Copyright	(c) Alberto Ruiz 2006-14
License	BSD3
Maintainer	Alberto Ruiz
Stability	provisional
Safe Haskell	None
Language	Haskell98

Numeric.LinearAlgebra.HMatrix

Contents

Basic types and data processing
Arithmetic and numeric classes
Products
- dot
- matrix-vector
- matrix-matrix
- other
Linear Systems
Inverse and pseudoinverse
Determinant and rank
Norms
Nullspace and range
SVD
Eigensystems
QR
Cholesky
Hessenberg
Schur
LU
Matrix functions
Correlation and convolution
Random arrays
Misc
Auxiliary classes

Description

Synopsis

Basic types and data processing

module Numeric.LinearAlgebra.Data

Arithmetic and numeric classes

The standard numeric classes are defined elementwise:

>>> vector [1,2,3] * vector [3,0,-2]
fromList [3.0,0.0,-6.0]

>>> matrix 3 [1..9] * ident 3
(3><3)
 [ 1.0, 0.0, 0.0
 , 0.0, 5.0, 0.0
 , 0.0, 0.0, 9.0 ]

In arithmetic operations single-element vectors and matrices (created from numeric literals or using scalar) automatically expand to match the dimensions of the other operand:

>>> 5 + 2*ident 3 :: Matrix Double
(3><3)
 [ 7.0, 5.0, 5.0
 , 5.0, 7.0, 5.0
 , 5.0, 5.0, 7.0 ]

>>> matrix 3 [1..9] + matrix 1 [10,20,30]
(3><3)
 [ 11.0, 12.0, 13.0
 , 24.0, 25.0, 26.0
 , 37.0, 38.0, 39.0 ]

Products

dot

dot :: Numeric t => Vector t -> Vector t -> t Source

<·> :: Numeric t => Vector t -> Vector t -> t infixr 8 Source

infix synonym for dot

>>> vector [1,2,3,4] <·> vector [-2,0,1,1]
5.0

>>> let 𝑖 = 0:+1 :: ℂ
>>> fromList [1+𝑖,1] <·> fromList [1,1+𝑖]
2.0 :+ 0.0

(the dot symbol "·" is obtained by Alt-Gr .)

matrix-vector

app :: Numeric t => Matrix t -> Vector t -> Vector t Source

dense matrix-vector product

(#>) :: Numeric t => Matrix t -> Vector t -> Vector t infixr 8 Source

infix synonym for app

>>> let m = (2><3) [1..]
>>> m
(2><3)
 [ 1.0, 2.0, 3.0
 , 4.0, 5.0, 6.0 ]

>>> let v = vector [10,20,30]

>>> m #> v
fromList [140.0,320.0]

(!#>) :: GMatrix -> Vector Double -> Vector Double infixr 8 Source

general matrix - vector product

>>> let m = mkSparse [((0,999),1.0),((1,1999),2.0)]
>>> m !#> vector [1..2000]
fromList [1000.0,4000.0]

matrix-matrix

mul :: Numeric t => Matrix t -> Matrix t -> Matrix t Source

dense matrix product

(<>) :: Numeric t => Matrix t -> Matrix t -> Matrix t infixr 8 Source

infix synonym of mul

>>> let a = (3><5) [1..]
>>> a
(3><5)
 [  1.0,  2.0,  3.0,  4.0,  5.0
 ,  6.0,  7.0,  8.0,  9.0, 10.0
 , 11.0, 12.0, 13.0, 14.0, 15.0 ]

>>> let b = (5><2) [1,3, 0,2, -1,5, 7,7, 6,0]
>>> b
(5><2)
 [  1.0, 3.0
 ,  0.0, 2.0
 , -1.0, 5.0
 ,  7.0, 7.0
 ,  6.0, 0.0 ]

>>> a <> b
(3><2)
 [  56.0,  50.0
 , 121.0, 135.0
 , 186.0, 220.0 ]

The matrix product is also implemented in the Data.Monoid instance, where single-element matrices (created from numeric literals or using scalar) are used for scaling.

>>> import Data.Monoid as M
>>> let m = matrix 3 [1..6]
>>> m M.<> 2 M.<> diagl[0.5,1,0]
(2><3)
 [ 1.0,  4.0, 0.0
 , 4.0, 10.0, 0.0 ]

mconcat uses optimiseMult to get the optimal association order.

other

outer :: Product 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 :: Product 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 ]

cross :: Vector Double -> Vector Double -> Vector Double Source

cross product (for three-element real vectors)

scale :: Container c e => e -> c e -> c e Source

multiplication by scalar

sumElements :: Container c e => c e -> e Source

the sum of elements

prodElements :: Container c e => c e -> e Source

the product of elements

Linear Systems

(<\>) :: (LSDiv c, Field t) => Matrix t -> c t -> c t infixl 7 Source

least squares solution of a linear system, similar to the \ operator of Matlab/Octave (based on linearSolveSVD)

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

Solve a linear system (for square coefficient matrix and several right-hand sides) using the LU decomposition, returning Nothing for a singular system. For underconstrained or overconstrained systems use linearSolveLS or linearSolveSVD.

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.

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.

cholSolve :: Field t => Matrix t -> Matrix t -> Matrix t Source

Solve a symmetric or Hermitian positive definite linear system using a precomputed Cholesky decomposition obtained by chol.

cgSolve Source

Arguments

:: Bool	is symmetric
-> GMatrix	coefficient matrix
-> Vector Double	right-hand side
-> Vector Double	solution

cgSolve' Source

Arguments

:: Bool	symmetric
-> R	relative tolerance for the residual (e.g. 1E-4)
-> R	relative tolerance for δx (e.g. 1E-3)
-> Int	maximum number of iterations
-> GMatrix	coefficient matrix
-> V	initial solution
-> V	right-hand side
-> [CGState]	solution

Inverse and pseudoinverse

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

Inverse of a square matrix. See also invlndet.

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

Pseudoinverse of a general matrix with default tolerance (pinvTol 1, similar to GNU-Octave).

pinvTol :: Field t => Double -> Matrix t -> Matrix t Source

pinvTol r computes the pseudoinverse of a matrix with tolerance tol=r*g*eps*(max rows cols), where g is the greatest singular value.

m = (3><3) [ 1, 0,    0
           , 0, 1,    0
           , 0, 0, 1e-10] :: Matrix Double

>>> pinv m
1. 0.           0.
0. 1.           0.
0. 0. 10000000000.

>>> pinvTol 1E8 m
1. 0. 0.
0. 1. 0.
0. 0. 1.

Determinant and rank

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

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

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

Number of linearly independent rows or columns. See also ranksv

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

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

invlndet Source

Arguments

:: 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.

Norms

class Normed a where Source

Methods

norm_0 :: a -> ℝ Source

norm_1 :: a -> ℝ Source

norm_2 :: a -> ℝ Source

norm_Inf :: a -> ℝ Source

Instances

Normed (Vector ℂ)
Normed (Vector ℝ)
Normed (Matrix ℂ)
Normed (Matrix ℝ)

norm_Frob :: (Normed (Vector t), Element t) => Matrix t -> ℝ Source

norm_nuclear :: Field t => Matrix t -> ℝ Source

Nullspace and range

orth :: Field t => Matrix t -> Matrix t Source

return an orthonormal basis of the range space of a matrix. See also orthSVD.

nullspace :: Field t => Matrix t -> Matrix t Source

return an orthonormal basis of the null space of a matrix. See also nullspaceSVD.

null1 :: Matrix Double -> Vector Double Source

solution of overconstrained homogeneous linear system

null1sym :: Matrix Double -> Vector Double Source

solution of overconstrained homogeneous symmetric linear system

SVD

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

Full singular value decomposition.

a = (5><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
 , 13.0, 14.0, 15.0 ] :: Matrix Double

>>> let (u,s,v) = svd a

>>> disp 3 u
5x5
-0.101   0.768   0.614   0.028  -0.149
-0.249   0.488  -0.503   0.172   0.646
-0.396   0.208  -0.405  -0.660  -0.449
-0.543  -0.072  -0.140   0.693  -0.447
-0.690  -0.352   0.433  -0.233   0.398

>>> s
fromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15]

>>> disp 3 v
3x3
-0.519  -0.751   0.408
-0.576  -0.046  -0.816
-0.632   0.659   0.408

>>> let d = diagRect 0 s 5 3
>>> disp 3 d
5x3
35.183  0.000  0.000
 0.000  1.477  0.000
 0.000  0.000  0.000
 0.000  0.000  0.000

>>> disp 3 $ u <> d <> tr v
5x3
 1.000   2.000   3.000
 4.000   5.000   6.000
 7.000   8.000   9.000
10.000  11.000  12.000
13.000  14.000  15.000

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 <> tr v.

a = (5><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
 , 13.0, 14.0, 15.0 ] :: Matrix Double

>>> let (u,s,v) = thinSVD a

>>> disp 3 u
5x3
-0.101   0.768   0.614
-0.249   0.488  -0.503
-0.396   0.208  -0.405
-0.543  -0.072  -0.140
-0.690  -0.352   0.433

>>> s
fromList [35.18264833189422,1.4769076999800903,1.089145439970417e-15]

>>> disp 3 v
3x3
-0.519  -0.751   0.408
-0.576  -0.046  -0.816
-0.632   0.659   0.408

>>> disp 3 $ u <> diag s <> tr v
5x3
 1.000   2.000   3.000
 4.000   5.000   6.000
 7.000   8.000   9.000
10.000  11.000  12.000
13.000  14.000  15.000

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.

a = (5><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
 , 13.0, 14.0, 15.0 ] :: Matrix Double

>>> let (u,s,v) = compactSVD a

>>> disp 3 u
5x2
-0.101   0.768
-0.249   0.488
-0.396   0.208
-0.543  -0.072
-0.690  -0.352

>>> s
fromList [35.18264833189422,1.4769076999800903]

>>> disp 3 u
5x2
-0.101   0.768
-0.249   0.488
-0.396   0.208
-0.543  -0.072
-0.690  -0.352

>>> disp 3 $ u <> diag s <> tr v
5x3
 1.000   2.000   3.000
 4.000   5.000   6.000
 7.000   8.000   9.000
10.000  11.000  12.000
13.000  14.000  15.000

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 left singular vectors (as columns).

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

Singular values and all right singular vectors (as columns).

Eigensystems

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

Eigenvalues (not ordered) and eigenvectors (as columns) of a general square matrix.

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

a = (3><3)
 [ 3, 0, -2
 , 4, 5, -1
 , 3, 1,  0 ] :: Matrix Double

>>> let (l, v) = eig a

>>> putStr . dispcf 3 . asRow $ l
1x3
1.925+1.523i  1.925-1.523i  4.151

>>> putStr . dispcf 3 $ v
3x3
-0.455+0.365i  -0.455-0.365i   0.181
        0.603          0.603  -0.978
 0.033+0.543i   0.033-0.543i  -0.104

>>> putStr . dispcf 3 $ complex a <> v
3x3
-1.432+0.010i  -1.432-0.010i   0.753
 1.160+0.918i   1.160-0.918i  -4.059
-0.763+1.096i  -0.763-1.096i  -0.433

>>> putStr . dispcf 3 $ v <> diag l
3x3
-1.432+0.010i  -1.432-0.010i   0.753
 1.160+0.918i   1.160-0.918i  -4.059
-0.763+1.096i  -0.763-1.096i  -0.433

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

Eigenvalues and eigenvectors (as columns) of a complex hermitian or real symmetric matrix, in descending order.

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

a = (3><3)
 [ 1.0, 2.0, 3.0
 , 2.0, 4.0, 5.0
 , 3.0, 5.0, 6.0 ]

>>> let (l, v) = eigSH a

>>> l
fromList [11.344814282762075,0.17091518882717918,-0.5157294715892575]

>>> disp 3 $ v <> diag l <> tr v
3x3
1.000  2.000  3.000
2.000  4.000  5.000
3.000  5.000  6.000

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 (not ordered) of a general square matrix.

eigenvaluesSH :: Field t => Matrix t -> Vector Double Source

Eigenvalues (in descending order) 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. It works with the upper triangular part.

geigSH' Source

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.

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

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

generate a matrix with k orthogonal columns from the output of qrRaw

Cholesky

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

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 t Source

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 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 :: (Complex Double -> Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) Source

Generic matrix functions for diagonalizable matrices. For instance:

logm = matFunc log

Correlation and convolution

corr Source

Arguments

:: (Container Vector t, Product t)
=> Vector t	kernel
-> Vector t	source
-> Vector t

correlation

>>> corr (fromList[1,2,3]) (fromList [1..10])
fromList [14.0,20.0,26.0,32.0,38.0,44.0,50.0,56.0]

conv :: (Container Vector t, Product t, Num t) => Vector t -> Vector t -> Vector t Source

convolution (corr with reversed kernel and padded input, equivalent to polynomial product)

>>> conv (fromList[1,1]) (fromList [-1,1])
fromList [-1.0,0.0,1.0]

corrMin :: (Container Vector t, RealElement t, Product t) => Vector t -> Vector t -> Vector t Source

similar to corr, using min instead of (*)

corr2 :: Product a => Matrix a -> Matrix a -> Matrix a Source

2D correlation (without padding)

>>> disp 5 $ corr2 (konst 1 (3,3)) (ident 10 :: Matrix Double)
8x8
3  2  1  0  0  0  0  0
2  3  2  1  0  0  0  0
1  2  3  2  1  0  0  0
0  1  2  3  2  1  0  0
0  0  1  2  3  2  1  0
0  0  0  1  2  3  2  1
0  0  0  0  1  2  3  2
0  0  0  0  0  1  2  3

conv2 Source

Arguments

:: (Num (Matrix a), Product a, Container Vector a)
=> Matrix a	kernel
-> Matrix a
-> Matrix a

2D convolution

>>> disp 5 $ conv2 (konst 1 (3,3)) (ident 10 :: Matrix Double)
12x12
1  1  1  0  0  0  0  0  0  0  0  0
1  2  2  1  0  0  0  0  0  0  0  0
1  2  3  2  1  0  0  0  0  0  0  0
0  1  2  3  2  1  0  0  0  0  0  0
0  0  1  2  3  2  1  0  0  0  0  0
0  0  0  1  2  3  2  1  0  0  0  0
0  0  0  0  1  2  3  2  1  0  0  0
0  0  0  0  0  1  2  3  2  1  0  0
0  0  0  0  0  0  1  2  3  2  1  0
0  0  0  0  0  0  0  1  2  3  2  1
0  0  0  0  0  0  0  0  1  2  2  1
0  0  0  0  0  0  0  0  0  1  1  1

Random arrays

type Seed = Int Source

data RandDist Source

Constructors

Uniform	uniform distribution in [0,1)
Gaussian	normal distribution with mean zero and standard deviation one

Instances

Enum RandDist

randomVector Source

Arguments

:: Seed
-> RandDist	distribution
-> Int	vector size
-> Vector Double

Obtains a vector of pseudorandom elements (use randomIO to get a random seed).

rand :: Int -> Int -> IO (Matrix Double) Source

pseudorandom matrix with uniform elements between 0 and 1

randn :: Int -> Int -> IO (Matrix Double) Source

pseudorandom matrix with normal elements

>>> disp 3 =<< randn 3 5
3x5
0.386  -1.141   0.491  -0.510   1.512
0.069  -0.919   1.022  -0.181   0.745
0.313  -0.670  -0.097  -1.575  -0.583

gaussianSample Source

Arguments

:: Seed
-> Int	number of rows
-> Vector Double	mean vector
-> Matrix Double	covariance matrix
-> Matrix Double	result

Obtains a matrix whose rows are pseudorandom samples from a multivariate Gaussian distribution.

uniformSample Source

Arguments

:: Seed
-> Int	number of rows
-> [(Double, Double)]	ranges for each column
-> Matrix Double	result

Obtains a matrix whose rows are pseudorandom samples from a multivariate uniform distribution.

Misc

meanCov :: Matrix Double -> (Vector Double, Matrix Double) Source

Compute mean vector and covariance matrix of the rows of a matrix.

>>> meanCov $ gaussianSample 666 1000 (fromList[4,5]) (diagl[2,3])
(fromList [4.010341078059521,5.0197204699640405],
(2><2)
 [     1.9862461923890056, -1.0127225830525157e-2
 , -1.0127225830525157e-2,     3.0373954915729318 ])

peps :: RealFloat x => x Source

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

relativeError :: (Normed c t, Num (c t)) => NormType -> c t -> c t -> Double Source

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

optimiseMult :: Monoid (Matrix t) => [Matrix t] -> Matrix t Source

udot :: Product e => Vector e -> Vector e -> e Source

unconjugated dot product

nullspaceSVD Source

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
-> Matrix t	nullspace

The nullspace of a matrix from its precomputed SVD decomposition.

orthSVD Source

Arguments

:: Field t
=> Either Double Int	Left "numeric" zero (eg. 1*`eps`), or Right "theoretical" matrix rank.
-> Matrix t	input matrix m
-> (Matrix t, Vector Double)	`leftSV` of m
-> Matrix t	orth

The range space a matrix from its precomputed SVD decomposition.

ranksv 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.

type ℝ = Double Source

type ℂ = Complex Double Source

iC :: ℂ Source

imaginary unit

Auxiliary classes

class Storable a => Element a Source

Supported matrix elements.

This class provides optimized internal operations for selected element types. It provides unoptimised defaults for any Storable type, so you can create instances simply as: instance Element Foo.

Instances

Element Double
Element Float
Element (Complex Double)
Element (Complex Float)

class (Complexable c, Fractional e, Element e) => Container c e Source

Basic element-by-element functions for numeric containers

Minimal complete definition

size', scalar', conj', scale', scaleRecip, addConstant, add, sub, mul, divide, equal, arctan2', cmap', konst', build', atIndex', minIndex', maxIndex', minElement', maxElement', sumElements', prodElements', step', cond', find', assoc', accum'

Instances

Container Vector Double
Container Vector Float
Container Vector a => Container Matrix a
Container Vector (Complex Double)
Container Vector (Complex Float)

class (Num e, Element e) => Product e Source

Matrix product and related functions

Minimal complete definition

multiply, absSum, norm1, norm2, normInf

Instances

Product Double
Product Float
Product (Complex Double)
Product (Complex Float)

class (Container Vector t, Container Matrix t, Konst t Int Vector, Konst t (Int, Int) Matrix, Product t) => Numeric t Source

Instances

Numeric Double
Numeric Float
Numeric (Complex Double)
Numeric (Complex Float)

class LSDiv c Source

Minimal complete definition

linSolve

Instances

LSDiv Vector
LSDiv Matrix

class Complexable c Source

Structures that may contain complex numbers

Minimal complete definition

toComplex', fromComplex', comp', single', double'

Instances

Complexable Vector
Complexable Matrix

class (Element t, Element (Complex t), RealFloat t) => RealElement t Source

Supported real types

Instances

RealElement Double
RealElement Float

type family RealOf x Source

Instances

type RealOf Double = Double
type RealOf Float = Float
type RealOf (Complex Double) = Double
type RealOf (Complex Float) = Float

type family ComplexOf x Source

Instances

type ComplexOf Double = Complex Double
type ComplexOf Float = Complex Float
type ComplexOf (Complex Double) = Complex Double
type ComplexOf (Complex Float) = Complex Float

type family SingleOf x Source

Instances

type SingleOf Double = Float
type SingleOf Float = Float
type SingleOf (Complex a) = Complex (SingleOf a)

type family DoubleOf x Source

Instances

type DoubleOf Double = Double
type DoubleOf Float = Double
type DoubleOf (Complex a) = Complex (DoubleOf a)

type family IndexOf c Source

Instances

type IndexOf Vector = Int
type IndexOf Matrix = (Int, Int)

class (Product t, Convert t, Container Vector t, Container Matrix t, Normed Matrix t, Normed Vector t, Floating t, RealOf t ~ Double) => Field t Source

Generic linear algebra functions for double precision real and complex matrices.

(Single precision data can be converted using single and double).

Minimal complete definition

svd', thinSVD', sv', luPacked', luSolve', mbLinearSolve', linearSolve', cholSolve', linearSolveSVD', linearSolveLS', eig', eigSH'', eigOnly, eigOnlySH, cholSH', mbCholSH', qr', qrgr', hess', schur'

Instances

Field Double
Field (Complex Double)

class Transposable m mt | m -> mt, mt -> m where Source

Methods

tr :: m -> mt Source

(conjugate) transpose

Instances

Transposable GMatrix GMatrix
Container Vector t => Transposable (Matrix t) (Matrix t)
(KnownNat n, KnownNat m) => Transposable (M m n) (M n m)
(KnownNat n, KnownNat m) => Transposable (L m n) (L n m)

data CGState Source

Constructors

CGState
Fields cgp :: V conjugate gradient cgr :: V residual cgr2 :: R squared norm of residual cgx :: V current solution cgdx :: R normalized size of correction

class Testable t where Source

Minimal complete definition

checkT

Methods

checkT :: t -> (Bool, IO ()) Source

ioCheckT :: t -> IO (Bool, IO ()) Source

Instances

Testable GMatrix
(KnownNat n', KnownNat m') => Testable (L n' m')