hmatrix-0.15.2.0: lib/Numeric/LinearAlgebra/Algorithms.hs
{-# LANGUAGE FlexibleContexts, FlexibleInstances #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE TypeFamilies #-}
-----------------------------------------------------------------------------
{- |
Module : Numeric.LinearAlgebra.Algorithms
Copyright : (c) Alberto Ruiz 2006-9
License : GPL-style
Maintainer : Alberto Ruiz (aruiz at um dot es)
Stability : provisional
Portability : uses ffi
High level generic interface to common matrix computations.
Specific functions for particular base types can also be explicitly
imported from "Numeric.LinearAlgebra.LAPACK".
-}
-----------------------------------------------------------------------------
module Numeric.LinearAlgebra.Algorithms (
-- * Supported types
Field(),
-- * Linear Systems
linearSolve,
luSolve,
cholSolve,
linearSolveLS,
linearSolveSVD,
inv, pinv, pinvTol,
det, invlndet,
rank, rcond,
-- * Matrix factorizations
-- ** Singular value decomposition
svd,
fullSVD,
thinSVD,
compactSVD,
singularValues,
leftSV, rightSV,
-- ** Eigensystems
eig, eigSH, eigSH',
eigenvalues, eigenvaluesSH, eigenvaluesSH',
geigSH',
-- ** QR
qr, rq,
-- ** Cholesky
chol, cholSH, mbCholSH,
-- ** Hessenberg
hess,
-- ** Schur
schur,
-- ** LU
lu, luPacked,
-- * Matrix functions
expm,
sqrtm,
matFunc,
-- * Nullspace
nullspacePrec,
nullVector,
nullspaceSVD,
orth,
-- * Norms
Normed(..), NormType(..),
relativeError,
-- * Misc
eps, peps, i,
-- * Util
haussholder,
unpackQR, unpackHess,
ranksv
) where
import Data.Packed.Internal hiding ((//))
import Data.Packed.Matrix
import Numeric.LinearAlgebra.LAPACK as LAPACK
import Data.List(foldl1')
import Data.Array
import Numeric.ContainerBoot
{- | Class used to define generic linear algebra computations for both real and complex matrices. Only double precision is supported in this version (we can
transform single precision objects using 'single' and 'double').
-}
class (Product t,
Convert t,
Container Vector t,
Container Matrix t,
Normed Matrix t,
Normed Vector t,
Floating t,
RealOf t ~ Double) => Field t where
svd' :: Matrix t -> (Matrix t, Vector Double, Matrix t)
thinSVD' :: Matrix t -> (Matrix t, Vector Double, Matrix t)
sv' :: Matrix t -> Vector Double
luPacked' :: Matrix t -> (Matrix t, [Int])
luSolve' :: (Matrix t, [Int]) -> Matrix t -> Matrix t
linearSolve' :: Matrix t -> Matrix t -> Matrix t
cholSolve' :: Matrix t -> Matrix t -> Matrix t
linearSolveSVD' :: Matrix t -> Matrix t -> Matrix t
linearSolveLS' :: Matrix t -> Matrix t -> Matrix t
eig' :: Matrix t -> (Vector (Complex Double), Matrix (Complex Double))
eigSH'' :: Matrix t -> (Vector Double, Matrix t)
eigOnly :: Matrix t -> Vector (Complex Double)
eigOnlySH :: Matrix t -> Vector Double
cholSH' :: Matrix t -> Matrix t
mbCholSH' :: Matrix t -> Maybe (Matrix t)
qr' :: Matrix t -> (Matrix t, Matrix t)
hess' :: Matrix t -> (Matrix t, Matrix t)
schur' :: Matrix t -> (Matrix t, Matrix t)
instance Field Double where
svd' = svdRd
thinSVD' = thinSVDRd
sv' = svR
luPacked' = luR
luSolve' (l_u,perm) = lusR l_u perm
linearSolve' = linearSolveR -- (luSolve . luPacked) ??
cholSolve' = cholSolveR
linearSolveLS' = linearSolveLSR
linearSolveSVD' = linearSolveSVDR Nothing
eig' = eigR
eigSH'' = eigS
eigOnly = eigOnlyR
eigOnlySH = eigOnlyS
cholSH' = cholS
mbCholSH' = mbCholS
qr' = unpackQR . qrR
hess' = unpackHess hessR
schur' = schurR
instance Field (Complex Double) where
#ifdef NOZGESDD
svd' = svdC
thinSVD' = thinSVDC
#else
svd' = svdCd
thinSVD' = thinSVDCd
#endif
sv' = svC
luPacked' = luC
luSolve' (l_u,perm) = lusC l_u perm
linearSolve' = linearSolveC
cholSolve' = cholSolveC
linearSolveLS' = linearSolveLSC
linearSolveSVD' = linearSolveSVDC Nothing
eig' = eigC
eigOnly = eigOnlyC
eigSH'' = eigH
eigOnlySH = eigOnlyH
cholSH' = cholH
mbCholSH' = mbCholH
qr' = unpackQR . qrC
hess' = unpackHess hessC
schur' = schurC
--------------------------------------------------------------
square m = rows m == cols m
vertical m = rows m >= cols m
exactHermitian m = m `equal` ctrans m
--------------------------------------------------------------
-- | Full singular value decomposition.
svd :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t)
svd = {-# SCC "svd" #-} svd'
-- | A version of 'svd' which returns only the @min (rows m) (cols m)@ singular vectors of @m@.
--
-- If @(u,s,v) = thinSVD m@ then @m == u \<> diag s \<> trans v@.
thinSVD :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t)
thinSVD = {-# SCC "thinSVD" #-} thinSVD'
-- | Singular values only.
singularValues :: Field t => Matrix t -> Vector Double
singularValues = {-# SCC "singularValues" #-} sv'
-- | A version of 'svd' which returns an appropriate diagonal matrix with the singular values.
--
-- If @(u,d,v) = fullSVD m@ then @m == u \<> d \<> trans v@.
fullSVD :: Field t => Matrix t -> (Matrix t, Matrix Double, Matrix t)
fullSVD m = (u,d,v) where
(u,s,v) = svd m
d = diagRect 0 s r c
r = rows m
c = cols m
-- | Similar to 'thinSVD', returning only the nonzero singular values and the corresponding singular vectors.
compactSVD :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t)
compactSVD m = (u', subVector 0 d s, v') where
(u,s,v) = thinSVD m
d = rankSVD (1*eps) m s `max` 1
u' = takeColumns d u
v' = takeColumns d v
-- | Singular values and all right singular vectors.
rightSV :: Field t => Matrix t -> (Vector Double, Matrix t)
rightSV m | vertical m = let (_,s,v) = thinSVD m in (s,v)
| otherwise = let (_,s,v) = svd m in (s,v)
-- | Singular values and all left singular vectors.
leftSV :: Field t => Matrix t -> (Matrix t, Vector Double)
leftSV m | vertical m = let (u,s,_) = svd m in (u,s)
| otherwise = let (u,s,_) = thinSVD m in (u,s)
--------------------------------------------------------------
-- | Obtains the LU decomposition of a matrix in a compact data structure suitable for 'luSolve'.
luPacked :: Field t => Matrix t -> (Matrix t, [Int])
luPacked = {-# SCC "luPacked" #-} luPacked'
-- | Solution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by 'luPacked'.
luSolve :: Field t => (Matrix t, [Int]) -> Matrix t -> Matrix t
luSolve = {-# SCC "luSolve" #-} luSolve'
-- | Solve a linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition. For underconstrained or overconstrained systems use 'linearSolveLS' or 'linearSolveSVD'.
-- It is similar to 'luSolve' . 'luPacked', but @linearSolve@ raises an error if called on a singular system.
linearSolve :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolve = {-# SCC "linearSolve" #-} linearSolve'
-- | Solve a symmetric or Hermitian positive definite linear system using a precomputed Cholesky decomposition obtained by 'chol'.
cholSolve :: Field t => Matrix t -> Matrix t -> Matrix t
cholSolve = {-# SCC "cholSolve" #-} cholSolve'
-- | Minimum norm solution of a general linear least squares problem Ax=B using the SVD. Admits rank-deficient systems but it is slower than 'linearSolveLS'. The effective rank of A is determined by treating as zero those singular valures which are less than 'eps' times the largest singular value.
linearSolveSVD :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolveSVD = {-# SCC "linearSolveSVD" #-} linearSolveSVD'
-- | Least squared error solution of an overconstrained linear system, or the minimum norm solution of an underconstrained system. For rank-deficient systems use 'linearSolveSVD'.
linearSolveLS :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolveLS = {-# SCC "linearSolveLS" #-} linearSolveLS'
--------------------------------------------------------------
-- | Eigenvalues and eigenvectors of a general square matrix.
--
-- If @(s,v) = eig m@ then @m \<> v == v \<> diag s@
eig :: Field t => Matrix t -> (Vector (Complex Double), Matrix (Complex Double))
eig = {-# SCC "eig" #-} eig'
-- | Eigenvalues of a general square matrix.
eigenvalues :: Field t => Matrix t -> Vector (Complex Double)
eigenvalues = {-# SCC "eigenvalues" #-} eigOnly
-- | Similar to 'eigSH' without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
eigSH' :: Field t => Matrix t -> (Vector Double, Matrix t)
eigSH' = {-# SCC "eigSH'" #-} eigSH''
-- | Similar to 'eigenvaluesSH' without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
eigenvaluesSH' :: Field t => Matrix t -> Vector Double
eigenvaluesSH' = {-# SCC "eigenvaluesSH'" #-} eigOnlySH
-- | Eigenvalues and Eigenvectors of a complex hermitian or real symmetric matrix.
--
-- If @(s,v) = eigSH m@ then @m == v \<> diag s \<> ctrans v@
eigSH :: Field t => Matrix t -> (Vector Double, Matrix t)
eigSH m | exactHermitian m = eigSH' m
| otherwise = error "eigSH requires complex hermitian or real symmetric matrix"
-- | Eigenvalues of a complex hermitian or real symmetric matrix.
eigenvaluesSH :: Field t => Matrix t -> Vector Double
eigenvaluesSH m | exactHermitian m = eigenvaluesSH' m
| otherwise = error "eigenvaluesSH requires complex hermitian or real symmetric matrix"
--------------------------------------------------------------
-- | QR factorization.
--
-- If @(q,r) = qr m@ then @m == q \<> r@, where q is unitary and r is upper triangular.
qr :: Field t => Matrix t -> (Matrix t, Matrix t)
qr = {-# SCC "qr" #-} qr'
-- | RQ factorization.
--
-- If @(r,q) = rq m@ then @m == r \<> q@, where q is unitary and r is upper triangular.
rq :: Field t => Matrix t -> (Matrix t, Matrix t)
rq m = {-# SCC "rq" #-} (r,q) where
(q',r') = qr $ trans $ rev1 m
r = rev2 (trans r')
q = rev2 (trans q')
rev1 = flipud . fliprl
rev2 = fliprl . flipud
-- | Hessenberg factorization.
--
-- If @(p,h) = hess m@ then @m == p \<> h \<> ctrans p@, where p is unitary
-- and h is in upper Hessenberg form (it has zero entries below the first subdiagonal).
hess :: Field t => Matrix t -> (Matrix t, Matrix t)
hess = hess'
-- | Schur factorization.
--
-- If @(u,s) = schur m@ then @m == u \<> s \<> ctrans u@, where u is unitary
-- and s is a Shur matrix. A complex Schur matrix is upper triangular. A real Schur matrix is
-- upper triangular in 2x2 blocks.
--
-- \"Anything that the Jordan decomposition can do, the Schur decomposition
-- can do better!\" (Van Loan)
schur :: Field t => Matrix t -> (Matrix t, Matrix t)
schur = schur'
-- | Similar to 'cholSH', but instead of an error (e.g., caused by a matrix not positive definite) it returns 'Nothing'.
mbCholSH :: Field t => Matrix t -> Maybe (Matrix t)
mbCholSH = {-# SCC "mbCholSH" #-} mbCholSH'
-- | Similar to 'chol', without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
cholSH :: Field t => Matrix t -> Matrix t
cholSH = {-# SCC "cholSH" #-} cholSH'
-- | Cholesky factorization of a positive definite hermitian or symmetric matrix.
--
-- If @c = chol m@ then @c@ is upper triangular and @m == ctrans c \<> c@.
chol :: Field t => Matrix t -> Matrix t
chol m | exactHermitian m = cholSH m
| otherwise = error "chol requires positive definite complex hermitian or real symmetric matrix"
-- | Joint computation of inverse and logarithm of determinant of a square matrix.
invlndet :: Field t
=> Matrix t
-> (Matrix t, (t, t)) -- ^ (inverse, (log abs det, sign or phase of det))
invlndet m | square m = (im,(ladm,sdm))
| otherwise = error $ "invlndet of nonsquare "++ shSize m ++ " matrix"
where
lp@(lup,perm) = luPacked m
s = signlp (rows m) perm
dg = toList $ takeDiag $ lup
ladm = sum $ map (log.abs) dg
sdm = s* product (map signum dg)
im = luSolve lp (ident (rows m))
-- | Determinant of a square matrix. To avoid possible overflow or underflow use 'invlndet'.
det :: Field t => Matrix t -> t
det m | square m = {-# SCC "det" #-} s * (product $ toList $ takeDiag $ lup)
| otherwise = error $ "det of nonsquare "++ shSize m ++ " matrix"
where (lup,perm) = luPacked m
s = signlp (rows m) perm
-- | Explicit LU factorization of a general matrix.
--
-- If @(l,u,p,s) = lu m@ then @m == p \<> l \<> u@, where l is lower triangular,
-- u is upper triangular, p is a permutation matrix and s is the signature of the permutation.
lu :: Field t => Matrix t -> (Matrix t, Matrix t, Matrix t, t)
lu = luFact . luPacked
-- | Inverse of a square matrix. See also 'invlndet'.
inv :: Field t => Matrix t -> Matrix t
inv m | square m = m `linearSolve` ident (rows m)
| otherwise = error $ "inv of nonsquare "++ shSize m ++ " matrix"
-- | Pseudoinverse of a general matrix with default tolerance ('pinvTol' 1, similar to GNU-Octave).
pinv :: Field t => Matrix t -> Matrix t
pinv = pinvTol 1
{- | @pinvTol r@ computes the pseudoinverse of a matrix with tolerance @tol=r*g*eps*(max rows cols)@, where g is the greatest singular value.
@\> let m = 'fromLists' [[1,0, 0]
,[0,1, 0]
,[0,0,1e-10]]
\ --
\> 'pinv' m
1. 0. 0.
0. 1. 0.
0. 0. 10000000000.
\ --
\> pinvTol 1E8 m
1. 0. 0.
0. 1. 0.
0. 0. 1.@
-}
pinvTol :: Field t => Double -> Matrix t -> Matrix t
pinvTol t m = conj v' `mXm` diag s' `mXm` ctrans u' where
(u,s,v) = thinSVD m
sl@(g:_) = toList s
s' = real . fromList . map rec $ sl
rec x = if x <= g*tol then x else 1/x
tol = (fromIntegral (max r c) * g * t * eps)
r = rows m
c = cols m
d = dim s
u' = takeColumns d u
v' = takeColumns d v
-- | Numeric rank of a matrix from the SVD decomposition.
rankSVD :: Element t
=> Double -- ^ numeric zero (e.g. 1*'eps')
-> Matrix t -- ^ input matrix m
-> Vector Double -- ^ 'sv' of m
-> Int -- ^ rank of m
rankSVD teps m s = ranksv teps (max (rows m) (cols m)) (toList s)
-- | Numeric rank of a matrix from its singular values.
ranksv :: Double -- ^ numeric zero (e.g. 1*'eps')
-> Int -- ^ maximum dimension of the matrix
-> [Double] -- ^ singular values
-> Int -- ^ rank of m
ranksv teps maxdim s = k where
g = maximum s
tol = fromIntegral maxdim * g * teps
s' = filter (>tol) s
k = if g > teps then length s' else 0
-- | The machine precision of a Double: @eps = 2.22044604925031e-16@ (the value used by GNU-Octave).
eps :: Double
eps = 2.22044604925031e-16
-- | 1 + 0.5*peps == 1, 1 + 0.6*peps /= 1
peps :: RealFloat x => x
peps = x where x = 2.0 ** fromIntegral (1 - floatDigits x)
-- | The imaginary unit: @i = 0.0 :+ 1.0@
i :: Complex Double
i = 0:+1
-----------------------------------------------------------------------
-- | The nullspace of a matrix from its SVD decomposition.
nullspaceSVD :: Field t
=> Either Double Int -- ^ Left \"numeric\" zero (eg. 1*'eps'),
-- or Right \"theoretical\" matrix rank.
-> Matrix t -- ^ input matrix m
-> (Vector Double, Matrix t) -- ^ 'rightSV' of m
-> [Vector t] -- ^ list of unitary vectors spanning the nullspace
nullspaceSVD hint a (s,v) = vs where
tol = case hint of
Left t -> t
_ -> eps
k = case hint of
Right t -> t
_ -> rankSVD tol a s
vs = drop k $ toRows $ ctrans v
-- | The nullspace of a matrix. See also 'nullspaceSVD'.
nullspacePrec :: Field t
=> Double -- ^ relative tolerance in 'eps' units (e.g., use 3 to get 3*'eps')
-> Matrix t -- ^ input matrix
-> [Vector t] -- ^ list of unitary vectors spanning the nullspace
nullspacePrec t m = nullspaceSVD (Left (t*eps)) m (rightSV m)
-- | The nullspace of a matrix, assumed to be one-dimensional, with machine precision.
nullVector :: Field t => Matrix t -> Vector t
nullVector = last . nullspacePrec 1
orth :: Field t => Matrix t -> [Vector t]
-- ^ Return an orthonormal basis of the range space of a matrix
orth m = take r $ toColumns u
where
(u,s,_) = compactSVD m
r = ranksv eps (max (rows m) (cols m)) (toList s)
------------------------------------------------------------------------
-- many thanks, quickcheck!
haussholder :: (Field a) => a -> Vector a -> Matrix a
haussholder tau v = ident (dim v) `sub` (tau `scale` (w `mXm` ctrans w))
where w = asColumn v
zh k v = fromList $ replicate (k-1) 0 ++ (1:drop k xs)
where xs = toList v
zt 0 v = v
zt k v = join [subVector 0 (dim v - k) v, konst 0 k]
unpackQR :: (Field t) => (Matrix t, Vector t) -> (Matrix t, Matrix t)
unpackQR (pq, tau) = {-# SCC "unpackQR" #-} (q,r)
where cs = toColumns pq
m = rows pq
n = cols pq
mn = min m n
r = fromColumns $ zipWith zt ([m-1, m-2 .. 1] ++ repeat 0) cs
vs = zipWith zh [1..mn] cs
hs = zipWith haussholder (toList tau) vs
q = foldl1' mXm hs
unpackHess :: (Field t) => (Matrix t -> (Matrix t,Vector t)) -> Matrix t -> (Matrix t, Matrix t)
unpackHess hf m
| rows m == 1 = ((1><1)[1],m)
| otherwise = (uH . hf) m
uH (pq, tau) = (p,h)
where cs = toColumns pq
m = rows pq
n = cols pq
mn = min m n
h = fromColumns $ zipWith zt ([m-2, m-3 .. 1] ++ repeat 0) cs
vs = zipWith zh [2..mn] cs
hs = zipWith haussholder (toList tau) vs
p = foldl1' mXm hs
--------------------------------------------------------------------------
-- | Reciprocal of the 2-norm condition number of a matrix, computed from the singular values.
rcond :: Field t => Matrix t -> Double
rcond m = last s / head s
where s = toList (singularValues m)
-- | Number of linearly independent rows or columns.
rank :: Field t => Matrix t -> Int
rank m = rankSVD eps m (singularValues m)
{-
expm' m = case diagonalize (complex m) of
Just (l,v) -> v `mXm` diag (exp l) `mXm` inv v
Nothing -> error "Sorry, expm not yet implemented for non-diagonalizable matrices"
where exp = vectorMapC Exp
-}
diagonalize m = if rank v == n
then Just (l,v)
else Nothing
where n = rows m
(l,v) = if exactHermitian m
then let (l',v') = eigSH m in (real l', v')
else eig m
-- | Generic matrix functions for diagonalizable matrices. For instance:
--
-- @logm = matFunc log@
--
matFunc :: (Complex Double -> Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double)
matFunc f m = case diagonalize m of
Just (l,v) -> v `mXm` diag (mapVector f l) `mXm` inv v
Nothing -> error "Sorry, matFunc requires a diagonalizable matrix"
--------------------------------------------------------------
golubeps :: Integer -> Integer -> Double
golubeps p q = a * fromIntegral b / fromIntegral c where
a = 2^^(3-p-q)
b = fact p * fact q
c = fact (p+q) * fact (p+q+1)
fact n = product [1..n]
epslist :: [(Int,Double)]
epslist = [ (fromIntegral k, golubeps k k) | k <- [1..]]
geps delta = head [ k | (k,g) <- epslist, g<delta]
{- | Matrix exponential. It uses a direct translation of Algorithm 11.3.1 in Golub & Van Loan,
based on a scaled Pade approximation.
-}
expm :: Field t => Matrix t -> Matrix t
expm = expGolub
expGolub :: Field t => Matrix t -> Matrix t
expGolub m = iterate msq f !! j
where j = max 0 $ floor $ logBase 2 $ pnorm Infinity m
a = m */ fromIntegral ((2::Int)^j)
q = geps eps -- 7 steps
eye = ident (rows m)
work (k,c,x,n,d) = (k',c',x',n',d')
where k' = k+1
c' = c * fromIntegral (q-k+1) / fromIntegral ((2*q-k+1)*k)
x' = a <> x
n' = n |+| (c' .* x')
d' = d |+| (((-1)^k * c') .* x')
(_,_,_,nf,df) = iterate work (1,1,eye,eye,eye) !! q
f = linearSolve df nf
msq x = x <> x
(<>) = multiply
v */ x = scale (recip x) v
(.*) = scale
(|+|) = add
--------------------------------------------------------------
{- | Matrix square root. Currently it uses a simple iterative algorithm described in Wikipedia.
It only works with invertible matrices that have a real solution. For diagonalizable matrices you can try @matFunc sqrt@.
@m = (2><2) [4,9
,0,4] :: Matrix Double@
@\>sqrtm m
(2><2)
[ 2.0, 2.25
, 0.0, 2.0 ]@
-}
sqrtm :: Field t => Matrix t -> Matrix t
sqrtm = sqrtmInv
sqrtmInv x = fst $ fixedPoint $ iterate f (x, ident (rows x))
where fixedPoint (a:b:rest) | pnorm PNorm1 (fst a |-| fst b) < peps = a
| otherwise = fixedPoint (b:rest)
fixedPoint _ = error "fixedpoint with impossible inputs"
f (y,z) = (0.5 .* (y |+| inv z),
0.5 .* (inv y |+| z))
(.*) = scale
(|+|) = add
(|-|) = sub
------------------------------------------------------------------
signlp r vals = foldl f 1 (zip [0..r-1] vals)
where f s (a,b) | a /= b = -s
| otherwise = s
swap (arr,s) (a,b) | a /= b = (arr // [(a, arr!b),(b,arr!a)],-s)
| otherwise = (arr,s)
fixPerm r vals = (fromColumns $ elems res, sign)
where v = [0..r-1]
s = toColumns (ident r)
(res,sign) = foldl swap (listArray (0,r-1) s, 1) (zip v vals)
triang r c h v = (r><c) [el s t | s<-[0..r-1], t<-[0..c-1]]
where el p q = if q-p>=h then v else 1 - v
luFact (l_u,perm) | r <= c = (l ,u ,p, s)
| otherwise = (l',u',p, s)
where
r = rows l_u
c = cols l_u
tu = triang r c 0 1
tl = triang r c 0 0
l = takeColumns r (l_u |*| tl) |+| diagRect 0 (konst 1 r) r r
u = l_u |*| tu
(p,s) = fixPerm r perm
l' = (l_u |*| tl) |+| diagRect 0 (konst 1 c) r c
u' = takeRows c (l_u |*| tu)
(|+|) = add
(|*|) = mul
---------------------------------------------------------------------------
data NormType = Infinity | PNorm1 | PNorm2 | Frobenius
class (RealFloat (RealOf t)) => Normed c t where
pnorm :: NormType -> c t -> RealOf t
instance Normed Vector Double where
pnorm PNorm1 = norm1
pnorm PNorm2 = norm2
pnorm Infinity = normInf
pnorm Frobenius = norm2
instance Normed Vector (Complex Double) where
pnorm PNorm1 = norm1
pnorm PNorm2 = norm2
pnorm Infinity = normInf
pnorm Frobenius = pnorm PNorm2
instance Normed Vector Float where
pnorm PNorm1 = norm1
pnorm PNorm2 = norm2
pnorm Infinity = normInf
pnorm Frobenius = pnorm PNorm2
instance Normed Vector (Complex Float) where
pnorm PNorm1 = norm1
pnorm PNorm2 = norm2
pnorm Infinity = normInf
pnorm Frobenius = pnorm PNorm2
instance Normed Matrix Double where
pnorm PNorm1 = maximum . map (pnorm PNorm1) . toColumns
pnorm PNorm2 = (@>0) . singularValues
pnorm Infinity = pnorm PNorm1 . trans
pnorm Frobenius = pnorm PNorm2 . flatten
instance Normed Matrix (Complex Double) where
pnorm PNorm1 = maximum . map (pnorm PNorm1) . toColumns
pnorm PNorm2 = (@>0) . singularValues
pnorm Infinity = pnorm PNorm1 . trans
pnorm Frobenius = pnorm PNorm2 . flatten
instance Normed Matrix Float where
pnorm PNorm1 = maximum . map (pnorm PNorm1) . toColumns
pnorm PNorm2 = realToFrac . (@>0) . singularValues . double
pnorm Infinity = pnorm PNorm1 . trans
pnorm Frobenius = pnorm PNorm2 . flatten
instance Normed Matrix (Complex Float) where
pnorm PNorm1 = maximum . map (pnorm PNorm1) . toColumns
pnorm PNorm2 = realToFrac . (@>0) . singularValues . double
pnorm Infinity = pnorm PNorm1 . trans
pnorm Frobenius = pnorm PNorm2 . flatten
-- | Approximate number of common digits in the maximum element.
relativeError :: (Normed c t, Container c t) => c t -> c t -> Int
relativeError x y = dig (norm (x `sub` y) / norm x)
where norm = pnorm Infinity
dig r = round $ -logBase 10 (realToFrac r :: Double)
----------------------------------------------------------------------
-- | Generalized symmetric positive definite eigensystem Av = lBv,
-- for A and B symmetric, B positive definite (conditions not checked).
geigSH' :: Field t
=> Matrix t -- ^ A
-> Matrix t -- ^ B
-> (Vector Double, Matrix t)
geigSH' a b = (l,v')
where
u = cholSH b
iu = inv u
c = ctrans iu <> a <> iu
(l,v) = eigSH' c
v' = iu <> v
(<>) = mXm