genVector @5) $ \(a, b) ->+    let x = luSolve a b+        residual = scaledResidualLinear @5 a x b+    in counterexample ("LU randsvd residual = " ++ show residual) $+       residual < 0.01++-- | Test 4: Cholesky solve with a randsvd SPD matrix (cond = 100).+prop_choleskySolveRandsvd :: Property+prop_choleskySolveRandsvd = withMaxSuccess 20 $+  forAll ((,) <$> genSPDMatrixWithCond @5 100.0 <*> genVector @5) $ \(a, b) ->+    let x = choleskySolve a b+        residual = scaledResidualLinear @5 a x b+    in counterexample ("Cholesky randsvd residual = " ++ show residual) $+       residual < 0.01++------------------------------------------------------------------------+-- Scaled residual bound tests+------------------------------------------------------------------------++-- | Test 5: LU residual bound for well-conditioned 3×3 random matrices.+prop_luResidualBound :: Property+prop_luResidualBound = forAll ((,) <$> genMatrix @3 @3 <*> genVector @3) $ \(a, b) ->+  let (luMat, _) = lu a+      diagOk = all (\i -> abs (luMat ! (i, i)) > 1e-6) [0..2]+  in diagOk ==>+    let x = luSolve a b+        residual = scaledResidualLinear @3 a x b+    in counterexample ("LU residual = " ++ show residual) $+       residual < 1e-6++-- | Test 6: Cholesky residual bound for well-conditioned 3×3 SPD matrices.+prop_choleskyResidualBound :: Property+prop_choleskyResidualBound = forAll ((,) <$> genSPDMatrix @3 <*> genVector @3) $ \(a, b) ->+  let x = choleskySolve a b+      residual = scaledResidualLinear @3 a x b+  in counterexample ("Cholesky residual = " ++ show residual) $+     residual < 1e-6++------------------------------------------------------------------------+-- Cross-module tests+------------------------------------------------------------------------++-- | Test 7: Determinant equals the product of eigenvalues for a known SPD matrix.+--+-- We use a diagonal-dominant matrix: diag(4,3,2,1) + 0.1*ones(4,4).+test_detEqualsEigenProduct :: Assertion+test_detEqualsEigenProduct = do+  let a = makeMatrix @4 @4 @M.P $ \i j ->+            let diag_ = case i of { 0 -> 4; 1 -> 3; 2 -> 2; _ -> 1 } :: Double+            in (if i == j then diag_ else 0) + 0.1+      d = det a+      (eigenvals, _) = symmetricEigen a 200 1e-12+      eigenProd = product [ eigenvals !. i | i <- [0..3] ]+      relErr = abs (d - eigenProd) / (abs d + 1e-15)+  assertBool ("det vs eigenproduct relative error too large: " ++ show relErr)+    (relErr < 1e-4)++------------------------------------------------------------------------+-- Larger-size tests+------------------------------------------------------------------------++-- | Test 8: LU reconstruction at 5×5 (like existing 3×3 test).+prop_luReconstruction5 :: Property+prop_luReconstruction5 = forAll (genMatrix @5 @5) $ \a ->+  let (luMat, pivArr) = lu a+      l = makeMatrix @5 @5 @M.P $ \i j ->+        if i == j then 1+        else if i > j then luMat ! (i, j)+        else 0 :: Double+      u = makeMatrix @5 @5 @M.P $ \i j ->+        if i <= j then luMat ! (i, j)+        else 0 :: Double+      lu_ = matMul l u+      pa = makeMatrix @5 @5 @M.P $ \i j ->+        a ! (M.index' pivArr i, j)+  in matApproxEq @5 @5 pa lu_++-- | Test 9: Cholesky reconstruction at 5×5 (like existing 3×3 test).+prop_choleskyReconstruction5 :: Property+prop_choleskyReconstruction5 = forAll (genSPDMatrix @5) $ \a ->+  let g = cholesky a+      gt = transpose g+      ggt = matMul g gt+  in matApproxEq @5 @5 a ggt++-- | Test 10: LU solve at 10×10 with diagonal guard.+prop_luSolve10 :: Property+prop_luSolve10 = withMaxSuccess 20 $+  forAll ((,) <$> genMatrix @10 @10 <*> genVector @10) $ \(a, b) ->+    let (luMat, _) = lu a+        diagOk = all (\i -> abs (luMat ! (i, i)) > 1e-6) [0..9]+    in diagOk ==>+      let x = luSolve a b+          residual = scaledResidualLinear @10 a x b+      in counterexample ("LU 10x10 residual = " ++ show residual) $+         residual < 1e-6
+ test/Test/Types.hs view
@@ -0,0 +1,256 @@+{-# LANGUAGE AllowAmbiguousTypes #-}++-- | Test helpers: Arbitrary instances, approximate equality, matrix generators.+module Test.Types+  ( -- * Approximate equality+    (~=)+  , matApproxEq+  , vecApproxEq+    -- * Matrix generators+  , genMatrix+  , genVector+  , genSPDMatrix+  , genUpperTriangular+  , genLowerTriangular+  , genSymmetric+    -- * Tolerance+  , defaultTol+    -- * Standard test matrices+  , hilbertMatrix+  , wilkinsonMatrix+  , frankMatrix+  , hadamardMatrix+    -- * Generators with controlled properties+  , genMatrixWithCond+  , genNearSingularMatrix+  , genClusteredEigenMatrix+  , genSPDMatrixWithCond+  ) where++import qualified Data.Massiv.Array as M+import Data.Massiv.Array (Ix2(..), Sz(..), Comp(..))+import GHC.TypeNats (KnownNat, natVal)+import Data.Proxy (Proxy(..))+import Data.Bits (popCount)+import qualified Data.Bits as Bits+import Test.QuickCheck++import Numeric.LinearAlgebra.Massiv.Types+import Numeric.LinearAlgebra.Massiv.Internal+import Numeric.LinearAlgebra.Massiv.BLAS.Level3 (matMul, transpose)+import Numeric.LinearAlgebra.Massiv.Orthogonal.QR (qr)++-- | Default tolerance for floating-point comparisons.+defaultTol :: Double+defaultTol = 1e-8++-- | Approximate equality for scalars.+(~=) :: Double -> Double -> Bool+x ~= y = abs (x - y) < defaultTol * (1 + abs x + abs y)++-- | Approximate equality for matrices.+matApproxEq :: forall m n. (KnownNat m, KnownNat n)+            => Matrix m n M.P Double -> Matrix m n M.P Double -> Bool+matApproxEq a b =+  let r = dimVal @m+      c = dimVal @n+  in all (\(i, j) -> (a ! (i, j)) ~= (b ! (i, j)))+     [(i, j) | i <- [0..r-1], j <- [0..c-1]]++-- | Approximate equality for vectors.+vecApproxEq :: forall n. KnownNat n+            => Vector n M.P Double -> Vector n M.P Double -> Bool+vecApproxEq a b =+  let nn = dimVal @n+  in all (\i -> (a !. i) ~= (b !. i)) [0..nn-1]++-- | Generate a random m×n matrix with entries in [-10, 10].+genMatrix :: forall m n. (KnownNat m, KnownNat n) => Gen (Matrix m n M.P Double)+genMatrix = do+  let r = fromIntegral (natVal (Proxy @m))+      c = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf (r * c) (choose (-10, 10))+  pure $ makeMatrix @m @n @M.P $ \i j -> entries !! (i * c + j)++-- | Generate a random n-element vector with entries in [-10, 10].+genVector :: forall n. KnownNat n => Gen (Vector n M.P Double)+genVector = do+  let nn = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf nn (choose (-10, 10))+  pure $ makeVector @n @M.P $ \i -> entries !! i++-- | Generate a symmetric positive definite matrix: A = BBᵀ + εI.+genSPDMatrix :: forall n. KnownNat n => Gen (Matrix n n M.P Double)+genSPDMatrix = do+  let nn = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf (nn * nn) (choose (-5, 5))+  let b = makeMatrix @n @n @M.P $ \i j -> entries !! (i * nn + j)+      -- BBᵀ + εI+      epsilon = 0.1 :: Double+  pure $ makeMatrix @n @n @M.P $ \i j ->+    let bbT = foldl' (\acc k -> acc + (b ! (i, k)) * (b ! (j, k))) 0 [0..nn-1]+    in bbT + if i == j then epsilon else 0++-- | Generate an upper triangular matrix with nonzero diagonal.+genUpperTriangular :: forall n. KnownNat n => Gen (Matrix n n M.P Double)+genUpperTriangular = do+  let nn = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf (nn * nn) (choose (-10, 10))+  diags <- vectorOf nn (choose (1, 10))  -- Ensure nonzero diagonal+  pure $ makeMatrix @n @n @M.P $ \i j ->+    if i == j then diags !! i+    else if i < j then entries !! (i * nn + j)+    else 0++-- | Generate a lower triangular matrix with nonzero diagonal.+genLowerTriangular :: forall n. KnownNat n => Gen (Matrix n n M.P Double)+genLowerTriangular = do+  let nn = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf (nn * nn) (choose (-10, 10))+  diags <- vectorOf nn (choose (1, 10))+  pure $ makeMatrix @n @n @M.P $ \i j ->+    if i == j then diags !! i+    else if i > j then entries !! (i * nn + j)+    else 0++-- | Generate a symmetric matrix.+genSymmetric :: forall n. KnownNat n => Gen (Matrix n n M.P Double)+genSymmetric = do+  let nn = fromIntegral (natVal (Proxy @n))+  entries <- vectorOf (nn * nn) (choose (-10, 10))+  pure $ makeMatrix @n @n @M.P $ \i j ->+    if i <= j then entries !! (i * nn + j)+    else entries !! (j * nn + i)++------------------------------------------------------------------------+-- Standard test matrices+------------------------------------------------------------------------++-- | Hilbert matrix of order /n/: \(H_{ij} = 1/(i+j+1)\).+--+-- The Hilbert matrix is symmetric positive definite with an+-- exponentially growing condition number. Its inverse has integer+-- entries.+--+-- /Reference:/ Higham (2002), Section 28.1; GVL4 p. 128.+hilbertMatrix :: forall n. KnownNat n => Matrix n n M.P Double+hilbertMatrix = makeMatrix @n @n @M.P $ \i j ->+  1 / fromIntegral (i + j + 1)++-- | Wilkinson matrix \(W_n\): symmetric tridiagonal with+-- diagonal \(|i - \lfloor n/2 \rfloor|\) and unit sub/superdiagonal.+--+-- Has near-degenerate eigenvalue pairs that stress eigenvalue solvers.+--+-- /Reference:/ Higham (2002), Section 28.6; MATLAB @gallery('wilk', n)@.+wilkinsonMatrix :: forall n. KnownNat n => Matrix n n M.P Double+wilkinsonMatrix =+  let nn = dimVal @n+      mid = nn `div` 2+  in makeMatrix @n @n @M.P $ \i j ->+    if i == j then fromIntegral (abs (i - mid))+    else if abs (i - j) == 1 then 1+    else 0++-- | Frank matrix of order /n/: upper Hessenberg with+-- \(\det(F) = 1\) and known positive real eigenvalues.+--+-- \(F_{ij} = n - \max(i,j)\) for \(j \ge i-1\), zero otherwise+-- (0-indexed).+--+-- /Reference:/ Higham (2002), Section 28.5; Frank (1958).+frankMatrix :: forall n. KnownNat n => Matrix n n M.P Double+frankMatrix =+  let nn = dimVal @n+  in makeMatrix @n @n @M.P $ \i j ->+    if j >= i - 1 && i - 1 >= 0 || j >= i+    then fromIntegral (nn - max i j)+    else 0++-- | Hadamard matrix of order /n/ via the Sylvester\/Walsh construction.+--+-- \(H_{ij} = (-1)^{\mathrm{popcount}(i \mathbin{\&} j)}\)+--+-- Produces a proper Hadamard matrix when /n/ is a power of 2,+-- satisfying \(H^T H = n I\). All entries are \(\pm 1\).+--+-- /Reference:/ Higham (2002), Section 28.3.+hadamardMatrix :: forall n. KnownNat n => Matrix n n M.P Double+hadamardMatrix = makeMatrix @n @n @M.P $ \i j ->+  if even (popCount ((Bits..&.) i j)) then 1 else -1++------------------------------------------------------------------------+-- Generators with controlled properties+------------------------------------------------------------------------++-- | Generate a matrix with prescribed 2-norm condition number /κ/.+--+-- Uses the @randsvd@ construction: \(A = U \Sigma V^T\) where+-- \(U\) and \(V\) are random orthogonal matrices obtained from+-- QR factorization of random matrices, and the singular values+-- are geometrically spaced from 1 to \(1/\kappa\).+--+-- /Reference:/ Higham (2002), Section 28.3; Fasi & Higham,+-- "Generating Extreme-Scale Matrices With Specified Singular+-- Values or Condition Number," SIAM J. Sci. Comput. 43(5), 2021.+genMatrixWithCond :: forall m n. (KnownNat m, KnownNat n)+  => Double -> Gen (Matrix m n M.P Double)+genMatrixWithCond kappa = do+  uRaw <- genMatrix @m @m+  vRaw <- genMatrix @n @n+  let (u, _) = qr uRaw+      (v, _) = qr vRaw+      mm = dimVal @m+      nn = dimVal @n+      minDim = min mm nn+      -- Singular values geometrically spaced: sigma_0 = 1, sigma_{minDim-1} = 1/kappa+      sigma = makeMatrix @m @n @M.P $ \i j ->+        if i == j && i < minDim+        then if minDim <= 1 then 1+             else let t = fromIntegral i / fromIntegral (minDim - 1)+                  in kappa ** (-t)+        else 0+  pure $ matMul u (matMul sigma (transpose v))++-- | Generate a near-singular matrix with smallest singular value /ε/.+--+-- Wrapper around 'genMatrixWithCond' with \(\kappa = 1/\varepsilon\).+genNearSingularMatrix :: forall n. KnownNat n+  => Double -> Gen (Matrix n n M.P Double)+genNearSingularMatrix smallSV = genMatrixWithCond @n @n (1 / smallSV)++-- | Generate a symmetric matrix with clustered eigenvalues near /c/.+--+-- Constructs \(A = Q \Lambda Q^T\) where \(\Lambda\) is diagonal+-- with entries \(c + i \cdot 10^{-6}\) and \(Q\) is a random+-- orthogonal matrix. This stresses iterative eigenvalue solvers+-- that must resolve near-degenerate eigenvalue pairs.+genClusteredEigenMatrix :: forall n. KnownNat n+  => Double -> Gen (Matrix n n M.P Double)+genClusteredEigenMatrix cluster = do+  qRaw <- genMatrix @n @n+  let (q, _) = qr qRaw+      qt = transpose q+      lambda = makeMatrix @n @n @M.P $ \i j ->+        if i == j then cluster + fromIntegral i * 1e-6 else 0+  pure $ matMul q (matMul lambda qt)++-- | Generate an SPD matrix with prescribed condition number /κ/.+--+-- Uses the @randsvd@ construction with \(U = V\) (ensuring symmetry):+-- \(A = Q \Lambda Q^T\) where eigenvalues are geometrically spaced+-- from 1 to \(1/\kappa\).+genSPDMatrixWithCond :: forall n. KnownNat n+  => Double -> Gen (Matrix n n M.P Double)+genSPDMatrixWithCond kappa = do+  qRaw <- genMatrix @n @n+  let (q, _) = qr qRaw+      qt = transpose q+      nn = dimVal @n+      lambda = makeMatrix @n @n @M.P $ \i j ->+        if i == j+        then if nn <= 1 then 1+             else let t = fromIntegral i / fromIntegral (nn - 1)+                  in kappa ** (-t)+        else 0+  pure $ matMul q (matMul lambda qt)