{-# LANGUAGE FlexibleContexts, FlexibleInstances #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE TypeFamilies #-}

-----------------------------------------------------------------------------
{- |
Module      :  Internal.Algorithms
Copyright   :  (c) Alberto Ruiz 2006-14
Maintainer  :  Alberto Ruiz
Stability   :  provisional

High level generic interface to common matrix computations.

Specific functions for particular base types can also be explicitly
imported from "Numeric.LinearAlgebra.LAPACK".

-}
-----------------------------------------------------------------------------

module Internal.Algorithms (
module Internal.Algorithms,
UpLo(..)
) where

import Internal.Vector
import Internal.Matrix
import Internal.Element
import Internal.Conversion
import Internal.LAPACK
import Internal.Numeric
import Data.List(foldl1')
import qualified Data.Array as A
import qualified Data.Vector.Storable as Vector
import Internal.ST
import Internal.Vectorized(range)
import Control.DeepSeq

{- | Generic linear algebra functions for double precision real and complex matrices.

(Single precision data can be converted using 'single' and 'double').

-}
class (Numeric t,
Convert t,
Normed Matrix t,
Normed Vector t,
Floating t,
Linear t Vector,
Linear t Matrix,
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
mbLinearSolve' :: Matrix t -> Matrix t -> Maybe (Matrix t)
linearSolve' :: Matrix t -> Matrix t -> Matrix t
cholSolve'   :: Matrix t -> Matrix t -> Matrix t
triSolve'   :: UpLo -> Matrix t -> Matrix t -> Matrix t
triDiagSolve' :: Vector t -> Vector t -> Vector t -> Matrix t -> Matrix t
ldlPacked'   :: Matrix t -> (Matrix t, [Int])
ldlSolve'    :: (Matrix t, [Int]) -> 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, Vector t)
qrgr'        :: Int -> (Matrix t, Vector 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) ??
mbLinearSolve' = mbLinearSolveR
cholSolve' = cholSolveR
triSolve' = triSolveR
triDiagSolve' = triDiagSolveR
linearSolveLS' = linearSolveLSR
linearSolveSVD' = linearSolveSVDR Nothing
eig' = eigR
eigSH'' = eigS
eigOnly = eigOnlyR
eigOnlySH = eigOnlyS
cholSH' = cholS
mbCholSH' = mbCholS
qr' = qrR
qrgr' = qrgrR
hess' = unpackHess hessR
schur' = schurR
ldlPacked' = ldlR
ldlSolve'= uncurry ldlsR

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
mbLinearSolve' = mbLinearSolveC
cholSolve' = cholSolveC
triSolve' = triSolveC
triDiagSolve' = triDiagSolveC
linearSolveLS' = linearSolveLSC
linearSolveSVD' = linearSolveSVDC Nothing
eig' = eigC
eigOnly = eigOnlyC
eigSH'' = eigH
eigOnlySH = eigOnlyH
cholSH' = cholH
mbCholSH' = mbCholH
qr' = qrC
qrgr' = qrgrC
hess' = unpackHess hessC
schur' = schurC
ldlPacked' = ldlC
ldlSolve' = uncurry ldlsC

--------------------------------------------------------------

square m = rows m == cols m

vertical m = rows m >= cols m

exactHermitian m = m equal ctrans m

--------------------------------------------------------------

{- | Full singular value decomposition.

@
a = (5><3)
[  1.0,  2.0,  3.0
,  4.0,  5.0,  6.0
,  7.0,  8.0,  9.0
, 10.0, 11.0, 12.0
, 13.0, 14.0, 15.0 ] :: Matrix Double
@

>>> let (u,s,v) = svd a

>>> disp 3 u
5x5
-0.101   0.768   0.614   0.028  -0.149
-0.249   0.488  -0.503   0.172   0.646
-0.396   0.208  -0.405  -0.660  -0.449
-0.543  -0.072  -0.140   0.693  -0.447
-0.690  -0.352   0.433  -0.233   0.398

>>> s
fromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15]

>>> disp 3 v
3x3
-0.519  -0.751   0.408
-0.576  -0.046  -0.816
-0.632   0.659   0.408

>>> let d = diagRect 0 s 5 3
>>> disp 3 d
5x3
35.183  0.000  0.000
0.000  1.477  0.000
0.000  0.000  0.000
0.000  0.000  0.000

>>> disp 3 $u <> d <> tr v 5x3 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 -} svd :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t) svd = {-# SCC "svd" #-} g . svd' where g (u,s,v) = (u,s,tr v) {- | A version of 'svd' which returns only the @min (rows m) (cols m)@ singular vectors of @m@. If @(u,s,v) = thinSVD m@ then @m == u \<> diag s \<> tr v@. @ a = (5><3) [ 1.0, 2.0, 3.0 , 4.0, 5.0, 6.0 , 7.0, 8.0, 9.0 , 10.0, 11.0, 12.0 , 13.0, 14.0, 15.0 ] :: Matrix Double @ >>> let (u,s,v) = thinSVD a >>> disp 3 u 5x3 -0.101 0.768 0.614 -0.249 0.488 -0.503 -0.396 0.208 -0.405 -0.543 -0.072 -0.140 -0.690 -0.352 0.433 >>> s fromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15] >>> disp 3 v 3x3 -0.519 -0.751 0.408 -0.576 -0.046 -0.816 -0.632 0.659 0.408 >>> disp 3$ u <> diag s <> tr v
5x3
1.000   2.000   3.000
4.000   5.000   6.000
7.000   8.000   9.000
10.000  11.000  12.000
13.000  14.000  15.000

-}
thinSVD :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t)
thinSVD = {-# SCC "thinSVD" #-} g . thinSVD'
where
g (u,s,v) = (u,s,tr v)

-- | 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 \<> tr 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.

@
a = (5><3)
[  1.0,  2.0,  3.0
,  4.0,  5.0,  6.0
,  7.0,  8.0,  9.0
, 10.0, 11.0, 12.0
, 13.0, 14.0, 15.0 ] :: Matrix Double
@

>>> let (u,s,v) = compactSVD a

>>> disp 3 u
5x2
-0.101   0.768
-0.249   0.488
-0.396   0.208
-0.543  -0.072
-0.690  -0.352

>>> s
fromList [35.18264833189422,1.4769076999800903]

>>> disp 3 u
5x2
-0.101   0.768
-0.249   0.488
-0.396   0.208
-0.543  -0.072
-0.690  -0.352

>>> disp 3 $u <> diag s <> tr v 5x3 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000 13.000 14.000 15.000 -} compactSVD :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t) compactSVD = compactSVDTol 1 -- | @compactSVDTol r@ is similar to 'compactSVD' (for which @r=1@), but uses tolerance @tol=r*g*eps*(max rows cols)@ to distinguish nonzero singular values, where @g@ is the greatest singular value. If @g<r*eps@, then only one singular value is returned. compactSVDTol :: Field t => Double -> Matrix t -> (Matrix t, Vector Double, Matrix t) compactSVDTol r m = (u', subVector 0 d s, v') where (u,s,v) = thinSVD m d = rankSVD (r*eps) m s max 1 u' = takeColumns d u v' = takeColumns d v -- | Singular values and all right singular vectors (as columns). 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 (as columns). 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) -------------------------------------------------------------- -- | LU decomposition of a matrix in a compact format. data LU t = LU (Matrix t) [Int] deriving Show instance (NFData t, Numeric t) => NFData (LU t) where rnf (LU m _) = rnf m -- | Obtains the LU decomposition of a matrix in a compact data structure suitable for 'luSolve'. luPacked :: Field t => Matrix t -> LU t luPacked x = {-# SCC "luPacked" #-} LU m p where (m,p) = luPacked' x -- | Solution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by 'luPacked'. luSolve :: Field t => LU t -> Matrix t -> Matrix t luSolve (LU m p) = {-# SCC "luSolve" #-} luSolve' (m,p) -- | 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 linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, returning Nothing for a singular system. For underconstrained or overconstrained systems use 'linearSolveLS' or 'linearSolveSVD'. mbLinearSolve :: Field t => Matrix t -> Matrix t -> Maybe (Matrix t) mbLinearSolve = {-# SCC "linearSolve" #-} mbLinearSolve' -- | Solve a symmetric or Hermitian positive definite linear system using a precomputed Cholesky decomposition obtained by 'chol'. cholSolve :: Field t => Matrix t -- ^ Cholesky decomposition of the coefficient matrix -> Matrix t -- ^ right hand sides -> Matrix t -- ^ solution cholSolve = {-# SCC "cholSolve" #-} cholSolve' -- | Solve a triangular linear system. If Upper is specified then -- all elements below the diagonal are ignored; if Lower is -- specified then all elements above the diagonal are ignored. triSolve :: Field t => UpLo -- ^ Lower or Upper -> Matrix t -- ^ coefficient matrix -> Matrix t -- ^ right hand sides -> Matrix t -- ^ solution triSolve = {-# SCC "triSolve" #-} triSolve' -- | Solve a tridiagonal linear system. Suppose you wish to solve $$Ax = b$$ where -- -- $-- A = -- \begin{bmatrix} -- 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 -- \\ 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 -- \\ 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 -- \\ 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 -- \\ 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 -- \\ 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 -- \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 -- \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 -- \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 -- \end{bmatrix} -- \quad -- b = -- \begin{bmatrix} -- 1.0 & 1.0 & 1.0 -- \\ 1.0 & -1.0 & 2.0 -- \\ 1.0 & 1.0 & 3.0 -- \\ 1.0 & -1.0 & 4.0 -- \\ 1.0 & 1.0 & 5.0 -- \\ 1.0 & -1.0 & 6.0 -- \\ 1.0 & 1.0 & 7.0 -- \\ 1.0 & -1.0 & 8.0 -- \\ 1.0 & 1.0 & 9.0 -- \end{bmatrix} --$ -- -- then -- -- @ -- dL = fromList [3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0] -- d = fromList [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0] -- dU = fromList [4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0] -- -- b = (9><3) -- [ -- 1.0, 1.0, 1.0, -- 1.0, -1.0, 2.0, -- 1.0, 1.0, 3.0, -- 1.0, -1.0, 4.0, -- 1.0, 1.0, 5.0, -- 1.0, -1.0, 6.0, -- 1.0, 1.0, 7.0, -- 1.0, -1.0, 8.0, -- 1.0, 1.0, 9.0 -- ] -- -- x = triDiagSolve dL d dU b -- @ -- triDiagSolve :: Field t => Vector t -- ^ lower diagonal: $$n - 1$$ elements -> Vector t -- ^ diagonal: $$n$$ elements -> Vector t -- ^ upper diagonal: $$n - 1$$ elements -> Matrix t -- ^ right hand sides -> Matrix t -- ^ solution triDiagSolve = {-# SCC "triDiagSolve" #-} triDiagSolve' -- | 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' -------------------------------------------------------------------------------- -- | LDL decomposition of a complex Hermitian or real symmetric matrix in a compact format. data LDL t = LDL (Matrix t) [Int] deriving Show instance (NFData t, Numeric t) => NFData (LDL t) where rnf (LDL m _) = rnf m -- | Similar to 'ldlPacked', without checking that the input matrix is hermitian or symmetric. It works with the lower triangular part. ldlPackedSH :: Field t => Matrix t -> LDL t ldlPackedSH x = {-# SCC "ldlPacked" #-} LDL m p where (m,p) = ldlPacked' x -- | Obtains the LDL decomposition of a matrix in a compact data structure suitable for 'ldlSolve'. ldlPacked :: Field t => Herm t -> LDL t ldlPacked (Herm m) = ldlPackedSH m -- | Solution of a linear system (for several right hand sides) from a precomputed LDL factorization obtained by 'ldlPacked'. -- -- Note: this can be slower than the general solver based on the LU decomposition. ldlSolve :: Field t => LDL t -> Matrix t -> Matrix t ldlSolve (LDL m p) = {-# SCC "ldlSolve" #-} ldlSolve' (m,p) -------------------------------------------------------------- {- | Eigenvalues (not ordered) and eigenvectors (as columns) of a general square matrix. If @(s,v) = eig m@ then @m \<> v == v \<> diag s@ @ a = (3><3) [ 3, 0, -2 , 4, 5, -1 , 3, 1, 0 ] :: Matrix Double @ >>> let (l, v) = eig a >>> putStr . dispcf 3 . asRow$ l
1x3
1.925+1.523i  1.925-1.523i  4.151

>>> putStr . dispcf 3 $v 3x3 -0.455+0.365i -0.455-0.365i 0.181 0.603 0.603 -0.978 0.033+0.543i 0.033-0.543i -0.104 >>> putStr . dispcf 3$ complex a <> v
3x3
-1.432+0.010i  -1.432-0.010i   0.753
1.160+0.918i   1.160-0.918i  -4.059
-0.763+1.096i  -0.763-1.096i  -0.433

>>> putStr . dispcf 3 $v <> diag l 3x3 -1.432+0.010i -1.432-0.010i 0.753 1.160+0.918i 1.160-0.918i -4.059 -0.763+1.096i -0.763-1.096i -0.433 -} eig :: Field t => Matrix t -> (Vector (Complex Double), Matrix (Complex Double)) eig = {-# SCC "eig" #-} eig' -- | Eigenvalues (not ordered) of a general square matrix. eigenvalues :: Field t => Matrix t -> Vector (Complex Double) eigenvalues = {-# SCC "eigenvalues" #-} eigOnly -- | Similar to 'eigSH' without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part. eigSH' :: Field t => Matrix t -> (Vector Double, Matrix t) eigSH' = {-# SCC "eigSH'" #-} eigSH'' -- | Similar to 'eigenvaluesSH' without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part. eigenvaluesSH' :: Field t => Matrix t -> Vector Double eigenvaluesSH' = {-# SCC "eigenvaluesSH'" #-} eigOnlySH {- | Eigenvalues and eigenvectors (as columns) of a complex hermitian or real symmetric matrix, in descending order. If @(s,v) = eigSH m@ then @m == v \<> diag s \<> tr v@ @ a = (3><3) [ 1.0, 2.0, 3.0 , 2.0, 4.0, 5.0 , 3.0, 5.0, 6.0 ] @ >>> let (l, v) = eigSH a >>> l fromList [11.344814282762075,0.17091518882717918,-0.5157294715892575] >>> disp 3$ v <> diag l <> tr v
3x3
1.000  2.000  3.000
2.000  4.000  5.000
3.000  5.000  6.000

-}
eigSH :: Field t => Herm t -> (Vector Double, Matrix t)
eigSH (Herm m) = eigSH' m

-- | Eigenvalues (in descending order) of a complex hermitian or real symmetric matrix.
eigenvaluesSH :: Field t => Herm t -> Vector Double
eigenvaluesSH (Herm m) = eigenvaluesSH' m

--------------------------------------------------------------

-- | QR decomposition of a matrix in compact form. (The orthogonal matrix is not explicitly formed.)
data QR t = QR (Matrix t) (Vector t)

instance (NFData t, Numeric t) => NFData (QR t)
where
rnf (QR m _) = rnf m

-- | QR factorization.
--
-- If @(q,r) = qr m@ then @m == q \<> r@, where q is unitary and r is upper triangular.
-- Note: the current implementation is very slow for large matrices. 'thinQR' is much faster.
qr :: Field t => Matrix t -> (Matrix t, Matrix t)
qr = {-# SCC "qr" #-} unpackQR . qr'

-- | A version of 'qr' which returns only the @min (rows m) (cols m)@ columns of @q@ and rows of @r@.
thinQR :: Field t => Matrix t -> (Matrix t, Matrix t)
thinQR = {-# SCC "thinQR" #-} thinUnpackQR . qr'

-- | Compute the QR decomposition of a matrix in compact form.
qrRaw :: Field t => Matrix t -> QR t
qrRaw m = QR x v
where
(x,v) = qr' m

-- | generate a matrix with k orthogonal columns from the compact QR decomposition obtained by 'qrRaw'.
--
qrgr :: Field t => Int -> QR t -> Matrix t
qrgr n (QR a t)
| dim t > min (cols a) (rows a) || n < 0 || n > dim t = error "qrgr expects k <= min(rows,cols)"
| otherwise = qrgr' n (a,t)

-- | RQ factorization.
--
-- If @(r,q) = rq m@ then @m == r \<> q@, where q is unitary and r is upper triangular.
-- Note: the current implementation is very slow for large matrices. 'thinRQ' is much faster.
rq :: Field t => Matrix t -> (Matrix t, Matrix t)
rq = {-# SCC "rq" #-} rqFromQR qr

-- | A version of 'rq' which returns only the @min (rows m) (cols m)@ columns of @r@ and rows of @q@.
thinRQ :: Field t => Matrix t -> (Matrix t, Matrix t)
thinRQ = {-# SCC "thinQR" #-} rqFromQR thinQR

rqFromQR :: Field t => (Matrix t -> (Matrix t, Matrix t)) -> Matrix t -> (Matrix t, Matrix t)
rqFromQR qr0 m = (r,q) where
(q',r') = qr0 $trans$ 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 \<> tr 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 \<> tr 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 = cholSH'

-- | Cholesky factorization of a positive definite hermitian or symmetric matrix.
--
-- If @c = chol m@ then @c@ is upper triangular and @m == tr c \<> c@.
chol :: Field t => Herm t ->  Matrix t
chol (Herm m) = {-# SCC "chol" #-} cholSH' m

-- | Similar to 'chol', but instead of an error (e.g., caused by a matrix not positive definite) it returns 'Nothing'.
mbChol :: Field t => Herm t -> Maybe (Matrix t)
mbChol (Herm m) = {-# SCC "mbChol" #-} mbCholSH' m

-- | 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 $"invlndet of nonsquare "++ shSize m ++ " matrix" where lp@(LU lup perm) = luPacked m s = signlp (rows m) perm dg = toList$ 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
LU 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

inv :: Field t => Matrix t -> Matrix t
inv m | square m = m linearSolve ident (rows m)
| otherwise = error $"inv of nonsquare "++ shSize m ++ " matrix" -- | Pseudoinverse of a general matrix with default tolerance ('pinvTol' 1, similar to GNU-Octave). pinv :: Field t => Matrix t -> Matrix t pinv = pinvTol 1 {- | @pinvTol r@ computes the pseudoinverse of a matrix with tolerance @tol=r*g*eps*(max rows cols)@, where g is the greatest singular value. @ m = (3><3) [ 1, 0, 0 , 0, 1, 0 , 0, 0, 1e-10] :: Matrix Double @ >>> pinv m 1. 0. 0. 0. 1. 0. 0. 0. 10000000000. >>> pinvTol 1E8 m 1. 0. 0. 0. 1. 0. 0. 0. 1. -} pinvTol :: Field t => Double -> Matrix t -> Matrix t pinvTol t m = v' mXm diag s' mXm ctrans u' where (u,s,v) = thinSVD m sl@(g:_) = toList s s' = real . fromList . map rec$ 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 nullspace of a matrix from its precomputed 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
-> Matrix t          -- ^ 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 = dropColumns k v

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 = toColumns $nullspaceSVD (Left (t*eps)) m (rightSV m) -- | The nullspace of a matrix, assumed to be one-dimensional, with machine precision. nullVector :: Field t => Matrix t -> Vector t nullVector = last . nullspacePrec 1 -- | The range space a matrix from its precomputed SVD decomposition. orthSVD :: Field t => Either Double Int -- ^ Left \"numeric\" zero (eg. 1*'eps'), -- or Right \"theoretical\" matrix rank. -> Matrix t -- ^ input matrix m -> (Matrix t, Vector Double) -- ^ 'leftSV' of m -> Matrix t -- ^ orth orthSVD hint a (v,s) = vs where tol = case hint of Left t -> t _ -> eps k = case hint of Right t -> t _ -> rankSVD tol a s vs = takeColumns k v orth :: Field t => Matrix t -> [Vector t] -- ^ Return an orthonormal basis of the range space of a matrix orth m = take r$ 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 $replicate (k-1) 0 ++ (1:drop k xs) where xs = toList v zt 0 v = v zt k v = vjoin [subVector 0 (dim v - k) v, konst' 0 k] unpackQR :: (Field t) => (Matrix t, Vector t) -> (Matrix t, Matrix t) unpackQR (pq, tau) = {-# SCC "unpackQR" #-} (q,r) where cs = toColumns pq m = rows pq n = cols pq mn = min m n r = fromColumns$ 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

thinUnpackQR :: (Field t) => (Matrix t, Vector t) -> (Matrix t, Matrix t)
thinUnpackQR (pq, tau) = (q, r)
where mn = uncurry min $size pq q = qrgr mn$ QR pq tau
r = fromRows $zipWith (\i v -> Vector.replicate i 0 Vector.++ Vector.drop i v) [0..mn-1] (toRows pq) unpackHess :: (Field t) => (Matrix t -> (Matrix t,Vector t)) -> Matrix t -> (Matrix t, Matrix t) unpackHess hf m | rows m == 1 = ((1><1)[1],m) | otherwise = (uH . hf) m uH (pq, tau) = (p,h) where cs = toColumns pq m = rows pq n = cols pq mn = min m n h = fromColumns$ 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. See also 'ranksv'
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 (trustSym 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 $pnorm Infinity m a = m */ fromIntegral ((2::Int)^j) q = geps eps -- 7 steps eye = ident (rows m) work (k,c,x,n,d) = (k',c',x',n',d') where k' = k+1 c' = c * fromIntegral (q-k+1) / fromIntegral ((2*q-k+1)*k) x' = a <> x n' = n |+| (c' .* x') d' = d |+| (((-1)^k * c') .* x') (_,_,_,nf,df) = iterate work (1,1,eye,eye,eye) !! q f = linearSolve df nf msq x = x <> x (<>) = multiply v */ x = scale (recip x) v (.*) = scale (|+|) = add -------------------------------------------------------------- {- | Matrix square root. Currently it uses a simple iterative algorithm described in Wikipedia. It only works with invertible matrices that have a real solution. @m = (2><2) [4,9 ,0,4] :: Matrix Double@ >>> sqrtm m (2><2) [ 2.0, 2.25 , 0.0, 2.0 ] For diagonalizable matrices you can try 'matFunc' @sqrt@: >>> matFunc sqrt ((2><2) [1,0,0,-1]) (2><2) [ 1.0 :+ 0.0, 0.0 :+ 0.0 , 0.0 :+ 0.0, 0.0 :+ 1.0 ] -} sqrtm :: Field t => Matrix t -> Matrix t sqrtm = sqrtmInv sqrtmInv x = fst$ fixedPoint $iterate f (x, ident (rows x)) where fixedPoint (a:b:rest) | pnorm PNorm1 (fst a |-| fst b) < peps = a | otherwise = fixedPoint (b:rest) fixedPoint _ = error "fixedpoint with impossible inputs" f (y,z) = (0.5 .* (y |+| inv z), 0.5 .* (inv y |+| z)) (.*) = scale (|+|) = add (|-|) = sub ------------------------------------------------------------------ signlp r vals = foldl f 1 (zip [0..r-1] vals) where f s (a,b) | a /= b = -s | otherwise = s fixPerm r vals = (fromColumns$ A.elems res, sign)
where
v = [0..r-1]
t = toColumns (ident r)
(res,sign) = foldl swap (A.listArray (0,r-1) t, 1) (zip v vals)
swap (arr,s) (a,b)
| a /= b    = (arr A.// [(a, arr A.! b),(b,arr A.! a)],-s)
| otherwise = (arr,s)

fixPerm' :: [Int] -> Vector I
fixPerm' s = res $mutable f s0 where s0 = reshape 1 (range (length s)) res = flatten . fst swap m i j = rowOper (SWAP i j AllCols) m f :: (Num t, Element t) => (Int, Int) -> STMatrix s t -> ST s () -- needed because of TypeFamilies f _ p = sequence_$ zipWith (swap p) [0..] s

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

-- | Compute the explicit LU decomposition from the compact one obtained by 'luPacked'.
luFact :: Numeric t => LU t -> (Matrix t, Matrix t, Matrix t, t)
luFact (LU 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)
(|*|) = 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)

relativeError :: Num a => (a -> Double) -> a -> a -> Double
relativeError norm a b = r
where
na = norm a
nb = norm b
nab = norm (a-b)
mx = max na nb
mn = min na nb
r = if mn < peps
then mx
else nab/mx

----------------------------------------------------------------------

-- | Generalized symmetric positive definite eigensystem Av = lBv,
-- for A and B symmetric, B positive definite.
geigSH :: Field t
=> Herm t -- ^ A
-> Herm t -- ^ B
-> (Vector Double, Matrix t)
geigSH (Herm a) (Herm b) = geigSH' a b

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

--------------------------------------------------------------------------------

-- | A matrix that, by construction, it is known to be complex Hermitian or real symmetric.
--
--   It can be created using 'sym', 'mTm', or 'trustSym', and the matrix can be extracted using 'unSym'.
newtype Herm t = Herm (Matrix t) deriving Show

instance (NFData t, Numeric t) => NFData (Herm t)
where
rnf (Herm m) = rnf m

-- | Extract the general matrix from a 'Herm' structure, forgetting its symmetric or Hermitian property.
unSym :: Herm t -> Matrix t
unSym (Herm x) = x

-- | Compute the complex Hermitian or real symmetric part of a square matrix (@(x + tr x)/2@).
sym :: Field t => Matrix t -> Herm t
sym x = Herm (scale 0.5 (tr x add x))

-- | Compute the contraction @tr x <> x@ of a general matrix.
mTm :: Numeric t => Matrix t -> Herm t
mTm x = Herm (tr x mXm x)

instance Field t => Linear t Herm where
scale  x (Herm m) = Herm (scale x m)

instance Field t => Additive (Herm t) where
add (Herm a) (Herm b) = Herm (a add b)

-- | At your own risk, declare that a matrix is complex Hermitian or real symmetric
--   for usage in 'chol', 'eigSH', etc. Only a triangular part of the matrix will be used.
trustSym :: Matrix t -> Herm t
trustSym x = (Herm x)