Numeric.LinearAlgebra.Algorithms
 Portability uses ffi Stability provisional Maintainer Alberto Ruiz (aruiz at um dot es)
 Contents Supported types Products Linear Systems Matrix factorizations Singular value decomposition Eigensystems QR Cholesky Hessenberg Schur LU Matrix functions Nullspace Norms Misc Util
Description

Generic interface for the most common functions. Using it we can write higher level algorithms and testing properties for both real and complex matrices.

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

Synopsis
class (Normed (Matrix t), Linear Vector t, Linear Matrix t) => Field t
multiply :: Field t => Matrix t -> Matrix t -> Matrix t
dot :: Field t => Vector t -> Vector t -> t
outer :: Field t => Vector t -> Vector t -> Matrix t
kronecker :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolve :: Field t => Matrix t -> Matrix t -> Matrix t
luSolve :: Field t => (Matrix t, [Int]) -> 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
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
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 :: Field t => (Complex Double -> Complex Double) -> Matrix t -> 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 Normed t where
 pnorm :: NormType -> t -> Double
data NormType
 = Infinity | PNorm1 | PNorm2
ctrans :: Field t => Matrix t -> Matrix t
eps :: Double
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 :: Element t3 => (Matrix t -> (t1, Vector t3, t2)) -> Matrix t -> (t1, Matrix t3, t2)
economy :: (Element t2, Element t1, Element t) => (Matrix t -> (Matrix t1, Vector Double, Matrix t2)) -> Matrix t -> (Matrix t1, Vector Double, Matrix t2)
Supported types
 class (Normed (Matrix t), Linear Vector t, Linear Matrix t) => Field t Source
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 Source
Matrix product.
 dot :: Field t => Vector t -> Vector t -> t Source
Euclidean inner product.
 outer :: Field t => Vector t -> Vector t -> Matrix t Source

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 Source

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 Source
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 t Source
Solution of a linear system (for several right hand sides) from the precomputed LU factorization obtained by luPacked.
 linearSolveLS :: Field t => Matrix t -> Matrix t -> Matrix t Source
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 t Source
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 t Source
Inverse of a square matrix.
 pinv :: Field t => Matrix t -> Matrix t Source
Pseudoinverse of a general matrix.
 det :: Field t => Matrix t -> t Source
Determinant of a square matrix.
 rank :: Field t => Matrix t -> Int Source
Number of linearly independent rows or columns.
 rcond :: Field t => Matrix t -> Double Source
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.
 singularValues :: Field t => Matrix t -> Vector Double Source
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.
 eigenvalues :: Field t => Matrix t -> Vector (Complex Double) Source
Eigenvalues of a general square matrix.
 eigenvaluesSH :: Field t => Matrix t -> Vector Double Source
Eigenvalues of a complex hermitian or real symmetric matrix.
 eigenvaluesSH' :: Field t => Matrix t -> Vector Double Source
Similar to eigenvaluesSH without checking that the input matrix is hermitian or symmetric.
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 t Source

Cholesky factorization of a positive definite hermitian or symmetric matrix.

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

 cholSH :: Field t => Matrix t -> Matrix t Source
Similar to chol without checking that the input matrix is hermitian or symmetric.
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 t Source
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 Source

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) Source

Generic matrix functions for diagonalizable matrices. For instance:

`logm = matFunc log`
Nullspace
 nullspacePrec Source
 :: Field t => Double input matrix -> Matrix t list of unitary vectors spanning the nullspace -> [Vector t] The nullspace of a matrix. See also nullspaceSVD.
 nullVector :: Field t => Matrix t -> Vector t Source
The nullspace of a matrix, assumed to be one-dimensional, with machine precision.
 nullspaceSVD Source
 :: Field t => Either Double Int input matrix m -> Matrix t rightSV of m -> (Vector Double, Matrix t) list of unitary vectors spanning the nullspace -> [Vector t] The nullspace of a matrix from its SVD decomposition.
Norms
 class Normed t where Source

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 Source
Instances
 Normed (Vector Double) Normed (Vector (Complex Double)) Normed (Matrix Double) Normed (Matrix (Complex Double))
 data NormType Source
Constructors
 Infinity PNorm1 PNorm2
Misc
 ctrans :: Field t => Matrix t -> Matrix t Source
Generic conjugate transpose.
 eps :: Double Source
The machine precision of a Double: eps = 2.22044604925031e-16 (the value used by GNU-Octave).
 i :: Complex Double Source
The imaginary unit: i = 0.0 :+ 1.0
Util
 haussholder :: Field a => a -> Vector a -> Matrix a Source
 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
 pinvTol :: Double -> Matrix Double -> Matrix Double Source
 ranksv Source
 :: 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 :: Element t3 => (Matrix t -> (t1, Vector t3, t2)) -> Matrix t -> (t1, Matrix t3, t2) Source
 economy :: (Element t2, Element t1, Element t) => (Matrix t -> (Matrix t1, Vector Double, Matrix t2)) -> Matrix t -> (Matrix t1, Vector Double, Matrix t2) Source
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