content: {-# 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,
      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,
      pinvTol,
      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) => 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 :: (Floating t, 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.
  pinv :: Field t => Matrix t -> Matrix t
  pinv m = linearSolveSVD m (ident (rows m))
  
  -- | 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)
  
  ------------------------------------------------------------------------
  
  {-  Pseudoinverse of a real matrix with the desired tolerance, expressed as a
  multiplicative factor of the default tolerance used by GNU-Octave (see 'pinv').
  
  @\> 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 :: Double -> Matrix Double -> Matrix Double
  pinvTol t m = v' `mXm` diag s' `mXm` trans u' where
      (u,s,v) = thinSVDRd m
      sl@(g:_) = toList s
      s' = fromList . map rec $ sl
      rec x = if x < g*tol then 1 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
  
  ---------------------------------------------------------------------
  
  -- 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 = [ (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 :: ( Fractional t, Element t, Field t
              , Normed Matrix t
              , RealFrac (RealOf t)
              , Floating (RealOf 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
  
  