Numeric.LinearAlgebra

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:

>>> 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

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

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

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

dense vector-matrix product

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

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

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

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

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

cross product (for three-element vectors)

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

the sum of elements

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

the product of elements

Linear systems

General

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

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

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

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)

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

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

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

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

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

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

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

Positive definite

cholSolve

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

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'

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

Inverse of a square matrix. See also invlndet.

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

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

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

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

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

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

Number of linearly independent rows or columns. See also ranksv

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

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

invlndet

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

Methods

norm_0 :: a -> R

norm_1 :: a -> R

norm_2 :: a -> R

norm_Inf :: a -> R

Instances

Normed (Vector Float)

Normed (Vector (Complex Float))

Normed (Vector C)

Normed (Vector R)

Normed (Vector Z)

Normed (Vector I)

KnownNat m => Normed (Vector (Mod m Z))

KnownNat m => Normed (Vector (Mod m I))

Normed (Matrix C)

Normed (Matrix R)

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

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

Nullspace and range

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

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

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

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

null1 :: Matrix R -> Vector R

solution of overconstrained homogeneous linear system

null1sym :: Herm R -> Vector R

solution of overconstrained homogeneous symmetric linear system

Singular value decomposition

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

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)

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)

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

Singular values only.

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

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

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

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

Eigendecomposition

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

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)

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)

Eigenvalues (not ordered) of a general square matrix.

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

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

geigSH

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)

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)

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

Compute the QR decomposition of a matrix in compact form.

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

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

Cholesky

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

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)

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)

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)

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

Hessenberg

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

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)

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

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.

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)

Generic matrix functions for diagonalizable matrices. For instance:

logm = matFunc log

Correlation and convolution

corr

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

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

similar to corr, using min instead of (*)

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

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

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

data RandDist

Constructors

Uniform

uniform distribution in [0,1)

Gaussian

normal distribution with mean zero and standard deviation one

Instances

Enum RandDist

randomVector

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)

pseudorandom matrix with uniform elements between 0 and 1

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

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

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

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)

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

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

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

unitary :: Vector Double -> Vector Double

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

peps :: RealFloat x => x

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

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

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

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

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

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

unconjugated dot product

nullspaceSVD

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

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

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

imaginary unit

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

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

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

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

trustSym :: Matrix t -> Herm t

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

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

Auxiliary classes

class Storable a => Element a

Supported matrix elements.

Minimal complete definition

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

Instances

Element Double

Element Float

Element CInt

Element Z

Element (Complex Double)

Element (Complex Float)

KnownNat m => Element (Mod m Z)

KnownNat m => Element (Mod m I)

class Element e => Container c e

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

Container Vector Float

Container Vector Z

Container Vector I

(Num a, Element a, Container Vector a) => Container Matrix a

Container Vector (Complex Double)

Container Vector (Complex Float)

KnownNat m => Container Vector (Mod m Z)

KnownNat m => Container Vector (Mod m I)

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

Matrix product and related functions

Minimal complete definition

multiply, absSum, norm1, norm2, normInf

Instances

Product Double

Product Float

Product Z

Product I

Product (Complex Double)

Product (Complex Float)

KnownNat m => Product (Mod m Z)

KnownNat m => Product (Mod m I)

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

Instances

Numeric Double

Numeric Float

Numeric Z

Numeric I

Numeric (Complex Double)

Numeric (Complex Float)

KnownNat m => Numeric (Mod m Z)

KnownNat m => Numeric (Mod m I)

class LSDiv c

Minimal complete definition

linSolve

Instances

LSDiv Vector

LSDiv Matrix

data Herm t

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

(Show t, Element t) => Show (Herm t)

(NFData t, Numeric t) => NFData (Herm t)

Field t => Additive (Herm t)

class Complexable c

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

Supported real types

Instances

RealElement Double

RealElement Float

type family RealOf x

Instances

type RealOf Double = Double

type RealOf Float = Float

type RealOf Z = Z

type RealOf I = I

type RealOf (Complex Double) = Double

type RealOf (Complex Float) = Float

type RealOf (Mod n Z) = Z

type RealOf (Mod n I) = I

type family ComplexOf x

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

Instances

type SingleOf Double = Float

type SingleOf Float = Float

type SingleOf (Complex a) = Complex (SingleOf a)

type family DoubleOf x

Instances

type DoubleOf Double = Double

type DoubleOf Float = Double

type DoubleOf (Complex a) = Complex (DoubleOf a)

type family IndexOf c

Instances

type IndexOf Vector = Int

type IndexOf Matrix = (Int, Int)

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

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

Field (Complex Double)

class Linear t c where

Methods

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

Instances

Container Matrix t => Linear t Matrix

Container Vector t => Linear t Vector

Field t => Linear t Herm

class Additive c where

Methods

add :: c -> c -> c

Instances

Container Vector t => Additive (Vector t)

Container Matrix t => Additive (Matrix t)

Field t => Additive (Herm t)

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

Methods

tr :: m -> mt

conjugate transpose

tr' :: m -> mt

transpose

Instances

Transposable GMatrix GMatrix

(CTrans t, 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 LU t

LU decomposition of a matrix in a compact format.

Constructors

LU (Matrix t) [Int]

Instances

(Show t, Element t) => Show (LU t)

(NFData t, Numeric t) => NFData (LU t)

data LDL t

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)

(NFData t, Numeric t) => NFData (LDL t)

data QR t

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)

data CGState

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

Minimal complete definition

checkT

Methods

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

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

Instances

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