Numeric.LinearAlgebra.Algorithms

A generic interface for some common functions. Using it we can write higher level algorithms and testing properties both for real and complex matrices.

In any case, the specific functions for particular base types can also be explicitly imported from Numeric.LinearAlgebra.LAPACK.

Supported types

class (Normed (Matrix t), Linear Vector t, Linear Matrix t) => Field t

Auxiliary typeclass used to define generic computations for both real and complex matrices.

Instances

Field Double

Field (Complex Double)

Products

multiply :: Field t => Matrix t -> Matrix t -> Matrix t

Matrix product.

dot :: Field t => Vector t -> Vector t -> t

Euclidean inner product.

outer :: Field t => Vector t -> Vector t -> Matrix t

Outer product of two vectors.

> fromList [1,2,3] `outer` fromList [5,2,3]
(3><3)
 [  5.0, 2.0, 3.0
 , 10.0, 4.0, 6.0
 , 15.0, 6.0, 9.0 ]

kronecker :: Field t => Matrix t -> Matrix t -> Matrix t

Kronecker product of two matrices.

m1=(2><3)
 [ 1.0,  2.0, 0.0
 , 0.0, -1.0, 3.0 ]
m2=(4><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 ]

> kronecker m1 m2
(8><9)
 [  1.0,  2.0,  3.0,   2.0,   4.0,   6.0,  0.0,  0.0,  0.0
 ,  4.0,  5.0,  6.0,   8.0,  10.0,  12.0,  0.0,  0.0,  0.0
 ,  7.0,  8.0,  9.0,  14.0,  16.0,  18.0,  0.0,  0.0,  0.0
 , 10.0, 11.0, 12.0,  20.0,  22.0,  24.0,  0.0,  0.0,  0.0
 ,  0.0,  0.0,  0.0,  -1.0,  -2.0,  -3.0,  3.0,  6.0,  9.0
 ,  0.0,  0.0,  0.0,  -4.0,  -5.0,  -6.0, 12.0, 15.0, 18.0
 ,  0.0,  0.0,  0.0,  -7.0,  -8.0,  -9.0, 21.0, 24.0, 27.0
 ,  0.0,  0.0,  0.0, -10.0, -11.0, -12.0, 30.0, 33.0, 36.0 ]

Linear Systems

linearSolve :: Field t => Matrix t -> Matrix t -> Matrix t

Solution of a general linear system (for several right-hand sides) using lapacks' dgesv or zgesv. It is similar to luSolve . luPacked, but linearSolve raises an error if called on a singular system. See also other versions of linearSolve in Numeric.LinearAlgebra.LAPACK.

luSolve :: Field t => (Matrix t, [Int]) -> Matrix t -> Matrix t

Solution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by luPacked.

linearSolveSVD :: Field t => Matrix t -> Matrix t -> Matrix t

inv :: Field t => Matrix t -> Matrix t

Inverse of a square matrix using lapacks' dgesv and zgesv.

pinv :: Field t => Matrix t -> Matrix t

Pseudoinverse of a general matrix using lapack's dgelss or zgelss.

det :: Field t => Matrix t -> t

determinant of a square matrix, computed from the LU decomposition.

rank :: Field t => Matrix t -> Int

Number of linearly independent rows or columns.

rcond :: Field t => Matrix t -> Double

Reciprocal of the 2-norm condition number of a matrix, computed from the SVD.

Matrix factorizations

Singular value decomposition

svd :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t)

Singular value decomposition using lapack's dgesvd or zgesvd.

full :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Matrix Double, Matrix t)

A version of svd which returns an appropriate diagonal matrix with the singular values.

If (u,d,v) = full svd m then m == u <> d <> trans v.

economy :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Vector Double, Matrix t)

A version of svd which returns only the nonzero singular values and the corresponding rows and columns of the rotations.

If (u,s,v) = economy svd m then m == u <> diag s <> trans v.

Eigensystems

eig :: Field t => Matrix t -> (Vector (Complex Double), Matrix (Complex Double))

Eigenvalues and eigenvectors of a general square matrix using lapack's dgeev or zgeev.

If (s,v) = eig m then m <> v == v <> diag s

eigSH :: Field t => Matrix t -> (Vector Double, Matrix t)

Eigenvalues and Eigenvectors of a complex hermitian or real symmetric matrix using lapack's dsyev or zheev.

If (s,v) = eigSH m then m == v <> diag s <> ctrans v

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

Similar to eigSH without checking that the input matrix is hermitian or symmetric.

QR

qr :: Field t => Matrix t -> (Matrix t, Matrix t)

QR factorization using lapack's dgeqr2 or zgeqr2.

If (q,r) = qr m then m == q <> r, where q is unitary and r is upper triangular.

Cholesky

chol :: Field t => Matrix t -> Matrix t

Cholesky factorization of a positive definite hermitian or symmetric matrix using lapack's dpotrf or zportrf.

If c = chol m then m == ctrans c <> c.

cholSH :: Field t => Matrix t -> Matrix t

Similar to chol without checking that the input matrix is hermitian or symmetric.

Hessenberg

hess :: Field t => Matrix t -> (Matrix t, Matrix t)

Hessenberg factorization using lapack's dgehrd or zgehrd.

If (p,h) = hess m then m == p <> h <> ctrans p, where p is unitary and h is in upper Hessenberg form.

Schur

schur :: Field t => Matrix t -> (Matrix t, Matrix t)

Schur factorization using lapack's dgees or zgees.

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)

Explicit LU factorization of a general matrix using lapack's dgetrf or zgetrf.

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])

Obtains the LU decomposition of a matrix in a compact data structure suitable for luSolve.

Matrix functions

expm :: Field t => Matrix t -> Matrix t

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 t

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 :: Field t => (Complex Double -> Complex Double) -> Matrix t -> Matrix (Complex Double)

Generic matrix functions for diagonalizable matrices. For instance:

logm = matFunc log

Nullspace

nullspacePrec

Arguments

:: Field t

=> Double

relative tolerance in eps units

-> Matrix t

input matrix

-> [Vector t]

list of unitary vectors spanning the nullspace

The nullspace of a matrix from its SVD decomposition.

nullVector :: Field t => Matrix t -> Vector t

The nullspace of a matrix, assumed to be one-dimensional, with default tolerance (shortcut for last . nullspacePrec 1).

Norms

class Normed t where

Objects which have a p-norm. Using it you can define convenient shortcuts:

norm2 x = pnorm PNorm2 x

frobenius m = norm2 . flatten $ m

Methods

pnorm :: NormType -> t -> Double

Instances

Normed (Vector Double)

Normed (Vector (Complex Double))

Normed (Matrix Double)

Normed (Matrix (Complex Double))

data NormType

Constructors

Infinity

PNorm1

PNorm2

Misc

ctrans :: Field t => Matrix t -> Matrix t

Generic conjugate transpose.

eps :: Double

The machine precision of a Double: eps = 2.22044604925031e-16 (the value used by GNU-Octave).

i :: Complex Double

The imaginary unit: i = 0.0 :+ 1.0

Util

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