hmatrix-0.13.1.0: Linear algebra and numerical computation

Portability uses ffi provisional Alberto Ruiz (aruiz at um dot es)

Numeric.LinearAlgebra.Algorithms

Description

High level generic interface to common matrix computations.

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

Synopsis

Supported types

class (Product t, Convert t, Container Vector t, Container Matrix t, Normed Matrix t, Normed Vector t) => Field t Source

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`).

Instances

 Field Double Field (Complex Double)

Linear Systems

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

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.

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 rank-deficient 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 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.

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

Inverse of a square matrix. See also `invlndet`.

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

Pseudoinverse of a general matrix.

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

Determinant of a square matrix. To avoid possible overflow or underflow use `invlndet`.

Arguments

 :: (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.

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

Number of linearly independent rows or columns.

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

Reciprocal of the 2-norm 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.

Singular values only.

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.

Eigenvalues of a complex hermitian or real symmetric matrix.

Similar to `eigenvaluesSH` without checking that the input matrix is hermitian or symmetric. It works with the upper triangular part.

Arguments

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

Generalized symmetric positive definite eigensystem Av = lBv, for A and B symmetric, B positive definite (conditions not checked).

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.

mbCholSH :: Field t => Matrix t -> Maybe (Matrix t)Source

Similar to `cholSH`, but instead of an error (e.g., caused by a matrix not positive definite) it returns `Nothing`.

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 ]```

Generic matrix functions for diagonalizable matrices. For instance:

`logm = matFunc log`

Nullspace

Arguments

 :: 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

The nullspace of a matrix. See also `nullspaceSVD`.

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

The nullspace of a matrix, assumed to be one-dimensional, with machine precision.

Arguments

 :: 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

The nullspace of a matrix from its SVD decomposition.

Norms

class RealFloat (RealOf t) => Normed c t whereSource

Methods

pnorm :: NormType -> c t -> RealOf tSource

Instances

 Normed Vector Double Normed Vector Float Normed Matrix Double Normed Matrix Float Normed Vector (Complex Double) Normed Vector (Complex Float) Normed Matrix (Complex Double) Normed Matrix (Complex Float)

data NormType Source

Constructors

 Infinity PNorm1 PNorm2 Frobenius

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 = 2.22044604925031e-16` (the value used by GNU-Octave).

peps :: RealFloat x => xSource

1 + 0.5*peps == 1, 1 + 0.6*peps /= 1

The imaginary unit: `i = 0.0 :+ 1.0`

Util

haussholder :: Field a => a -> Vector a -> Matrix aSource

unpackQR :: Field t => (Matrix t, Vector t) -> (Matrix t, Matrix t)Source

unpackHess :: Field t => (Matrix t -> (Matrix t, Vector t)) -> Matrix t -> (Matrix t, Matrix t)Source

Arguments

 :: Double numeric zero (e.g. 1*`eps`) -> Int maximum dimension of the matrix -> [Double] singular values -> Int rank of m

Numeric rank of a matrix from its singular values.

full :: (Num t1, Storable t1) => (Matrix t -> (t2, Vector t1, t3)) -> Matrix t -> (t2, Matrix t1, t3)Source

economy :: (Element t, Element t1, Element t2) => (Matrix t -> (Matrix t1, Vector Double, Matrix t2)) -> Matrix t -> (Matrix t1, Vector Double, Matrix t2)Source