Copyright	(c) Alberto Ruiz 2006-15
License	BSD3
Maintainer	Alberto Ruiz
Stability	provisional
Safe Haskell	None
Language	Haskell98

Numeric.LinearAlgebra

Contents

Basic types and data manipulation
Numeric classes
Autoconformable dimensions
Products
- Dot
- Matrix-vector
- Matrix-matrix
- Other
Linear systems
- General
- Determined
- Symmetric indefinite
- Positive definite
- Sparse
Inverse and pseudoinverse
Determinant and rank
Norms
Nullspace and range
Singular value decomposition
Eigendecomposition
QR
Cholesky
LU
Hessenberg
Schur
Matrix functions
Correlation and convolution
Random arrays
Misc
Auxiliary classes

Description

Synopsis

Basic types and data manipulation

This package works with 2D (Matrix) and 1D (Vector) arrays of real (R) or complex (C) double precision numbers. Single precision and machine integers are also supported for basic arithmetic and data manipulation.

module Numeric.LinearAlgebra.Data

Numeric classes

The standard numeric classes are defined elementwise (commonly referred to as the Hadamard product or the Schur product):

>>> 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 ]

Autoconformable dimensions

In most operations, single-element vectors and matrices (created from numeric literals or using scalar), and matrices with just one row or column, 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 ]

>>> (4><3) [1..] + row [10,20,30]
(4><3)
 [ 11.0, 22.0, 33.0
 , 14.0, 25.0, 36.0
 , 17.0, 28.0, 39.0
 , 20.0, 31.0, 42.0 ]

Products

Dot

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

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

An 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

Matrix-vector

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

dense matrix-vector product

>>> 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]

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

dense vector-matrix product

(!#>) :: 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

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

dense matrix product

>>> 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 :: Product t => Vector t -> Vector t -> Vector t Source

cross product (for three-element vectors)

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

General

(<\>) :: (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)

a = (3><2)
 [ 1.0,  2.0
 , 2.0,  4.0
 , 2.0, -1.0 ]

v = vector [13.0,27.0,1.0]

>>> let x = a <\> v
>>> x
fromList [3.0799999999999996,5.159999999999999]

>>> a #> x
fromList [13.399999999999999,26.799999999999997,1.0]

It also admits multiple right-hand sides stored as columns in a matrix.

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.

Determined

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.

a = (2><2)
 [ 1.0, 2.0
 , 3.0, 5.0 ]

b = (2><3)
 [  6.0, 1.0, 10.0
 , 15.0, 3.0, 26.0 ]

>>> linearSolve a b
Just (2><3)
 [ -1.4802973661668753e-15,     0.9999999999999997, 1.999999999999997
 ,       3.000000000000001, 1.6653345369377348e-16, 4.000000000000002 ]

>>> let Just x = it
>>> disp 5 x
2x3
-0.00000  1.00000  2.00000
 3.00000  0.00000  4.00000

>>> a <> x
(2><3)
 [  6.0, 1.0, 10.0
 , 15.0, 3.0, 26.0 ]

luSolve :: Field t => LU t -> Matrix t -> Matrix t Source

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

luPacked :: Field t => Matrix t -> LU t Source

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

luSolve' :: (Fractional a, Container Vector a) => LU a -> Matrix a -> Matrix a Source

Experimental implementation of luSolve for any Fractional element type, including Mod n I and Mod n Z.

>>> let a = (2><2) [1,2,3,5] :: Matrix (Z ./. 13)
(2><2)
 [ 1, 2
 , 3, 5 ]
>>> b
(2><3)
 [ 5, 1, 3
 , 8, 6, 3 ]

>>> luSolve' (luPacked' a) b
(2><3)
 [ 4,  7, 4
 , 7, 10, 6 ]

luPacked' :: (Fractional t, Num (Vector t), Container Vector t, Normed (Vector t)) => Matrix t -> LU t Source

Experimental implementation of luPacked for any Fractional element type, including Mod n I and Mod n Z.

>>> let m = ident 5 + (5><5) [0..] :: Matrix (Z ./. 17)
(5><5)
 [  1,  1,  2,  3,  4
 ,  5,  7,  7,  8,  9
 , 10, 11, 13, 13, 14
 , 15, 16,  0,  2,  2
 ,  3,  4,  5,  6,  8 ]

>>> let (l,u,p,s) = luFact $ luPacked' m
>>> l
(5><5)
 [  1,  0, 0,  0, 0
 ,  6,  1, 0,  0, 0
 , 12,  7, 1,  0, 0
 ,  7, 10, 7,  1, 0
 ,  8,  2, 6, 11, 1 ]
>>> u
(5><5)
 [ 15, 16,  0,  2,  2
 ,  0, 13,  7, 13, 14
 ,  0,  0, 15,  0, 11
 ,  0,  0,  0, 15, 15
 ,  0,  0,  0,  0,  1 ]

Symmetric indefinite

ldlSolve :: Field t => LDL t -> Matrix t -> Matrix t Source

Solution of a linear system (for several right hand sides) from a precomputed LDL factorization obtained by ldlPacked.

Note: this can be slower than the general solver based on the LU decomposition.

ldlPacked :: Field t => Herm t -> LDL t Source

Obtains the LDL decomposition of a matrix in a compact data structure suitable for ldlSolve.

Positive definite

cholSolve Source

Arguments

:: Field t
=> Matrix t	Cholesky decomposition of the coefficient matrix
-> Matrix t	right hand sides
-> Matrix t	solution

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

Sparse

cgSolve Source

Arguments

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

Solve a sparse linear system using the conjugate gradient method with default parameters.

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
-> Vector R	initial solution
-> Vector R	right-hand side
-> [CGState]	solution

Solve a sparse linear system using the conjugate gradient method with default parameters.

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 Float) Source
Normed (Vector (Complex Float)) Source
Normed (Vector C) Source
Normed (Vector R) Source
Normed (Vector Z) Source
Normed (Vector I) Source
KnownNat m => Normed (Vector (Mod m Z)) Source
KnownNat m => Normed (Vector (Mod m I)) Source
Normed (Matrix C) Source
Normed (Matrix R) Source

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 R -> Vector R Source

solution of overconstrained homogeneous linear system

null1sym :: Herm R -> Vector R Source

solution of overconstrained homogeneous symmetric linear system

Singular value decomposition

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).

Eigendecomposition

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 => Herm 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

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

Eigenvalues (not ordered) of a general square matrix.

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

Eigenvalues (in descending order) of a complex hermitian or real symmetric matrix.

geigSH Source

Arguments

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

Generalized symmetric positive definite eigensystem Av = lBv, for A and B symmetric, B positive definite.

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 -> QR t Source

Compute the QR decomposition of a matrix in compact form.

qrgr :: Field t => Int -> QR t -> Matrix t Source

generate a matrix with k orthogonal columns from the compact QR decomposition obtained by qrRaw.

Cholesky

chol :: Field t => Herm 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 == tr c <> c.

mbChol :: Field t => Herm t -> Maybe (Matrix t) Source

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

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.

luFact :: Numeric t => LU t -> (Matrix t, Matrix t, Matrix t, t) Source

Compute the explicit LU decomposition from the compact one obtained by luPacked.

Hessenberg

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

Hessenberg factorization.

If (p,h) = hess m then m == p <> h <> tr 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 <> tr 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)

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.

m = (2><2) [4,9
           ,0,4] :: Matrix Double

>>> sqrtm m
(2><2)
 [ 2.0, 2.25
 , 0.0,  2.0 ]

For diagonalizable matrices you can try matFunc sqrt:

>>> matFunc sqrt ((2><2) [1,0,0,-1])
(2><2)
 [ 1.0 :+ 0.0, 0.0 :+ 0.0
 , 0.0 :+ 0.0, 0.0 :+ 1.0 ]

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 Source

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
-> Herm 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, Herm 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 ])

rowOuters :: Matrix Double -> Matrix Double -> Matrix Double Source

outer products of rows

>>> a
(3><2)
 [   1.0,   2.0
 ,  10.0,  20.0
 , 100.0, 200.0 ]
>>> b
(3><3)
 [ 1.0, 2.0, 3.0
 , 4.0, 5.0, 6.0
 , 7.0, 8.0, 9.0 ]

>>> rowOuters a (b ||| 1)
(3><8)
 [   1.0,   2.0,   3.0,   1.0,    2.0,    4.0,    6.0,   2.0
 ,  40.0,  50.0,  60.0,  10.0,   80.0,  100.0,  120.0,  20.0
 , 700.0, 800.0, 900.0, 100.0, 1400.0, 1600.0, 1800.0, 200.0 ]

pairwiseD2 :: Matrix Double -> Matrix Double -> Matrix Double Source

Matrix of pairwise squared distances of row vectors (using the matrix product trick in blog.smola.org)

unitary :: Vector Double -> Vector Double Source

Obtains a vector in the same direction with 2-norm=1

peps :: RealFloat x => x Source

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

relativeError :: Num a => (a -> Double) -> a -> a -> Double Source

magnit :: (Element t, Normed (Vector t)) => R -> t -> Bool Source

Check if the absolute value or complex magnitude is greater than a given threshold

>>> magnit 1E-6 (1E-12 :: R)
False
>>> magnit 1E-6 (3+iC :: C)
True
>>> magnit 0 (3 :: I ./. 5)
True

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.

iC :: C Source

imaginary unit

sym :: Field t => Matrix t -> Herm t Source

Compute the complex Hermitian or real symmetric part of a square matrix ((x + tr x)/2).

mTm :: Numeric t => Matrix t -> Herm t Source

Compute the contraction tr x <> x of a general matrix.

trustSym :: Matrix t -> Herm t Source

At your own risk, declare that a matrix is complex Hermitian or real symmetric for usage in chol, eigSH, etc. Only a triangular part of the matrix will be used.

unSym :: Herm t -> Matrix t Source

Extract the general matrix from a Herm structure, forgetting its symmetric or Hermitian property.

Auxiliary classes

class Storable a => Element a Source

Supported matrix elements.

Minimal complete definition

constantD, extractR, setRect, sortI, sortV, compareV, selectV, remapM, rowOp, gemm

Instances

Element Double Source
Element Float Source
Element CInt Source
Element Z Source
Element (Complex Double) Source
Element (Complex Float) Source
KnownNat m => Element (Mod m Z) Source
KnownNat m => Element (Mod m I) Source

class Element e => Container c e Source

Basic element-by-element functions for numeric containers

Minimal complete definition

conj', size', scalar', scale', addConstant, add', sub, mul, equal, cmap', konst', build', atIndex', minIndex', maxIndex', minElement', maxElement', sumElements', prodElements', step', ccompare', cselect', find', assoc', accum', scaleRecip, divide, arctan2', cmod', fromInt', toInt', fromZ', toZ'

Instances

Container Vector Double Source
Container Vector Float Source
Container Vector Z Source
Container Vector I Source
(Num a, Element a, Container Vector a) => Container Matrix a Source
Container Vector (Complex Double) Source
Container Vector (Complex Float) Source
KnownNat m => Container Vector (Mod m Z) Source
KnownNat m => Container Vector (Mod m I) Source

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

Matrix product and related functions

Minimal complete definition

multiply, absSum, norm1, norm2, normInf

Instances

Product Double Source
Product Float Source
Product Z Source
Product I Source
Product (Complex Double) Source
Product (Complex Float) Source
KnownNat m => Product (Mod m Z) Source
KnownNat m => Product (Mod m I) Source

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

Instances

Numeric Double Source
Numeric Float Source
Numeric Z Source
Numeric I Source
Numeric (Complex Double) Source
Numeric (Complex Float) Source
KnownNat m => Numeric (Mod m Z) Source
KnownNat m => Numeric (Mod m I) Source

class LSDiv c Source

Minimal complete definition

linSolve

Instances

LSDiv Vector Source
LSDiv Matrix Source

data Herm t Source

A matrix that, by construction, it is known to be complex Hermitian or real symmetric.

It can be created using sym, mTm, or trustSym, and the matrix can be extracted using unSym.

Instances

Field t => Linear t Herm Source
(Show t, Element t) => Show (Herm t) Source
(NFData t, Numeric t) => NFData (Herm t) Source
Field t => Additive (Herm t) Source

class Complexable c Source

Structures that may contain complex numbers

Minimal complete definition

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

Instances

Complexable Vector Source
Complexable Matrix Source

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

Supported real types

Instances

RealElement Double Source
RealElement Float Source

type family RealOf x Source

Instances

type RealOf Double = Double Source
type RealOf Float = Float Source
type RealOf Z = Z Source
type RealOf I = I Source
type RealOf (Complex Double) = Double Source
type RealOf (Complex Float) = Float Source
type RealOf (Mod n Z) = Z Source
type RealOf (Mod n I) = I Source

type family ComplexOf x Source

Instances

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

type family SingleOf x Source

Instances

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

type family DoubleOf x Source

Instances

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

type family IndexOf c Source

Instances

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

class (Numeric t, Convert t, Normed Matrix t, Normed Vector t, Floating t, Linear t Vector, Linear t Matrix, Additive (Vector t), Additive (Matrix 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', ldlPacked', ldlSolve', linearSolveSVD', linearSolveLS', eig', eigSH'', eigOnly, eigOnlySH, cholSH', mbCholSH', qr', qrgr', hess', schur'

Instances

Field Double Source
Field (Complex Double) Source

class Linear t c where Source

Methods

scale :: t -> c t -> c t Source

Instances

Container Matrix t => Linear t Matrix Source
Container Vector t => Linear t Vector Source
Field t => Linear t Herm Source

class Additive c where Source

Methods

add :: c -> c -> c Source

Instances

Container Vector t => Additive (Vector t) Source
Container Matrix t => Additive (Matrix t) Source
Field t => Additive (Herm t) Source
Additive (C n) Source
Additive (R n) Source
KnownNat n => Additive (Sym n) Source
(KnownNat m, KnownNat n) => Additive (M m n) Source
(KnownNat m, KnownNat n) => Additive (L m n) Source

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

Methods

tr :: m -> mt Source

conjugate transpose

tr' :: m -> mt Source

transpose

Instances

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

data LU t Source

LU decomposition of a matrix in a compact format.

Constructors

LU (Matrix t) [Int]

Instances

(Show t, Element t) => Show (LU t) Source
(NFData t, Numeric t) => NFData (LU t) Source

data LDL t Source

LDL decomposition of a complex Hermitian or real symmetric matrix in a compact format.

Constructors

LDL (Matrix t) [Int]

Instances

(Show t, Element t) => Show (LDL t) Source
(NFData t, Numeric t) => NFData (LDL t) Source

data QR t Source

QR decomposition of a matrix in compact form. (The orthogonal matrix is not explicitly formed.)

Constructors

QR (Matrix t) (Vector t)

Instances

(NFData t, Numeric t) => NFData (QR t) Source

data CGState Source

Constructors

CGState
Fields cgp :: Vector R conjugate gradient cgr :: Vector R residual cgr2 :: R squared norm of residual cgx :: Vector R 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

KnownNat m => Testable (Matrix (Mod m I)) Source