Portability  uses ffi 

Stability  provisional 
Maintainer  Alberto Ruiz (aruiz at um dot es) 
High level generic interface to common matrix computations.
Specific functions for particular base types can also be explicitly imported from Numeric.LinearAlgebra.LAPACK.
 class (Product t, Convert t, Container Vector t, Container Matrix t, Normed Matrix t, Normed Vector t) => Field t
 linearSolve :: Field t => Matrix t > Matrix t > Matrix t
 luSolve :: Field t => (Matrix t, [Int]) > Matrix t > Matrix t
 cholSolve :: Field t => Matrix t > Matrix t > Matrix t
 linearSolveLS :: Field t => Matrix t > Matrix t > Matrix t
 linearSolveSVD :: Field t => Matrix t > Matrix t > Matrix t
 inv :: Field t => Matrix t > Matrix t
 pinv :: Field t => Matrix t > Matrix t
 det :: Field t => Matrix t > t
 invlndet :: (Floating t, Field t) => Matrix t > (Matrix t, (t, t))
 rank :: Field t => Matrix t > Int
 rcond :: Field t => Matrix t > Double
 svd :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)
 fullSVD :: Field t => Matrix t > (Matrix t, Matrix Double, Matrix t)
 thinSVD :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)
 compactSVD :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)
 singularValues :: Field t => Matrix t > Vector Double
 leftSV :: Field t => Matrix t > (Matrix t, Vector Double)
 rightSV :: Field t => Matrix t > (Vector Double, Matrix t)
 eig :: Field t => Matrix t > (Vector (Complex Double), Matrix (Complex Double))
 eigSH :: Field t => Matrix t > (Vector Double, Matrix t)
 eigSH' :: Field t => Matrix t > (Vector Double, Matrix t)
 eigenvalues :: Field t => Matrix t > Vector (Complex Double)
 eigenvaluesSH :: Field t => Matrix t > Vector Double
 eigenvaluesSH' :: Field t => Matrix t > Vector Double
 qr :: Field t => Matrix t > (Matrix t, Matrix t)
 rq :: Field t => Matrix t > (Matrix t, Matrix t)
 chol :: Field t => Matrix t > Matrix t
 cholSH :: Field t => Matrix t > Matrix t
 mbCholSH :: Field t => Matrix t > Maybe (Matrix t)
 hess :: Field t => Matrix t > (Matrix t, Matrix t)
 schur :: Field t => Matrix t > (Matrix t, Matrix t)
 lu :: Field t => Matrix t > (Matrix t, Matrix t, Matrix t, t)
 luPacked :: Field t => Matrix t > (Matrix t, [Int])
 expm :: Field t => Matrix t > Matrix t
 sqrtm :: Field t => Matrix t > Matrix t
 matFunc :: (Complex Double > Complex Double) > Matrix (Complex Double) > Matrix (Complex Double)
 nullspacePrec :: Field t => Double > Matrix t > [Vector t]
 nullVector :: Field t => Matrix t > Vector t
 nullspaceSVD :: Field t => Either Double Int > Matrix t > (Vector Double, Matrix t) > [Vector t]
 class RealFloat (RealOf t) => Normed c t where
 data NormType
 relativeError :: (Normed c t, Container c t) => c t > c t > Int
 eps :: Double
 peps :: RealFloat x => x
 i :: Complex Double
 haussholder :: Field a => a > Vector a > Matrix a
 unpackQR :: Field t => (Matrix t, Vector t) > (Matrix t, Matrix t)
 unpackHess :: Field t => (Matrix t > (Matrix t, Vector t)) > Matrix t > (Matrix t, Matrix t)
 pinvTol :: Double > Matrix Double > Matrix Double
 ranksv :: Double > Int > [Double] > Int
 full :: (Storable t1, Num t1) => (Matrix t > (t2, Vector t1, t3)) > Matrix t > (t2, Matrix t1, t3)
 economy :: (Element t1, Element t2, Element t) => (Matrix t > (Matrix t1, Vector Double, Matrix t2)) > Matrix t > (Matrix t1, Vector Double, Matrix t2)
Supported types
class (Product t, Convert t, Container Vector t, Container Matrix t, Normed Matrix t, Normed Vector t) => Field t Source
Linear Systems
linearSolve :: Field t => Matrix t > Matrix t > Matrix tSource
Solve a linear system (for square coefficient matrix and several righthand 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.
luSolve :: Field t => (Matrix t, [Int]) > Matrix t > Matrix tSource
Solution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by luPacked
.
cholSolve :: Field t => Matrix t > Matrix t > Matrix tSource
Solve a symmetric or Hermitian positive definite linear system using a precomputed Cholesky decomposition obtained by chol
.
linearSolveLS :: Field t => Matrix t > Matrix t > Matrix tSource
Least squared error solution of an overconstrained linear system, or the minimum norm solution of an underconstrained system. For rankdeficient systems use linearSolveSVD
.
linearSolveSVD :: Field t => Matrix t > Matrix t > Matrix tSource
Minimum norm solution of a general linear least squares problem Ax=B using the SVD. Admits rankdeficient 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.
det :: Field t => Matrix t > tSource
Determinant of a square matrix. To avoid possible overflow or underflow use invlndet
.
:: (Floating t, Field t)  
=> Matrix t  
> (Matrix t, (t, t))  (inverse, (log abs det, sign or phase of det)) 
Joint computation of inverse and logarithm of determinant of a square matrix.
rcond :: Field t => Matrix t > DoubleSource
Reciprocal of the 2norm condition number of a matrix, computed from the singular values.
Matrix factorizations
Singular value decomposition
svd :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)Source
Full singular value decomposition.
fullSVD :: Field t => Matrix t > (Matrix t, Matrix Double, Matrix t)Source
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
.
thinSVD :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)Source
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
.
compactSVD :: Field t => Matrix t > (Matrix t, Vector Double, Matrix t)Source
Similar to thinSVD
, returning only the nonzero singular values and the corresponding singular vectors.
leftSV :: Field t => Matrix t > (Matrix t, Vector Double)Source
Singular values and all right singular vectors.
rightSV :: Field t => Matrix t > (Vector Double, Matrix t)Source
Singular values and all right singular vectors.
Eigensystems
eig :: Field t => Matrix t > (Vector (Complex Double), Matrix (Complex Double))Source
Eigenvalues and eigenvectors of a general square matrix.
If (s,v) = eig m
then m <> v == v <> diag s
eigSH :: Field t => Matrix t > (Vector Double, Matrix t)Source
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)Source
Similar to eigSH
without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
eigenvalues :: Field t => Matrix t > Vector (Complex Double)Source
Eigenvalues of a general square matrix.
eigenvaluesSH :: Field t => Matrix t > Vector DoubleSource
Eigenvalues of a complex hermitian or real symmetric matrix.
eigenvaluesSH' :: Field t => Matrix t > Vector DoubleSource
Similar to eigenvaluesSH
without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
QR
qr :: Field t => Matrix t > (Matrix t, Matrix t)Source
QR factorization.
If (q,r) = qr m
then m == q <> r
, where q is unitary and r is upper triangular.
rq :: Field t => Matrix t > (Matrix t, Matrix t)Source
RQ factorization.
If (r,q) = rq m
then m == r <> q
, where q is unitary and r is upper triangular.
Cholesky
chol :: Field t => Matrix t > Matrix tSource
Cholesky factorization of a positive definite hermitian or symmetric matrix.
If c = chol m
then c
is upper triangular and m == ctrans c <> c
.
cholSH :: Field t => Matrix t > Matrix tSource
Similar to chol
, without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.
Hessenberg
hess :: Field t => Matrix t > (Matrix t, Matrix t)Source
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).
Schur
schur :: Field t => Matrix t > (Matrix t, Matrix t)Source
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)
LU
lu :: Field t => Matrix t > (Matrix t, Matrix t, Matrix t, t)Source
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.
luPacked :: Field t => Matrix t > (Matrix t, [Int])Source
Obtains the LU decomposition of a matrix in a compact data structure suitable for luSolve
.
Matrix functions
expm :: Field t => Matrix t > Matrix tSource
Matrix exponential. It uses a direct translation of Algorithm 11.3.1 in Golub & Van Loan, based on a scaled Pade approximation.
sqrtm :: Field t => Matrix t > Matrix tSource
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 ]
matFunc :: (Complex Double > Complex Double) > Matrix (Complex Double) > Matrix (Complex Double)Source
Generic matrix functions for diagonalizable matrices. For instance:
logm = matFunc log
Nullspace
:: Field t  
=> Double  
> Matrix t  input matrix 
> [Vector t]  list of unitary vectors spanning the nullspace 
The nullspace of a matrix. See also nullspaceSVD
.
nullVector :: Field t => Matrix t > Vector tSource
The nullspace of a matrix, assumed to be onedimensional, with machine precision.
:: Field t  
=> Either Double Int  Left "numeric" zero (eg. 1* 
> Matrix t  input matrix m 
> (Vector Double, Matrix t) 

> [Vector t]  list of unitary vectors spanning the nullspace 
The nullspace of a matrix from its SVD decomposition.
Norms
relativeError :: (Normed c t, Container c t) => c t > c t > IntSource
Approximate number of common digits in the maximum element.
Misc
The machine precision of a Double: eps = 2.22044604925031e16
(the value used by GNUOctave).
Util
haussholder :: Field a => a > Vector a > Matrix aSource
unpackHess :: Field t => (Matrix t > (Matrix t, Vector t)) > Matrix t > (Matrix t, Matrix t)Source
:: Double  numeric zero (e.g. 1* 
> Int  maximum dimension of the matrix 
> [Double]  singular values 
> Int  rank of m 
Numeric rank of a matrix from its singular values.