{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE FlexibleContexts, TypeFamilies, MultiParamTypeClasses, FlexibleInstances #-}
{-# language ApplicativeDo #-}
-- {-# OPTIONS_GHC -O2 -rtsopts -with-rtsopts=-K32m -prof#-}
{-|
This module exposes the user interface to the library.
-}
module Numeric.LinearAlgebra.Sparse
(
-- * Linear solvers
-- ** Iterative methods
(<\>),
-- -- ** Preconditioners
-- jacobiPre, ilu0Pre, mSsorPre,
-- ** Moore-Penrose pseudoinverse
pinv,
-- ** Direct methods
luSolve,
-- *** Forward substitution
triLowerSolve,
-- *** Backward substitution
triUpperSolve,
-- * Eigensolvers
eigsQR,
-- eigRayleigh,
eigsArnoldi,
-- * Matrix factorization algorithms
-- ** QR
qr,
-- ** LU
lu,
-- ** Cholesky
chol,
-- ** Arnoldi iteration
arnoldi,
-- * Utilities
-- ** Givens' rotation
givens,
-- ** Condition number
conditionNumberSM,
-- ** Householder reflection
hhRefl,
-- -- * Householder bidiagonalization
-- ** Matrix partitioning
diagPartitions,
-- -- * Random arrays
-- randArray,
-- -- * Random matrices and vectors
-- randMat, randVec,
-- -- ** Sparse "
-- randSpMat, randSpVec,
-- * Creation and conversion of sparse data
-- ** SpVector
-- *** Sparse
fromListSV, toListSV,
-- *** Dense
-- **** " from a list of entries
vr, vc,
-- **** " from/to a Vector of entries
fromVector, toVectorDense,
-- **** " having constant elements
constv,
-- ** SpMatrix
fromListSM, toListSM,
-- ** Packing and unpacking, rows or columns of a sparse matrix
-- *** ", using lists as container
fromRowsL, toRowsL,
fromColsL, toColsL,
-- *** ", using Vector as container
fromRowsV, fromColsV,
-- ** Block operations
(-=-), (-||-), fromBlocksDiag,
-- ** Special matrices
eye, mkDiagonal, mkSubDiagonal,
permutationSM, permutPairsSM,
-- * Predicates
isOrthogonalSM, isDiagonalSM,
-- * Manipulation of sparse data
filterSV, ifilterSV,
-- * Sparsity-related predicates
nearZero, nearOne, isNz,
-- * Operators
-- ** Scaling
(.*), (./),
-- ** Inner product
(<.>),
-- ** Matrix-vector product
(#>), (<#),
-- ** Matrix-matrix product
(##), (#^#), (##^),
-- *** Sparsifying matrix-matrix product
(#~#), (#~^#), (#~#^),
-- ** Vector outer product
(><),
-- * Common operations
dim, nnz, spy,
-- ** Vector spaces
cvx,
-- *** Norms and normalization
norm, norm2, norm2', normalize, normalize2, normalize2',
norm1, hilbertDistSq,
-- ** Matrix-related
transpose, trace, normFrobenius,
-- * Pretty-printing
prd, prd0,
-- * Iteration combinators
untilConvergedG0, untilConvergedG, untilConvergedGM,
modifyInspectGuarded, modifyInspectGuardedM, IterationConfig (..),
modifyUntil, modifyUntilM,
-- * Internal
linSolve0, LinSolveMethod(..),
-- * Exceptions
PartialFunctionError,InputError, OutOfBoundsIndexError,
OperandSizeMismatch, MatrixException, IterationException
)
where
import Control.Exception.Common
import Control.Iterative
import Data.Sparse.Common
import Control.Monad.Catch
import Data.Typeable
-- import Control.Applicative ((<|>))
import Control.Monad.State.Strict
import qualified Control.Monad.Trans.State as MTS
import Data.Complex
import qualified Data.Sparse.Internal.IntM as I
import Data.Maybe
import qualified Data.Vector as V
-- | A lumped constraint for numerical types
type Num' x = (Epsilon x, Elt x, Show x, Ord x, Typeable x)
-- * Matrix condition number
-- | Matrix condition number: computes the QR factorization and extracts the extremal eigenvalues from the R factor
conditionNumberSM :: (MonadThrow m, MonadIO m, MatrixRing (SpMatrix a),
PrintDense (SpMatrix a), Num' a) =>
SpMatrix a -> m a
conditionNumberSM m = do
(_, r) <- qr m
let
u = extractDiagDense r
lmax = abs (maximum u)
lmin = abs (minimum u)
kappa = lmax / lmin
if nearZero lmin
then throwM (HugeConditionNumber "conditionNumberSM" kappa)
else return kappa
-- * Householder transformation
hhMat :: (Num a, AdditiveGroup a) => a -> SpVector a -> SpMatrix a
hhMat beta x = eye n ^-^ beta `scale` (x >< x) where
n = dim x
-- | Householder reflection: a vector `x` uniquely defines an orthogonal (hyper)plane, i.e. an orthogonal subspace; the Householder operator reflects any point `v` through this subspace: v' = (I - 2 x >< x) v
hhRefl :: (Num a, AdditiveGroup a) => SpVector a -> SpMatrix a
hhRefl = hhMat 2
-- * Givens rotation matrix
-- -- -- | Givens coefficients (using stable algorithm shown in Anderson, Edward (4 December 2000). "Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem". LAPACK Working Note)
-- -- -- givensCoef0 :: (Ord b, Floating a, Eq a) => (a -> b) -> a -> a -> (a, a, a)
-- -- givensCoef0
-- -- :: (Signum t, Ord a, Floating t, Eq t) =>
-- -- (t -> a) -> t -> t -> (t, t, t)
-- givensCoef0 ff a b -- returns (c, s, r) where r = norm (a, b)
-- | b==0 = (signum' a, 0, abs a)
-- | a==0 = (0, signum' b, abs b)
-- | ff a > ff b = let t = b/a
-- u = signum' a * abs ( sqrt (1 + t**2))
-- in (1/u, - t/u, a*u)
-- | otherwise = let t = a/b
-- u = signum' b * abs ( sqrt (1 + t**2))
-- in (t/u, - 1/u, b*u)
-- -- | Givens coefficients, real-valued
-- givensCoef :: (Floating a, Ord a) => a -> a -> (a, a, a)
-- givensCoef = givensCoef0 abs
-- -- | Givens coefficients, complex-valued
-- givensCoefC :: RealFloat a => Complex a -> Complex a -> (Complex a, Complex a, Complex a)
-- givensCoefC = givensCoef0 magnitude
-- | Givens matrix : a planar rotation embedded in R^n.
--
-- >>> aa = fromListSM
-- >>> g <- givens aa 1 0
--
-- Row version of the method: given a matrix element below the diagonal, indexed (i,j), choose a row index i' that is below the diagonal as well and distinct from i such that the corresponding element is nonzero.
--
-- To zero out entry A(i, j) we must find row i' such that A(i', j) is non-zero but A has zeros in row i' for all column indices < j.
--
-- NB: The Givens' matrix differs from Identity in 4 entries
--
-- NB2: The form of a Complex rotation matrix in R^2 is as follows (`*` indicates complex conjugation):
--
-- @
-- ( c s )
-- G =( )
-- ( -s* c*)
-- @
{-# inline givens #-}
givens :: (Elt a, MonadThrow m) => SpMatrix a -> IxRow -> IxCol -> m (SpMatrix a)
givens aa i j
| isValidIxSM aa (i,j) && nrows aa >= ncols aa = do
i' <- candidateRows' (immSM aa) i j
return $ givensMat aa i i' j
| otherwise = throwM (OOBIxsError "Givens" [i, j])
where
givensMat mm i i' j =
fromListSM'
[(i,i, conj c), (i,j, - conj s),
(j,i, s), (j,j, c)]
(eye (nrows mm))
where
(c, s, _) = givensCoef a b
a = mm @@ (i', j)
b = mm @@ (i, j) -- element to zero out
candidateRows' mm i j | null u = throwM (OOBNoCompatRows "Givens" (i,j))
| otherwise = return $ head (I.keys u) where
u = I.filterWithKey (\irow row -> irow /= i &&
firstNZColumn row j) mm
firstNZColumn m k = isJust (I.lookup k m) &&
isNothing (I.lookupLT k m)
-- | Givens coefficients and norm of associated vector
givensCoef :: Elt t => t -> t -> (t, t, t)
givensCoef u v = (c0/r, s0/r, r) where
c0 = conj u
s0 = conj v
r = hypot u v
hypot :: Elt a => a -> a -> a
hypot x y = sqrt (mag2 x + mag2 y) where
mag2 i = i * conj i
-- * QR decomposition
{-|
Given a matrix A, returns a pair of matrices (Q, R) such that Q R = A, where Q is orthogonal and R is upper triangular. Applies Givens rotation iteratively to zero out sub-diagonal elements.
NB: at each iteration `i` we multiply the Givens matrix `G_i` by the previous partial result `M`. Since this corresponds to a rotation, and the `givensCoef` function already computes the value of the resulting non-zero component (output `r`), `G_i ## M` can be simplified by just updating two entries of `M` (i.e. zeroing one out and changing the other into `r`).
However, we must also accumulate the `G_i` in order to build `Q`, and the present formulation follows this definition closely.
-}
{-# inline qr #-}
qr :: (Elt a, MatrixRing (SpMatrix a), PrintDense (SpMatrix a),
Epsilon a, MonadThrow m, MonadIO m) =>
SpMatrix a
-> m (SpMatrix a, SpMatrix a) -- ^ Q, R
qr mm = do
(qt, r, _) <- modifyUntilM' config haltf qrstepf gminit
return (transpose qt, r)
where
gminit = (eye (nrows mm), mm, subdiagIndicesSM mm)
haltf (_, _, iis) = null iis
config = IterConf 0 False fst2 prd2 where
fst2 (x,y,_) = (x,y)
prd2 (x,y) = do
prd0 x
prd0 y
qrstepf (qmatt, m, iis) = do
let (i, j) = head iis
g <- givens m i j
let
qmatt' = g #~# qmatt -- update Q'
m' = g #~# m -- update R
return (qmatt', m', tail iis)
-- * Eigenvalue algorithms
-- ** QR algorithm
-- | `eigsQR n mm` performs at most `n` iterations of the QR algorithm on matrix `mm`, and returns a SpVector containing all eigenvalues.
eigsQR :: (MonadThrow m, MonadIO m, Num' a, Normed (SpVector a), MatrixRing (SpMatrix a), Typeable (Magnitude (SpVector a)), PrintDense (SpVector a), PrintDense (SpMatrix a)) =>
Int -- ^ Max. # of iterations
-> Bool -- ^ Print debug information
-> SpMatrix a -- ^ Operand matrix
-> m (SpVector a) -- ^ Eigenvalues {λ_i}
eigsQR nitermax debq m = pf <$> untilConvergedGM "eigsQR" c (const True) stepf m
where
pf = extractDiagDense
c = IterConf nitermax debq pf prd
stepf mm = do
(q, _) <- qr mm
return $ q #~^# (m ## q) -- r #~# q
-- ** Arnoldi-QR
-- | `eigsArnoldi n aa b` computes at most n iterations of the Arnoldi algorithm to find a Krylov subspace of (A, b), denoted Q, along with a Hessenberg matrix of coefficients H. After that, it computes the QR decomposition of H, denoted (O, R) and the eigenvalues {λ_i} of A are listed on the diagonal of the R factor.
eigsArnoldi :: (Scalar (SpVector t) ~ t, MatrixType (SpVector t) ~ SpMatrix t,
Elt t, V (SpVector t), Epsilon t, PrintDense (SpMatrix t),
MatrixRing (SpMatrix t), MonadThrow m, MonadIO m) =>
Int
-> SpMatrix t
-> SpVector t
-> m (SpMatrix t, SpMatrix t, SpVector t) -- ^ Q, O, {λ_i}
eigsArnoldi nitermax aa b = do
(q, h) <- arnoldi aa b nitermax
(o, r) <- qr h
return (q, o, extractDiagDense r)
-- * Householder vector
-- (Golub & Van Loan, Alg. 5.1.1, function `house`)
hhV :: (Scalar (SpVector t) ~ t, Elt t, InnerSpace (SpVector t), Epsilon t) =>
SpVector t -> (SpVector t, t)
hhV x = (v, beta) where
tx = tailSV x
sigma = tx <.> tx
vtemp = singletonSV 1 `concatSV` tx
(v, beta) | nearZero sigma = (vtemp, 0)
| otherwise = let mu = sqrt (headSV x**2 + sigma)
xh = headSV x
vh | mag xh <= 1 = xh - mu
| otherwise = - sigma / (xh + mu)
vnew = (1 / vh) `scale` insertSpVector 0 vh vtemp
in (vnew, 2 * xh**2 / (sigma + vh**2))
-- -- * Bidiagonalization
-- -- * SVD
{- Golub & Van Loan, sec 8.6.2 (p 452 segg.)
SVD of A, Golub-Kahan method
* reduce A to upper bidiagonal form B (Alg. 5.4.2, Householder bidiagonalization)
* compute SVD of B (implicit-shift QR step applied to B^T B, Alg. 8.3.2)
-}
-- * Cholesky factorization
-- | Given a positive semidefinite matrix A, returns a lower-triangular matrix L such that L L^T = A . This is an implementation of the Cholesky–Banachiewicz algorithm, i.e. proceeding row by row from the upper-left corner.
-- | NB: The algorithm throws an exception if some diagonal element of A is zero.
chol :: (Elt a, Epsilon a, MonadThrow m, MonadIO m, PrintDense (SpMatrix a)) =>
SpMatrix a
-> m (SpMatrix a) -- ^ L
chol aa = do
let n = nrows aa
q (i, _) = i == n
config = IterConf 0 False snd prd0
l0 <- cholUpd aa (0, zeroSM n n)
(_, lfin) <- modifyUntilM' config q (cholUpd aa) l0
return lfin
where
oops i = throwM (NeedsPivoting "chol" (unwords ["L", show (i,i)]) :: MatrixException Double)
cholUpd aa (i, ll) = do
sd <- cholSDRowUpd aa ll i -- update subdiagonal entries
ll' <- cholDiagUpd aa sd i -- update diagonal entries
return (i + 1, ll')
cholSDRowUpd aa ll i = do
lrs <- fromListSV (i + 1) <$> onRangeSparseM cholSubDiag [0 .. i-1]
return $ insertRow ll lrs i where
cholSubDiag j | isNz ljj = return $ 1/ljj*(aij - inn)
| otherwise = oops j
where
ljj = ll @@! (j, j)
aij = aa @@! (i, j)
inn = contractSub ll ll i j (j - 1)
cholDiagUpd aa ll i = do
cd <- cholDiag
return $ insertSpMatrix i i cd ll where
cholDiag | i == 0 = sqrt <$> aai
| otherwise = do
a <- aai
let l = sum (fmap (**2) lrow)
return $ sqrt (a - l)
where
lrow = ifilterSV (\j _ -> j < i) (extractRow ll i) -- sub-diagonal elems of L
aai | isNz aaii = return aaii
| otherwise = oops i
where
aaii = aa @@! (i,i)
-- * LU factorization
-- | Given a matrix A, returns a pair of matrices (L, U) where L is lower triangular and U is upper triangular such that L U = A . Implements the Doolittle algorithm, which sets the diagonal of the L matrix to ones and expects all diagonal entries of A to be nonzero. Apply pivoting (row or column permutation) to enforce a nonzero diagonal of the A matrix (the algorithm throws an appropriate exception otherwise).
lu :: (Scalar (SpVector t) ~ t, Elt t, VectorSpace (SpVector t), Epsilon t,
MonadThrow m) =>
SpMatrix t
-> m (SpMatrix t, SpMatrix t) -- ^ L, U
lu aa = do
let oops j = throwM (NeedsPivoting "solveForLij" ("U" ++ show (j, j)) :: MatrixException Double)
n = nrows aa
q (i, _, _) = i == n - 1
luInit | isNz u00 = return (1, l0, u0)
| otherwise = oops (0 :: Int)
where
l0 = insertCol (eye n) ((extractSubCol aa 0 (1, n-1)) ./ u00 ) 0
u0 = insertRow (zeroSM n n) (extractRow aa 0) 0 -- initial U
u00 = u0 @@! (0,0) -- make sure this is non-zero by applying permutation
luUpd (i, l, u) = do -- (i + 1, l', u')
u' <- uUpd aa n (i, l, u) -- update U
l' <- lUpd (i, l, u') -- update L
return (i + 1, l', u')
uUpd aa n (ix, lmat, umat) = do
let us = onRangeSparse (solveForUij ix) [ix .. n - 1]
solveForUij i j = a - p where
a = aa @@! (i, j)
p = contractSub lmat umat i j (i - 1)
return $ insertRow umat (fromListSV n us) ix
lUpd (ix, lmat, umat) = do -- insertCol lmat (fromListSV n ls) ix
ls <- lsm
return $ insertCol lmat (fromListSV n ls) ix
where
lsm = onRangeSparseM (`solveForLij` ix) [ix + 1 .. n - 1]
solveForLij i j
| isNz ujj = return $ (a - p)/ujj
| otherwise = oops j
where
a = aa @@! (i, j)
ujj = umat @@! (j , j) -- NB this must be /= 0
p = contractSub lmat umat i j (i - 1)
s0 <- luInit
(ixf, lf, uf) <- MTS.execStateT (modifyUntilM q luUpd) s0
ufin <- uUpd aa n (ixf, lf, uf) -- final U update
return (lf, ufin)
lu' aa = do
let oops j = throwM (NeedsPivoting "solveForLij" ("U" ++ show (j, j)) :: MatrixException Double)
n = nrows aa
q (i, _, _) = i == n - 1
luInit | isNz u00 = return (1, l0, u0)
| otherwise = oops (0 :: Int)
where
l0 = insertCol (eye n) ((extractSubCol aa 0 (1, n-1)) ./ u00 ) 0
u0 = insertRow (zeroSM n n) (extractRow aa 0) 0 -- initial U
u00 = u0 @@! (0,0) -- make sure this is non-zero by applying permutation
luUpd (i, l, u) = do -- (i + 1, l', u')
u' <- uUpd aa n (i, l, u) -- update U
l' <- lUpd (i, l, u') -- update L
return (i + 1, l', u')
uUpd aa n (ix, lmat, umat) = do
let us = onRangeSparse (solveForUij ix) [ix .. n - 1]
solveForUij i j = a - p where
a = aa @@! (i, j)
p = contractSub lmat umat i j (i - 1)
return $ insertRow umat (fromListSV n us) ix
lUpd (ix, lmat, umat) = do -- insertCol lmat (fromListSV n ls) ix
ls <- lsm
return $ insertCol lmat (fromListSV n ls) ix
where
lsm = onRangeSparseM (`solveForLij` ix) [ix + 1 .. n - 1]
solveForLij i j
| isNz ujj = return $ (a - p)/ujj
| otherwise = oops j
where
a = aa @@! (i, j)
ujj = umat @@! (j , j) -- NB this must be /= 0
p = contractSub lmat umat i j (i - 1)
s0 <- luInit
let config = IterConf 0 True vf prd2 where
vf (_, l, u) = (l, u)
prd2 (x, y) = do
prd0 x
prd0 y
(ixf, lf, uf) <- modifyUntilM' config q luUpd s0
ufin <- uUpd aa n (ixf, lf, uf) -- final U update
return (lf, ufin)
tmc4, tmc5, tmc6 :: SpMatrix (Complex Double)
tmc4 = fromListDenseSM 3 [3:+1, 4:+(-1), (-5):+3, 2:+2, 3:+(-2), 5:+0.2, 7:+(-2), 9:+(-1), 2:+3]
tvc4 = vc [1:+3,2:+2,1:+9]
tmc5 = fromListDenseSM 4 $ zipWith (:+) [16..31] [17,14..]
tmc6 = fromListDenseSM 2 $ zipWith (:+) [1,2,3,4] [4,3,2,1]
-- -- Produces the permutation matrix necessary to have a nonzero in position (iref, jref). This is used in the LU factorization
-- permutAA :: Num b => IxRow -> IxCol -> SpMatrix a -> Maybe (SpMatrix b)
-- permutAA iref jref (SM (nro,_) mm)
-- | isJust (lookupIM2 iref jref mm) = Nothing -- eye nro
-- | otherwise = Just $ permutationSM nro [head u] where
-- u = IM.keys (ifilterIM2 ff mm)
-- ff i j _ = i /= iref &&
-- j == jref
-- * Arnoldi iteration
-- | Given a matrix A, a vector b and a positive integer `n`, this procedure finds the basis of an order `n` Krylov subspace (as the columns of matrix Q), along with an upper Hessenberg matrix H, such that A = Q^T H Q.
-- At the i`th iteration, it finds (i + 1) coefficients (the i`th column of the Hessenberg matrix H) and the (i + 1)`th Krylov vector.
arnoldi :: (MatrixType (SpVector a) ~ SpMatrix a, V (SpVector a) ,
Scalar (SpVector a) ~ a, Epsilon a, MonadThrow m) =>
SpMatrix a -- ^ System matrix
-> SpVector a -- ^ r.h.s.
-> Int -- ^ Max. # of iterations
-> m (SpMatrix a, SpMatrix a) -- ^ Q, H
arnoldi aa b kn | n == nb = return (fromColsV qvfin, fromListSM (nmax + 1, nmax) hhfin)
| otherwise = throwM (MatVecSizeMismatchException "arnoldi" (m,n) nb)
where
(qvfin, hhfin, nmax, _) = execState (modifyUntil tf arnoldiStep) arnInit
tf (_, _, ii, fbreak) = ii == kn || fbreak -- termination criterion
(m, n) = (nrows aa, ncols aa)
nb = dim b
arnInit = (qv1, hh1, 1, False) where
q0 = normalize2 b -- starting basis vector
aq0 = aa #> q0 -- A q0
h11 = q0 `dot` aq0
q1nn = aq0 ^-^ (h11 .* q0)
hh1 = V.fromList [(0, 0, h11), (1, 0, h21)] where
h21 = norm2' q1nn
q1 = normalize2 q1nn -- q1 `dot` q0 ~ 0
qv1 = V.fromList [q0, q1]
arnoldiStep (qv, hh, i, _) = (qv', hh', i + 1, breakf) where
qi = V.last qv
aqi = aa #> qi
hhcoli = fmap (`dot` aqi) qv -- H_{1, i}, H_{2, i}, .. , H_{m + 1, i}
zv = zeroSV m
qipnn =
aqi ^-^ V.foldl' (^+^) zv (V.zipWith (.*) hhcoli qv) -- unnormalized q_{i+1}
qipnorm = norm2' qipnn -- normalization factor H_{i+1, i}
qip = normalize2 qipnn -- q_{i + 1}
hh' = (V.++) hh (indexed2 $ V.snoc hhcoli qipnorm) where -- update H
indexed2 v = V.zip3 ii jj v
ii = V.fromList [0 .. n] -- nth col of upper Hessenberg has `n+1` nz
jj = V.replicate (n + 1) i -- `n+1` replicas of `i`
qv' = V.snoc qv qip -- append q_{i+1} to Krylov basis Q_i
breakf | nearZero qipnorm = True -- breakdown condition
| otherwise = False
-- * Preconditioning
-- | Partition a matrix into strictly subdiagonal, diagonal and strictly superdiagonal parts
diagPartitions :: SpMatrix a
-> (SpMatrix a, SpMatrix a, SpMatrix a) -- ^ Subdiagonal, diagonal, superdiagonal partitions
diagPartitions aa = (e,d,f) where
e = extractSubDiag aa
d = extractDiag aa
f = extractSuperDiag aa
-- ** Jacobi preconditioner
-- | The Jacobi preconditioner is just the reciprocal of the diagonal
jacobiPre :: Fractional a => SpMatrix a -> SpMatrix a
jacobiPre x = recip <$> extractDiag x
-- ** Incomplete LU
-- | Used for Incomplete LU : remove entries in the output matrix corresponding to zero entries in the input matrix (this is called ILU(0) in the preconditioner literature)
ilu0Pre :: (Scalar (SpVector t) ~ t, Elt t, VectorSpace (SpVector t),
Epsilon t, MonadThrow m) =>
SpMatrix t
-> m (SpMatrix t, SpMatrix t) -- ^ L, U (with holes)
ilu0Pre aa = do
(l, u) <- lu aa
let lh = sparsifyLU l aa
uh = sparsifyLU u aa
sparsifyLU m m2 = ifilterSM f m where
f i j _ = isJust (lookupSM m2 i j)
return (lh, uh)
-- ** SSOR
-- | Symmetric Successive Over-Relaxation. `mSsor aa omega` : if `omega = 1` it returns the symmetric Gauss-Seidel preconditioner. When ω = 1, the SOR reduces to Gauss-Seidel; when ω > 1 and ω < 1, it corresponds to over-relaxation and under-relaxation, respectively.
mSsorPre :: (MatrixRing (SpMatrix b), Fractional b) =>
SpMatrix b
-> b -- ^ relaxation factor
-> (SpMatrix b, SpMatrix b) -- ^ Left, right factors
mSsorPre aa omega = (l, r) where
(e, d, f) = diagPartitions aa
n = nrows e
l = (eye n ^-^ scale omega e) ## reciprocal d
r = d ^-^ scale omega f
-- * Linear solver, LU-based
luSolveConfig :: PrintDense (SpVector t) => IterationConfig (SpVector t, IxRow) (SpVector t)
luSolveConfig = IterConf 0 False fst prd0
-- | Direct solver based on a triangular factorization of the system matrix.
luSolve :: (Scalar (SpVector t) ~ t, MonadThrow m, Elt t, InnerSpace (SpVector t),
Epsilon t, PrintDense (SpVector t), MonadIO m) =>
SpMatrix t -- ^ Lower triangular
-> SpMatrix t -- ^ Upper triangular
-> SpVector t -- ^ r.h.s.
-> m (SpVector t)
luSolve ll uu b
| isLowerTriSM ll && isUpperTriSM uu = do
w <- triLowerSolve0 luSolveConfig ll b
triUpperSolve0 luSolveConfig uu w
| otherwise = throwM (NonTriangularException "luSolve")
-- | Forward substitution solver
triLowerSolve
:: (Scalar (SpVector t) ~ t, Elt t, InnerSpace (SpVector t),
PrintDense (SpVector t), Epsilon t, MonadThrow m, MonadIO m) =>
SpMatrix t -> SpVector t -> m (SpVector t)
triLowerSolve = triLowerSolve0 luSolveConfig
triLowerSolve0 :: (Scalar (SpVector t) ~ t, Elt t, InnerSpace (SpVector t),
Epsilon t, MonadThrow m, MonadIO m) =>
IterationConfig (SpVector t, IxRow) b
-> SpMatrix t -> SpVector t -> m (SpVector t)
triLowerSolve0 config ll b = do
let q (_, i) = i == nb
nb = svDim b
oops i = throwM (NeedsPivoting "triLowerSolve" (unwords ["L", show (i, i)]) :: MatrixException Double)
lStep (ww, i) = do -- (ww', i + 1) where
let
lii = ll @@ (i, i)
bi = b @@ i
wi = (bi - r)/lii where
r = extractSubRow ll i (0, i-1) `dot` takeSV i ww
if isNz lii
then return (insertSpVector i wi ww, i + 1)
else oops i
lInit = do -- (ww0, 1) where
let
l00 = ll @@ (0, 0)
b0 = b @@ 0
w0 = b0 / l00
if isNz l00
then return (insertSpVector 0 w0 $ zeroSV (dim b), 1)
else oops (0 :: Int)
l0 <- lInit
(v, _) <- modifyUntilM' config q lStep l0
return $ sparsifySV v
-- NB in the computation of `xi` we must rebalance the subrow indices (extractSubRow_RK) because `dropSV` does that too, in order to take the inner product with consistent index pairs
-- | Backward substitution solver
triUpperSolve
:: (Scalar (SpVector t) ~ t, Elt t, InnerSpace (SpVector t),
PrintDense (SpVector t), Epsilon t, MonadThrow m, MonadIO m) =>
SpMatrix t -> SpVector t -> m (SpVector t)
triUpperSolve = triUpperSolve0 luSolveConfig
triUpperSolve0 :: (Scalar (SpVector t) ~ t, Elt t, InnerSpace (SpVector t),
Epsilon t, MonadThrow m, MonadIO m) =>
IterationConfig (SpVector t, IxRow) b
-> SpMatrix t -> SpVector t -> m (SpVector t)
triUpperSolve0 conf uu w = do
let q (_, i) = i == (- 1)
nw = svDim w
oops i = throwM (NeedsPivoting "triUpperSolve" (unwords ["U", show (i, i)]) :: MatrixException Double)
uStep (xx, i) = do
let uii = uu @@ (i, i)
wi = w @@ i
r = extractSubRow_RK uu i (i + 1, nw - 1) `dot` dropSV (i + 1) xx
xi = (wi - r) / uii
if isNz uii
then return (insertSpVector i xi xx, i - 1)
else oops i
uInit = do
let i = nw - 1
u00 = uu @@! (i, i)
w0 = w @@ i
x0 = w0 / u00
if isNz u00
then return (insertSpVector i x0 (zeroSV nw), i - 1)
else oops (0 :: Int)
u0 <- uInit
(x, _) <- modifyUntilM' conf q uStep u0
return $ sparsifySV x
-- * Iterative linear solvers
-- ** GMRES
-- | Given a linear system `A x = b` where `A` is an (m x m) real-valued matrix, the GMRES method finds an approximate solution `xhat` such that the Euclidean norm of the residual `A xhat - b` is minimized. `xhat` is spanned by the order-`n` Krylov subspace of (A, b).
-- In this implementation:
-- 1) the Arnoldi iteration is carried out until numerical breakdown (therefore yielding _at_most_ `m+1` Krylov basis vectors)
-- 2) the resulting Hessenberg matrix H is factorized in QR form (H = Q R)
-- 3) the Krylov-subspace solution `yhat` is found by backsubstitution (since R is upper-triangular)
-- 4) the approximate solution in the original space `xhat` is computed using the Krylov basis, `xhat = Q_n yhat`
--
-- A common optimization involves interleaving the QR factorization (and the subsequent triangular solve) with the Arnoldi process (and employing an updating QR factorization which only requires one Givens' rotation at every update).
-- gmres :: (Scalar (SpVector t) ~ t, MatrixType (SpVector t) ~ SpMatrix t,
-- Elt t, Normed (SpVector t), LinearVectorSpace (SpVector t), Epsilon t,
-- MonadThrow m) =>
-- SpMatrix t -> SpVector t -> m (SpVector t)
gmres aa b = do
let m = ncols aa
(qa, ha) <- arnoldi aa b m -- at most m steps of Arnoldi (aa, b)
-- b' = (transposeSe qa) #> b
let b' = norm2' b .* ei mp1 1 -- b rotated back to canonical basis by Q^T
where mp1 = nrows ha -- = 1 + (# Arnoldi iterations)
(qh, rh) <- qr ha
let rhs' = takeSV (dim b' - 1) (transpose qh #> b')
rh' = takeRows (nrows rh - 1) rh -- last row of `rh` is 0
yhat <- triUpperSolve rh' rhs'
let qa' = takeCols (ncols qa - 1) qa -- we don't use last column of Krylov basis
return $ qa' #> yhat
-- ** CGNE
data CGNE a =
CGNE {_xCgne , _rCgne, _pCgne :: SpVector a} deriving Eq
instance Show a => Show (CGNE a) where
show (CGNE x r p) = "x = " ++ show x ++ "\n" ++
"r = " ++ show r ++ "\n" ++
"p = " ++ show p ++ "\n"
cgneInit :: (MatrixType (SpVector a) ~ SpMatrix a,
LinearVectorSpace (SpVector a)) =>
SpMatrix a -> SpVector a -> SpVector a -> CGNE a
cgneInit aa b x0 = CGNE x0 r0 p0 where
r0 = b ^-^ (aa #> x0) -- residual of initial guess solution
p0 = transposeSM aa #> r0
cgneStep :: (MatrixType (SpVector a) ~ SpMatrix a,
LinearVectorSpace (SpVector a), InnerSpace (SpVector a),
MatrixRing (SpMatrix a), Fractional (Scalar (SpVector a))) =>
SpMatrix a -> CGNE a -> CGNE a
cgneStep aa (CGNE x r p) = CGNE x1 r1 p1 where
alphai = (r `dot` r) / (p `dot` p)
x1 = x ^+^ (alphai .* p)
r1 = r ^-^ (alphai .* (aa #> p))
beta = (r1 `dot` r1) / (r `dot` r)
p1 = transpose aa #> r ^+^ (beta .* p)
-- ** BCG
data BCG a =
BCG { _xBcg, _rBcg, _rHatBcg, _pBcg, _pHatBcg :: SpVector a } deriving Eq
bcgInit :: LinearVectorSpace (SpVector a) =>
MatrixType (SpVector a) -> SpVector a -> SpVector a -> BCG a
bcgInit aa b x0 = BCG x0 r0 r0hat p0 p0hat where
r0 = b ^-^ (aa #> x0)
r0hat = r0
p0 = r0
p0hat = r0
bcgStep :: (MatrixType (SpVector a) ~ SpMatrix a,
LinearVectorSpace (SpVector a), InnerSpace (SpVector a),
MatrixRing (SpMatrix a), Fractional (Scalar (SpVector a))) =>
SpMatrix a -> BCG a -> BCG a
bcgStep aa (BCG x r rhat p phat) = BCG x1 r1 rhat1 p1 phat1 where
aap = aa #> p
alpha = (r `dot` rhat) / (aap `dot` phat)
x1 = x ^+^ (alpha .* p)
r1 = r ^-^ (alpha .* aap)
rhat1 = rhat ^-^ (alpha .* (transpose aa #> phat))
beta = (r1 `dot` rhat1) / (r `dot` rhat)
p1 = r1 ^+^ (beta .* p)
phat1 = rhat1 ^+^ (beta .* phat)
instance Show a => Show (BCG a) where
show (BCG x r rhat p phat) = "x = " ++ show x ++ "\n" ++
"r = " ++ show r ++ "\n" ++
"r_hat = " ++ show rhat ++ "\n" ++
"p = " ++ show p ++ "\n" ++
"p_hat = " ++ show phat ++ "\n"
-- ** CGS
data CGS a = CGS { _x, _r, _p, _u :: SpVector a} deriving Eq
cgsInit :: LinearVectorSpace (SpVector a) =>
MatrixType (SpVector a) -> SpVector a -> SpVector a -> CGS a
cgsInit aa b x0 = CGS x0 r0 r0 r0 where
r0 = b ^-^ (aa #> x0) -- residual of initial guess solution
cgsStep :: (V (SpVector a), Fractional (Scalar (SpVector a))) =>
MatrixType (SpVector a) -> SpVector a -> CGS a -> CGS a
cgsStep aa rhat (CGS x r p u) = CGS xj1 rj1 pj1 uj1
where
aap = aa #> p
alphaj = (r `dot` rhat) / (aap `dot` rhat)
q = u ^-^ (alphaj .* aap)
xj1 = x ^+^ (alphaj .* (u ^+^ q)) -- updated solution
rj1 = r ^-^ (alphaj .* (aa #> (u ^+^ q))) -- updated residual
betaj = (rj1 `dot` rhat) / (r `dot` rhat)
uj1 = rj1 ^+^ (betaj .* q)
pj1 = uj1 ^+^ (betaj .* (q ^+^ (betaj .* p)))
instance (Show a) => Show (CGS a) where
show (CGS x r p u) = "x = " ++ show x ++ "\n" ++
"r = " ++ show r ++ "\n" ++
"p = " ++ show p ++ "\n" ++
"u = " ++ show u ++ "\n"
-- ** BiCGSTAB
data BICGSTAB a =
BICGSTAB { _xBicgstab, _rBicgstab, _pBicgstab :: SpVector a} deriving Eq
bicgsInit :: LinearVectorSpace (SpVector a) =>
MatrixType (SpVector a) -> SpVector a -> SpVector a -> BICGSTAB a
bicgsInit aa b x0 = BICGSTAB x0 r0 r0 where
r0 = b ^-^ (aa #> x0) -- residual of initial guess solution
bicgstabStep :: (V (SpVector a), Fractional (Scalar (SpVector a))) =>
MatrixType (SpVector a) -> SpVector a -> BICGSTAB a -> BICGSTAB a
bicgstabStep aa r0hat (BICGSTAB x r p) = BICGSTAB xj1 rj1 pj1 where
aap = aa #> p
alphaj = (r <.> r0hat) / (aap <.> r0hat)
sj = r ^-^ (alphaj .* aap)
aasj = aa #> sj
omegaj = (aasj <.> sj) / (aasj <.> aasj)
xj1 = x ^+^ (alphaj .* p) ^+^ (omegaj .* sj) -- updated solution
rj1 = sj ^-^ (omegaj .* aasj)
betaj = (rj1 <.> r0hat)/(r <.> r0hat) * alphaj / omegaj
pj1 = rj1 ^+^ (betaj .* (p ^-^ (omegaj .* aap)))
instance Show a => Show (BICGSTAB a) where
show (BICGSTAB x r p) = "x = " ++ show x ++ "\n" ++
"r = " ++ show r ++ "\n" ++
"p = " ++ show p ++ "\n"
-- * Moore-Penrose pseudoinverse
-- | Least-squares approximation of a rectangular system of equations.
pinv :: (LinearSystem v, MatrixRing (MatrixType v), MonadThrow m, MonadIO m) =>
MatrixType v -> v -> m v
pinv aa b = (aa #^# aa) <\> atb where
atb = transpose aa #> b
-- * Linear solver interface
-- -- TFQMR is in LinearSolvers.Experimental for now
-- | Iterative methods for linear systems
data LinSolveMethod = GMRES_ -- ^ Generalized Minimal RESidual
| CGNE_ -- ^ Conjugate Gradient on the Normal Equations
| BCG_ -- ^ BiConjugate Gradient
| CGS_ -- ^ Conjugate Gradient Squared
| BICGSTAB_ -- ^ BiConjugate Gradient Stabilized
deriving (Eq, Show)
-- | Interface method to individual linear solvers
linSolve0 method aa b x0
| m /= nb = throwM (MatVecSizeMismatchException "linSolve0" dm nb)
| otherwise = solve aa b where
solve aa' b' | isDiagonalSM aa' = return $ reciprocal aa' #> b' -- diagonal solve
| otherwise = xHat
xHat = case method of
BICGSTAB_ -> solver "BICGSTAB" nits _xBicgstab (bicgstabStep aa r0hat) (bicgsInit aa b x0)
BCG_ -> solver "BCG" nits _xBcg (bcgStep aa) (bcgInit aa b x0)
CGS_ -> solver "CGS" nits _x (cgsStep aa r0hat) (cgsInit aa b x0)
GMRES_ -> gmres aa b
CGNE_ -> solver "CGNE" nits _xCgne (cgneStep aa) (cgneInit aa b x0)
r0hat = b ^-^ (aa #> x0)
nits = 200
dm@(m,n) = dim aa
nb = dim b
solver fname nitermax fproj stepf initf = do
xf <- untilConvergedG fname config (const True) stepf initf
return $ fproj xf
where
config = IterConf nitermax False fproj prd0
-- | <\> uses the GMRES method as default
instance LinearSystem (SpVector Double) where
aa <\> b = linSolve0 GMRES_ aa b (mkSpVR n $ replicate n 0.1)
where n = ncols aa
-- instance LinearSystem (SpVector (Complex Double)) where
-- aa <\> b = linSolve0 GMRES_ aa b (mkSpVC n $ replicate n 0.1)
-- where n = ncols aa
-- | TODO : if system is poorly conditioned, is it better to warn the user or just switch solvers (e.g. via the pseudoinverse) ?
-- linSolveQR aa b init f1 stepf
-- | isInfinite k = do
-- tell "linSolveQR : rank-deficient system"
-- | otherwise = do
-- solv aa b init
-- where
-- (q, r) = qr aa
-- k = conditionNumberSM r
-- solv aa b init = execState (untilConverged f1 stepf) init
-- test data
-- -- aa4 : eigenvalues 1 (mult.=2) and -1
-- aa4 :: SpMatrix Double
-- aa4 = fromListDenseSM 3 [3,2,-2,2,2,-1,6,5,-4]
-- aa4c :: SpMatrix (Complex Double)
-- aa4c = toC <$> aa4
-- -- * Random arrays
-- randArray :: PrimMonad m => Int -> Double -> Double -> m [Double]
-- randArray n mu sig = do
-- g <- MWC.create
-- replicateM n (MWC.normal mu sig g)
-- -- * Random matrices and vectors
-- -- |Dense SpMatrix
-- randMat :: PrimMonad m => Int -> m (SpMatrix Double)
-- randMat n = do
-- g <- MWC.create
-- aav <- replicateM (n^2) (MWC.normal 0 1 g)
-- let ii_ = [0 .. n-1]
-- (ix_,iy_) = unzip $ concatMap (zip ii_ . replicate n) ii_
-- return $ fromListSM (n,n) $ zip3 ix_ iy_ aav
-- -- | Dense SpVector
-- randVec :: PrimMonad m => Int -> m (SpVector Double)
-- randVec n = do
-- g <- MWC.create
-- bv <- replicateM n (MWC.normal 0 1 g)
-- let ii_ = [0..n-1]
-- return $ fromListSV n $ zip ii_ bv
-- -- | Sparse SpMatrix
-- randSpMat :: Int -> Int -> IO (SpMatrix Double)
-- randSpMat n nsp | nsp > n^2 = error "randSpMat : nsp must be < n^2 "
-- | otherwise = do
-- g <- MWC.create
-- aav <- replicateM nsp (MWC.normal 0 1 g)
-- ii <- replicateM nsp (MWC.uniformR (0, n-1) g :: IO Int)
-- jj <- replicateM nsp (MWC.uniformR (0, n-1) g :: IO Int)
-- return $ fromListSM (n,n) $ zip3 ii jj aav
-- -- | Sparse SpVector
-- randSpVec :: Int -> Int -> IO (SpVector Double)
-- randSpVec n nsp | nsp > n = error "randSpVec : nsp must be < n"
-- | otherwise = do
-- g <- MWC.create
-- aav <- replicateM nsp (MWC.normal 0 1 g)
-- ii <- replicateM nsp (MWC.uniformR (0, n-1) g :: IO Int)
-- return $ fromListSV n $ zip ii aav