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
- Triangular
- Tridiagonal
- 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

module Numeric.LinearAlgebra.Data
dot :: Numeric t => Vector t -> Vector t -> t
(<.>) :: Numeric t => Vector t -> Vector t -> t
(#>) :: Numeric t => Matrix t -> Vector t -> Vector t
(<#) :: Numeric t => Vector t -> Matrix t -> Vector t
(!#>) :: GMatrix -> Vector Double -> Vector Double
(<>) :: Numeric t => Matrix t -> Matrix t -> Matrix t
outer :: Product t => Vector t -> Vector t -> Matrix t
kronecker :: Product t => Matrix t -> Matrix t -> Matrix t
cross :: Product t => Vector t -> Vector t -> Vector t
scale :: Linear t c => t -> c t -> c t
add :: Additive c => c -> c -> c
sumElements :: Container c e => c e -> e
prodElements :: Container c e => c e -> e
(<\>) :: (LSDiv c, Field t) => Matrix t -> c t -> c t
linearSolveLS :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolveSVD :: Field t => Matrix t -> Matrix t -> Matrix t
linearSolve :: Field t => Matrix t -> Matrix t -> Maybe (Matrix t)
luSolve :: Field t => LU t -> Matrix t -> Matrix t
luPacked :: Field t => Matrix t -> LU t
luSolve' :: (Fractional t, Container Vector t) => LU t -> Matrix t -> Matrix t
luPacked' :: (Container Vector t, Fractional t, Normed (Vector t), Num (Vector t)) => Matrix t -> LU t
ldlSolve :: Field t => LDL t -> Matrix t -> Matrix t
ldlPacked :: Field t => Herm t -> LDL t
cholSolve :: Field t => Matrix t -> Matrix t -> Matrix t
data UpLo
- = Lower
- | Upper
triSolve :: Field t => UpLo -> Matrix t -> Matrix t -> Matrix t
triDiagSolve :: Field t => Vector t -> Vector t -> Vector t -> Matrix t -> Matrix t
cgSolve :: Bool -> GMatrix -> Vector R -> Vector R
cgSolve' :: Bool -> R -> R -> Int -> GMatrix -> Vector R -> Vector R -> [CGState]
inv :: Field t => Matrix t -> Matrix t
pinv :: Field t => Matrix t -> Matrix t
pinvTol :: Field t => Double -> Matrix t -> Matrix t
rcond :: Field t => Matrix t -> Double
rank :: Field t => Matrix t -> Int
det :: Field t => Matrix t -> t
invlndet :: Field t => Matrix t -> (Matrix t, (t, t))
class Normed a where
- norm_0 :: a -> R
- norm_1 :: a -> R
- norm_2 :: a -> R
- norm_Inf :: a -> R
norm_Frob :: (Normed (Vector t), Element t) => Matrix t -> R
norm_nuclear :: Field t => Matrix t -> R
orth :: Field t => Matrix t -> Matrix t
nullspace :: Field t => Matrix t -> Matrix t
null1 :: Matrix R -> Vector R
null1sym :: Herm R -> Vector R
svd :: Field t => Matrix t -> (Matrix t, Vector 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)
compactSVDTol :: Field t => Double -> 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))
geig :: Field t => Matrix t -> Matrix t -> (Vector (Complex Double), Vector t, Matrix (Complex Double))
eigSH :: Field t => Herm t -> (Vector Double, Matrix t)
eigenvalues :: Field t => Matrix t -> Vector (Complex Double)
geigenvalues :: Field t => Matrix t -> Matrix t -> (Vector (Complex Double), Vector t)
eigenvaluesSH :: Field t => Herm t -> Vector Double
geigSH :: Field t => Herm t -> Herm t -> (Vector Double, Matrix t)
qr :: Field t => Matrix t -> (Matrix t, Matrix t)
thinQR :: Field t => Matrix t -> (Matrix t, Matrix t)
rq :: Field t => Matrix t -> (Matrix t, Matrix t)
thinRQ :: Field t => Matrix t -> (Matrix t, Matrix t)
qrRaw :: Field t => Matrix t -> QR t
qrgr :: Field t => Int -> QR t -> Matrix t
chol :: Field t => Herm t -> Matrix t
mbChol :: Field t => Herm t -> Maybe (Matrix t)
lu :: Field t => Matrix t -> (Matrix t, Matrix t, Matrix t, t)
luFact :: Numeric t => LU t -> (Matrix t, Matrix t, Matrix t, t)
hess :: Field t => Matrix t -> (Matrix t, Matrix t)
schur :: Field t => Matrix t -> (Matrix t, Matrix t)
expm :: Field t => Matrix t -> Matrix t
sqrtm :: Field t => Matrix t -> Matrix t
matFunc :: (Complex Double -> Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double)
corr :: (Container Vector t, Product t) => Vector t -> Vector t -> Vector t
conv :: (Container Vector t, Product t, Num t) => Vector t -> Vector t -> Vector t
corrMin :: (Container Vector t, RealElement t, Product t) => Vector t -> Vector t -> Vector t
corr2 :: Product a => Matrix a -> Matrix a -> Matrix a
conv2 :: (Num (Matrix a), Product a, Container Vector a) => Matrix a -> Matrix a -> Matrix a
type Seed = Int
data RandDist
- = Uniform
- | Gaussian
randomVector :: Seed -> RandDist -> Int -> Vector Double
rand :: Int -> Int -> IO (Matrix Double)
randn :: Int -> Int -> IO (Matrix Double)
gaussianSample :: Seed -> Int -> Vector Double -> Herm Double -> Matrix Double
uniformSample :: Seed -> Int -> [(Double, Double)] -> Matrix Double
meanCov :: Matrix Double -> (Vector Double, Herm Double)
rowOuters :: Matrix Double -> Matrix Double -> Matrix Double
pairwiseD2 :: Matrix Double -> Matrix Double -> Matrix Double
normalize :: (Normed (Vector t), Num (Vector t), Field t) => Vector t -> Vector t
peps :: RealFloat x => x
relativeError :: Num a => (a -> Double) -> a -> a -> Double
magnit :: (Element t, Normed (Vector t)) => R -> t -> Bool
haussholder :: Field a => a -> Vector a -> Matrix a
optimiseMult :: Monoid (Matrix t) => [Matrix t] -> Matrix t
udot :: Product e => Vector e -> Vector e -> e
nullspaceSVD :: Field t => Either Double Int -> Matrix t -> (Vector Double, Matrix t) -> Matrix t
orthSVD :: Field t => Either Double Int -> Matrix t -> (Matrix t, Vector Double) -> Matrix t
ranksv :: Double -> Int -> [Double] -> Int
iC :: C
sym :: Field t => Matrix t -> Herm t
mTm :: Numeric t => Matrix t -> Herm t
trustSym :: Matrix t -> Herm t
unSym :: Herm t -> Matrix t
class Storable a => Element a
class Element e => Container c e
class (Num e, Element e) => Product e
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
class LSDiv c
data Herm t
class Complexable c
class (Element t, Element (Complex t), RealFloat t) => RealElement t
type family RealOf x
type ComplexOf x = Complex (RealOf x)
type family SingleOf x
type family DoubleOf x
type family IndexOf (c :: * -> *)
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
class Linear t c
class Additive c
class Transposable m mt | m -> mt, mt -> m
data LU t = LU (Matrix t) [Int]
data LDL t = LDL (Matrix t) [Int]
data QR t = QR (Matrix t) (Vector t)
data CGState = CGState {
- cgp :: Vector R
- cgr :: Vector R
- cgr2 :: R
- cgx :: Vector R
- cgdx :: R
}
class Testable t where
- checkT :: t -> (Bool, IO ())
- ioCheckT :: t -> IO (Bool, IO ())

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]
[3.0,0.0,-6.0]
it :: Vector R

>>> 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
[140.0,320.0]
it :: Vector Numeric.LinearAlgebra.Data.R

(<#) :: 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 :: GMatrix
>>> m !#> vector [1..2000]
[1000.0,4000.0]
it :: Vector Double

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)

scale :: Linear t c => t -> c t -> c t Source #

add :: Additive c => c -> c -> c Source #

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
[3.0799999999999996,5.159999999999999]
it :: Vector Numeric.LinearAlgebra.Data.R

>>> a #> x
[13.399999999999999,26.799999999999997,0.9999999999999991]
it :: Vector Numeric.LinearAlgebra.Data.R

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

Triangular

data UpLo Source #

Constructors

Lower
Upper

triSolve Source #

Arguments

:: Field t
=> UpLo	`Lower` or `Upper`
-> Matrix t	coefficient matrix
-> Matrix t	right hand sides
-> Matrix t	solution

Solve a triangular linear system. If Upper is specified then all elements below the diagonal are ignored; if Lower is specified then all elements above the diagonal are ignored.

Tridiagonal

triDiagSolve Source #

Arguments

:: Field t
=> Vector t	lower diagonal: \(n - 1\) elements
-> Vector t	diagonal: \(n\) elements
-> Vector t	upper diagonal: \(n - 1\) elements
-> Matrix t	right hand sides
-> Matrix t	solution

Solve a tridiagonal linear system. Suppose you wish to solve \(Ax = b\) where

\[ A = \begin{bmatrix} 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 & 4.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 3.0 & 1.0 \end{bmatrix} \quad b = \begin{bmatrix} 1.0 & 1.0 & 1.0 \\ 1.0 & -1.0 & 2.0 \\ 1.0 & 1.0 & 3.0 \\ 1.0 & -1.0 & 4.0 \\ 1.0 & 1.0 & 5.0 \\ 1.0 & -1.0 & 6.0 \\ 1.0 & 1.0 & 7.0 \\ 1.0 & -1.0 & 8.0 \\ 1.0 & 1.0 & 9.0 \end{bmatrix} \]

then

dL =  fromList [3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0]
d  =  fromList [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
dU =  fromList [4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0]

b = (9><3)
    [
      1.0,   1.0,   1.0,
      1.0,  -1.0,   2.0,
      1.0,   1.0,   3.0,
      1.0,  -1.0,   4.0,
      1.0,   1.0,   5.0,
      1.0,  -1.0,   6.0,
      1.0,   1.0,   7.0,
      1.0,  -1.0,   8.0,
      1.0,   1.0,   9.0
    ]

x = triDiagSolve dL d dU b

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 #

p-norm for vectors, operator norm for matrices

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 #
Instance details Defined in Internal.Util Methods norm_0 :: Vector Float -> R Source # norm_1 :: Vector Float -> R Source # norm_2 :: Vector Float -> R Source # norm_Inf :: Vector Float -> R Source #
Normed (Vector (Complex Float)) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Vector (Complex Float) -> R Source # norm_1 :: Vector (Complex Float) -> R Source # norm_2 :: Vector (Complex Float) -> R Source # norm_Inf :: Vector (Complex Float) -> R Source #
Normed (Vector C) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Vector C -> R Source # norm_1 :: Vector C -> R Source # norm_2 :: Vector C -> R Source # norm_Inf :: Vector C -> R Source #
Normed (Vector R) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Vector R -> R Source # norm_1 :: Vector R -> R Source # norm_2 :: Vector R -> R Source # norm_Inf :: Vector R -> R Source #
Normed (Vector Z) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Vector Z -> R Source # norm_1 :: Vector Z -> R Source # norm_2 :: Vector Z -> R Source # norm_Inf :: Vector Z -> R Source #
Normed (Vector I) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Vector I -> R Source # norm_1 :: Vector I -> R Source # norm_2 :: Vector I -> R Source # norm_Inf :: Vector I -> R Source #
KnownNat m => Normed (Vector (Mod m Z)) Source #
Instance details Defined in Internal.Modular Methods norm_0 :: Vector (Mod m Z) -> R Source # norm_1 :: Vector (Mod m Z) -> R Source # norm_2 :: Vector (Mod m Z) -> R Source # norm_Inf :: Vector (Mod m Z) -> R Source #
KnownNat m => Normed (Vector (Mod m I)) Source #
Instance details Defined in Internal.Modular Methods norm_0 :: Vector (Mod m I) -> R Source # norm_1 :: Vector (Mod m I) -> R Source # norm_2 :: Vector (Mod m I) -> R Source # norm_Inf :: Vector (Mod m I) -> R Source #
Normed (Matrix C) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Matrix C -> R Source # norm_1 :: Matrix C -> R Source # norm_2 :: Matrix C -> R Source # norm_Inf :: Matrix C -> R Source #
Normed (Matrix R) Source #
Instance details Defined in Internal.Util Methods norm_0 :: Matrix R -> R Source # norm_1 :: Matrix R -> R Source # norm_2 :: Matrix R -> R Source # norm_Inf :: Matrix R -> R Source #
KnownNat n => Normed (R n) Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods norm_0 :: R n -> R0 Source # norm_1 :: R n -> R0 Source # norm_2 :: R n -> R0 Source # norm_Inf :: R n -> R0 Source #
(KnownNat m, KnownNat n) => Normed (L m n) Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods norm_0 :: L m n -> R Source # norm_1 :: L m n -> R Source # norm_2 :: L m n -> R Source # norm_Inf :: L m n -> R Source #

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

Frobenius norm (Schatten p-norm with p=2)

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

Sum of singular values (Schatten p-norm with p=1)

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
[35.18264833189422,1.4769076999800903,1.089145439970417e-15]
it :: Vector Double

>>> 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
[35.18264833189422,1.4769076999800903,1.089145439970417e-15]
it :: Vector Double

>>> 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
[35.18264833189422,1.476907699980091]
it :: Vector Double

>>> 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

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

compactSVDTol r is similar to compactSVD (for which r=1), but uses tolerance tol=r*g*eps*(max rows cols) to distinguish nonzero singular values, where g is the greatest singular value. If g<r*eps, then only one singular value is returned.

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

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

Generalized eigenvalues (not ordered) and eigenvectors (as columns) of a pair of nonsymmetric matrices. Eigenvalues are represented as pairs of alpha, beta, where eigenvalue = alpha / beta. Alpha is always complex, but betas has the same type as the input matrix.

If (alphas, betas, v) = geig a b, then a <> v == b <> v <> diag (alphas / betas)

Note that beta can be 0 and that has reasonable interpretation.

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
[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.

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

Generalized eigenvalues of a pair of matrices. Represented as pairs of alpha, beta, where eigenvalue is alpha / beta as in geig.

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. Note: the current implementation is very slow for large matrices. thinQR is much faster.

thinQR :: Field t => Matrix t -> (Matrix t, Matrix t) Source #

A version of qr which returns only the min (rows m) (cols m) columns of q and rows of r.

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. Note: the current implementation is very slow for large matrices. thinRQ is much faster.

thinRQ :: Field t => Matrix t -> (Matrix t, Matrix t) Source #

A version of rq which returns only the min (rows m) (cols m) columns of r and rows of q.

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])
[14.0,20.0,26.0,32.0,38.0,44.0,50.0,56.0]
it :: (Enum t, Product t, Container Vector t) => Vector t

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])
[-1.0,0.0,1.0]
it :: (Product t, Container Vector t) => Vector t

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 #
Instance details Defined in Internal.Vectorized Methods succ :: RandDist -> RandDist # pred :: RandDist -> RandDist # toEnum :: Int -> RandDist # fromEnum :: RandDist -> Int # enumFrom :: RandDist -> [RandDist] # enumFromThen :: RandDist -> RandDist -> [RandDist] # enumFromTo :: RandDist -> RandDist -> [RandDist] # enumFromThenTo :: RandDist -> RandDist -> RandDist -> [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
-> 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]) (trustSym $ diagl [2,3])
([3.9933155655086696,5.061409102770331],Herm (2><2)
 [    1.9963242906624408, -4.227815571404954e-2
 , -4.227815571404954e-2,    3.2003833097832857 ])
it :: (Vector Double, Herm Double)

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)

normalize :: (Normed (Vector t), Num (Vector t), Field t) => Vector t -> Vector t 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, reorderV

Instances

Element Double Source #
Instance details Defined in Internal.Matrix Methods constantD :: Double -> Int -> Vector Double extractR :: MatrixOrder -> Matrix Double -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix Double) setRect :: Int -> Int -> Matrix Double -> Matrix Double -> IO () sortI :: Vector Double -> Vector CInt sortV :: Vector Double -> Vector Double compareV :: Vector Double -> Vector Double -> Vector CInt selectV :: Vector CInt -> Vector Double -> Vector Double -> Vector Double -> Vector Double remapM :: Matrix CInt -> Matrix CInt -> Matrix Double -> Matrix Double rowOp :: Int -> Double -> Int -> Int -> Int -> Int -> Matrix Double -> IO () gemm :: Vector Double -> Matrix Double -> Matrix Double -> Matrix Double -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector Double -> Vector Double
Element Float Source #
Instance details Defined in Internal.Matrix Methods constantD :: Float -> Int -> Vector Float extractR :: MatrixOrder -> Matrix Float -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix Float) setRect :: Int -> Int -> Matrix Float -> Matrix Float -> IO () sortI :: Vector Float -> Vector CInt sortV :: Vector Float -> Vector Float compareV :: Vector Float -> Vector Float -> Vector CInt selectV :: Vector CInt -> Vector Float -> Vector Float -> Vector Float -> Vector Float remapM :: Matrix CInt -> Matrix CInt -> Matrix Float -> Matrix Float rowOp :: Int -> Float -> Int -> Int -> Int -> Int -> Matrix Float -> IO () gemm :: Vector Float -> Matrix Float -> Matrix Float -> Matrix Float -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector Float -> Vector Float
Element CInt Source #
Instance details Defined in Internal.Matrix Methods constantD :: CInt -> Int -> Vector CInt extractR :: MatrixOrder -> Matrix CInt -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix CInt) setRect :: Int -> Int -> Matrix CInt -> Matrix CInt -> IO () sortI :: Vector CInt -> Vector CInt sortV :: Vector CInt -> Vector CInt compareV :: Vector CInt -> Vector CInt -> Vector CInt selectV :: Vector CInt -> Vector CInt -> Vector CInt -> Vector CInt -> Vector CInt remapM :: Matrix CInt -> Matrix CInt -> Matrix CInt -> Matrix CInt rowOp :: Int -> CInt -> Int -> Int -> Int -> Int -> Matrix CInt -> IO () gemm :: Vector CInt -> Matrix CInt -> Matrix CInt -> Matrix CInt -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector CInt -> Vector CInt
Element Z Source #
Instance details Defined in Internal.Matrix Methods constantD :: Z -> Int -> Vector Z extractR :: MatrixOrder -> Matrix Z -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix Z) setRect :: Int -> Int -> Matrix Z -> Matrix Z -> IO () sortI :: Vector Z -> Vector CInt sortV :: Vector Z -> Vector Z compareV :: Vector Z -> Vector Z -> Vector CInt selectV :: Vector CInt -> Vector Z -> Vector Z -> Vector Z -> Vector Z remapM :: Matrix CInt -> Matrix CInt -> Matrix Z -> Matrix Z rowOp :: Int -> Z -> Int -> Int -> Int -> Int -> Matrix Z -> IO () gemm :: Vector Z -> Matrix Z -> Matrix Z -> Matrix Z -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector Z -> Vector Z
Element (Complex Double) Source #
Instance details Defined in Internal.Matrix Methods constantD :: Complex Double -> Int -> Vector (Complex Double) extractR :: MatrixOrder -> Matrix (Complex Double) -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix (Complex Double)) setRect :: Int -> Int -> Matrix (Complex Double) -> Matrix (Complex Double) -> IO () sortI :: Vector (Complex Double) -> Vector CInt sortV :: Vector (Complex Double) -> Vector (Complex Double) compareV :: Vector (Complex Double) -> Vector (Complex Double) -> Vector CInt selectV :: Vector CInt -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) remapM :: Matrix CInt -> Matrix CInt -> Matrix (Complex Double) -> Matrix (Complex Double) rowOp :: Int -> Complex Double -> Int -> Int -> Int -> Int -> Matrix (Complex Double) -> IO () gemm :: Vector (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector (Complex Double) -> Vector (Complex Double)
Element (Complex Float) Source #
Instance details Defined in Internal.Matrix Methods constantD :: Complex Float -> Int -> Vector (Complex Float) extractR :: MatrixOrder -> Matrix (Complex Float) -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix (Complex Float)) setRect :: Int -> Int -> Matrix (Complex Float) -> Matrix (Complex Float) -> IO () sortI :: Vector (Complex Float) -> Vector CInt sortV :: Vector (Complex Float) -> Vector (Complex Float) compareV :: Vector (Complex Float) -> Vector (Complex Float) -> Vector CInt selectV :: Vector CInt -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) remapM :: Matrix CInt -> Matrix CInt -> Matrix (Complex Float) -> Matrix (Complex Float) rowOp :: Int -> Complex Float -> Int -> Int -> Int -> Int -> Matrix (Complex Float) -> IO () gemm :: Vector (Complex Float) -> Matrix (Complex Float) -> Matrix (Complex Float) -> Matrix (Complex Float) -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector (Complex Float) -> Vector (Complex Float)
KnownNat m => Element (Mod m Z) Source #
Instance details Defined in Internal.Modular Methods constantD :: Mod m Z -> Int -> Vector (Mod m Z) extractR :: MatrixOrder -> Matrix (Mod m Z) -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix (Mod m Z)) setRect :: Int -> Int -> Matrix (Mod m Z) -> Matrix (Mod m Z) -> IO () sortI :: Vector (Mod m Z) -> Vector CInt sortV :: Vector (Mod m Z) -> Vector (Mod m Z) compareV :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector CInt selectV :: Vector CInt -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) remapM :: Matrix CInt -> Matrix CInt -> Matrix (Mod m Z) -> Matrix (Mod m Z) rowOp :: Int -> Mod m Z -> Int -> Int -> Int -> Int -> Matrix (Mod m Z) -> IO () gemm :: Vector (Mod m Z) -> Matrix (Mod m Z) -> Matrix (Mod m Z) -> Matrix (Mod m Z) -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector (Mod m Z) -> Vector (Mod m Z)
KnownNat m => Element (Mod m I) Source #
Instance details Defined in Internal.Modular Methods constantD :: Mod m I -> Int -> Vector (Mod m I) extractR :: MatrixOrder -> Matrix (Mod m I) -> CInt -> Vector CInt -> CInt -> Vector CInt -> IO (Matrix (Mod m I)) setRect :: Int -> Int -> Matrix (Mod m I) -> Matrix (Mod m I) -> IO () sortI :: Vector (Mod m I) -> Vector CInt sortV :: Vector (Mod m I) -> Vector (Mod m I) compareV :: Vector (Mod m I) -> Vector (Mod m I) -> Vector CInt selectV :: Vector CInt -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) remapM :: Matrix CInt -> Matrix CInt -> Matrix (Mod m I) -> Matrix (Mod m I) rowOp :: Int -> Mod m I -> Int -> Int -> Int -> Int -> Matrix (Mod m I) -> IO () gemm :: Vector (Mod m I) -> Matrix (Mod m I) -> Matrix (Mod m I) -> Matrix (Mod m I) -> IO () reorderV :: Vector CInt -> Vector CInt -> Vector (Mod m I) -> Vector (Mod m I)

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 #
Instance details Defined in Internal.Numeric Methods conj' :: Vector Double -> Vector Double size' :: Vector Double -> IndexOf Vector scalar' :: Double -> Vector Double scale' :: Double -> Vector Double -> Vector Double addConstant :: Double -> Vector Double -> Vector Double add' :: Vector Double -> Vector Double -> Vector Double sub :: Vector Double -> Vector Double -> Vector Double mul :: Vector Double -> Vector Double -> Vector Double equal :: Vector Double -> Vector Double -> Bool cmap' :: Element b => (Double -> b) -> Vector Double -> Vector b konst' :: Double -> IndexOf Vector -> Vector Double build' :: IndexOf Vector -> ArgOf Vector Double -> Vector Double atIndex' :: Vector Double -> IndexOf Vector -> Double minIndex' :: Vector Double -> IndexOf Vector maxIndex' :: Vector Double -> IndexOf Vector minElement' :: Vector Double -> Double maxElement' :: Vector Double -> Double sumElements' :: Vector Double -> Double prodElements' :: Vector Double -> Double step' :: Vector Double -> Vector Double ccompare' :: Vector Double -> Vector Double -> Vector I cselect' :: Vector I -> Vector Double -> Vector Double -> Vector Double -> Vector Double find' :: (Double -> Bool) -> Vector Double -> [IndexOf Vector] assoc' :: IndexOf Vector -> Double -> [(IndexOf Vector, Double)] -> Vector Double accum' :: Vector Double -> (Double -> Double -> Double) -> [(IndexOf Vector, Double)] -> Vector Double scaleRecip :: Double -> Vector Double -> Vector Double divide :: Vector Double -> Vector Double -> Vector Double arctan2' :: Vector Double -> Vector Double -> Vector Double cmod' :: Double -> Vector Double -> Vector Double fromInt' :: Vector I -> Vector Double toInt' :: Vector Double -> Vector I fromZ' :: Vector Z -> Vector Double toZ' :: Vector Double -> Vector Z
Container Vector Float Source #
Instance details Defined in Internal.Numeric Methods conj' :: Vector Float -> Vector Float size' :: Vector Float -> IndexOf Vector scalar' :: Float -> Vector Float scale' :: Float -> Vector Float -> Vector Float addConstant :: Float -> Vector Float -> Vector Float add' :: Vector Float -> Vector Float -> Vector Float sub :: Vector Float -> Vector Float -> Vector Float mul :: Vector Float -> Vector Float -> Vector Float equal :: Vector Float -> Vector Float -> Bool cmap' :: Element b => (Float -> b) -> Vector Float -> Vector b konst' :: Float -> IndexOf Vector -> Vector Float build' :: IndexOf Vector -> ArgOf Vector Float -> Vector Float atIndex' :: Vector Float -> IndexOf Vector -> Float minIndex' :: Vector Float -> IndexOf Vector maxIndex' :: Vector Float -> IndexOf Vector minElement' :: Vector Float -> Float maxElement' :: Vector Float -> Float sumElements' :: Vector Float -> Float prodElements' :: Vector Float -> Float step' :: Vector Float -> Vector Float ccompare' :: Vector Float -> Vector Float -> Vector I cselect' :: Vector I -> Vector Float -> Vector Float -> Vector Float -> Vector Float find' :: (Float -> Bool) -> Vector Float -> [IndexOf Vector] assoc' :: IndexOf Vector -> Float -> [(IndexOf Vector, Float)] -> Vector Float accum' :: Vector Float -> (Float -> Float -> Float) -> [(IndexOf Vector, Float)] -> Vector Float scaleRecip :: Float -> Vector Float -> Vector Float divide :: Vector Float -> Vector Float -> Vector Float arctan2' :: Vector Float -> Vector Float -> Vector Float cmod' :: Float -> Vector Float -> Vector Float fromInt' :: Vector I -> Vector Float toInt' :: Vector Float -> Vector I fromZ' :: Vector Z -> Vector Float toZ' :: Vector Float -> Vector Z
Container Vector Z Source #
Instance details Defined in Internal.Numeric Methods conj' :: Vector Z -> Vector Z size' :: Vector Z -> IndexOf Vector scalar' :: Z -> Vector Z scale' :: Z -> Vector Z -> Vector Z addConstant :: Z -> Vector Z -> Vector Z add' :: Vector Z -> Vector Z -> Vector Z sub :: Vector Z -> Vector Z -> Vector Z mul :: Vector Z -> Vector Z -> Vector Z equal :: Vector Z -> Vector Z -> Bool cmap' :: Element b => (Z -> b) -> Vector Z -> Vector b konst' :: Z -> IndexOf Vector -> Vector Z build' :: IndexOf Vector -> ArgOf Vector Z -> Vector Z atIndex' :: Vector Z -> IndexOf Vector -> Z minIndex' :: Vector Z -> IndexOf Vector maxIndex' :: Vector Z -> IndexOf Vector minElement' :: Vector Z -> Z maxElement' :: Vector Z -> Z sumElements' :: Vector Z -> Z prodElements' :: Vector Z -> Z step' :: Vector Z -> Vector Z ccompare' :: Vector Z -> Vector Z -> Vector I cselect' :: Vector I -> Vector Z -> Vector Z -> Vector Z -> Vector Z find' :: (Z -> Bool) -> Vector Z -> [IndexOf Vector] assoc' :: IndexOf Vector -> Z -> [(IndexOf Vector, Z)] -> Vector Z accum' :: Vector Z -> (Z -> Z -> Z) -> [(IndexOf Vector, Z)] -> Vector Z scaleRecip :: Z -> Vector Z -> Vector Z divide :: Vector Z -> Vector Z -> Vector Z arctan2' :: Vector Z -> Vector Z -> Vector Z cmod' :: Z -> Vector Z -> Vector Z fromInt' :: Vector I -> Vector Z toInt' :: Vector Z -> Vector I fromZ' :: Vector Z -> Vector Z toZ' :: Vector Z -> Vector Z
Container Vector I Source #
Instance details Defined in Internal.Numeric Methods conj' :: Vector I -> Vector I size' :: Vector I -> IndexOf Vector scalar' :: I -> Vector I scale' :: I -> Vector I -> Vector I addConstant :: I -> Vector I -> Vector I add' :: Vector I -> Vector I -> Vector I sub :: Vector I -> Vector I -> Vector I mul :: Vector I -> Vector I -> Vector I equal :: Vector I -> Vector I -> Bool cmap' :: Element b => (I -> b) -> Vector I -> Vector b konst' :: I -> IndexOf Vector -> Vector I build' :: IndexOf Vector -> ArgOf Vector I -> Vector I atIndex' :: Vector I -> IndexOf Vector -> I minIndex' :: Vector I -> IndexOf Vector maxIndex' :: Vector I -> IndexOf Vector minElement' :: Vector I -> I maxElement' :: Vector I -> I sumElements' :: Vector I -> I prodElements' :: Vector I -> I step' :: Vector I -> Vector I ccompare' :: Vector I -> Vector I -> Vector I cselect' :: Vector I -> Vector I -> Vector I -> Vector I -> Vector I find' :: (I -> Bool) -> Vector I -> [IndexOf Vector] assoc' :: IndexOf Vector -> I -> [(IndexOf Vector, I)] -> Vector I accum' :: Vector I -> (I -> I -> I) -> [(IndexOf Vector, I)] -> Vector I scaleRecip :: I -> Vector I -> Vector I divide :: Vector I -> Vector I -> Vector I arctan2' :: Vector I -> Vector I -> Vector I cmod' :: I -> Vector I -> Vector I fromInt' :: Vector I -> Vector I toInt' :: Vector I -> Vector I fromZ' :: Vector Z -> Vector I toZ' :: Vector I -> Vector Z
(Num a, Element a, Container Vector a) => Container Matrix a Source #
Instance details Defined in Internal.Numeric Methods conj' :: Matrix a -> Matrix a size' :: Matrix a -> IndexOf Matrix scalar' :: a -> Matrix a scale' :: a -> Matrix a -> Matrix a addConstant :: a -> Matrix a -> Matrix a add' :: Matrix a -> Matrix a -> Matrix a sub :: Matrix a -> Matrix a -> Matrix a mul :: Matrix a -> Matrix a -> Matrix a equal :: Matrix a -> Matrix a -> Bool cmap' :: Element b => (a -> b) -> Matrix a -> Matrix b konst' :: a -> IndexOf Matrix -> Matrix a build' :: IndexOf Matrix -> ArgOf Matrix a -> Matrix a atIndex' :: Matrix a -> IndexOf Matrix -> a minIndex' :: Matrix a -> IndexOf Matrix maxIndex' :: Matrix a -> IndexOf Matrix minElement' :: Matrix a -> a maxElement' :: Matrix a -> a sumElements' :: Matrix a -> a prodElements' :: Matrix a -> a step' :: Matrix a -> Matrix a ccompare' :: Matrix a -> Matrix a -> Matrix I cselect' :: Matrix I -> Matrix a -> Matrix a -> Matrix a -> Matrix a find' :: (a -> Bool) -> Matrix a -> [IndexOf Matrix] assoc' :: IndexOf Matrix -> a -> [(IndexOf Matrix, a)] -> Matrix a accum' :: Matrix a -> (a -> a -> a) -> [(IndexOf Matrix, a)] -> Matrix a scaleRecip :: a -> Matrix a -> Matrix a divide :: Matrix a -> Matrix a -> Matrix a arctan2' :: Matrix a -> Matrix a -> Matrix a cmod' :: a -> Matrix a -> Matrix a fromInt' :: Matrix I -> Matrix a toInt' :: Matrix a -> Matrix I fromZ' :: Matrix Z -> Matrix a toZ' :: Matrix a -> Matrix Z
Container Vector (Complex Double) Source #
Instance details Defined in Internal.Numeric Methods conj' :: Vector (Complex Double) -> Vector (Complex Double) size' :: Vector (Complex Double) -> IndexOf Vector scalar' :: Complex Double -> Vector (Complex Double) scale' :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) addConstant :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) add' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) sub :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) mul :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) equal :: Vector (Complex Double) -> Vector (Complex Double) -> Bool cmap' :: Element b => (Complex Double -> b) -> Vector (Complex Double) -> Vector b konst' :: Complex Double -> IndexOf Vector -> Vector (Complex Double) build' :: IndexOf Vector -> ArgOf Vector (Complex Double) -> Vector (Complex Double) atIndex' :: Vector (Complex Double) -> IndexOf Vector -> Complex Double minIndex' :: Vector (Complex Double) -> IndexOf Vector maxIndex' :: Vector (Complex Double) -> IndexOf Vector minElement' :: Vector (Complex Double) -> Complex Double maxElement' :: Vector (Complex Double) -> Complex Double sumElements' :: Vector (Complex Double) -> Complex Double prodElements' :: Vector (Complex Double) -> Complex Double step' :: Vector (Complex Double) -> Vector (Complex Double) ccompare' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector I cselect' :: Vector I -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) find' :: (Complex Double -> Bool) -> Vector (Complex Double) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Complex Double -> [(IndexOf Vector, Complex Double)] -> Vector (Complex Double) accum' :: Vector (Complex Double) -> (Complex Double -> Complex Double -> Complex Double) -> [(IndexOf Vector, Complex Double)] -> Vector (Complex Double) scaleRecip :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) divide :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) arctan2' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) cmod' :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) fromInt' :: Vector I -> Vector (Complex Double) toInt' :: Vector (Complex Double) -> Vector I fromZ' :: Vector Z -> Vector (Complex Double) toZ' :: Vector (Complex Double) -> Vector Z
Container Vector (Complex Float) Source #
Instance details Defined in Internal.Numeric Methods conj' :: Vector (Complex Float) -> Vector (Complex Float) size' :: Vector (Complex Float) -> IndexOf Vector scalar' :: Complex Float -> Vector (Complex Float) scale' :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) addConstant :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) add' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) sub :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) mul :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) equal :: Vector (Complex Float) -> Vector (Complex Float) -> Bool cmap' :: Element b => (Complex Float -> b) -> Vector (Complex Float) -> Vector b konst' :: Complex Float -> IndexOf Vector -> Vector (Complex Float) build' :: IndexOf Vector -> ArgOf Vector (Complex Float) -> Vector (Complex Float) atIndex' :: Vector (Complex Float) -> IndexOf Vector -> Complex Float minIndex' :: Vector (Complex Float) -> IndexOf Vector maxIndex' :: Vector (Complex Float) -> IndexOf Vector minElement' :: Vector (Complex Float) -> Complex Float maxElement' :: Vector (Complex Float) -> Complex Float sumElements' :: Vector (Complex Float) -> Complex Float prodElements' :: Vector (Complex Float) -> Complex Float step' :: Vector (Complex Float) -> Vector (Complex Float) ccompare' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector I cselect' :: Vector I -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) find' :: (Complex Float -> Bool) -> Vector (Complex Float) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Complex Float -> [(IndexOf Vector, Complex Float)] -> Vector (Complex Float) accum' :: Vector (Complex Float) -> (Complex Float -> Complex Float -> Complex Float) -> [(IndexOf Vector, Complex Float)] -> Vector (Complex Float) scaleRecip :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) divide :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) arctan2' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) cmod' :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) fromInt' :: Vector I -> Vector (Complex Float) toInt' :: Vector (Complex Float) -> Vector I fromZ' :: Vector Z -> Vector (Complex Float) toZ' :: Vector (Complex Float) -> Vector Z
KnownNat m => Container Vector (Mod m Z) Source #
Instance details Defined in Internal.Modular Methods conj' :: Vector (Mod m Z) -> Vector (Mod m Z) size' :: Vector (Mod m Z) -> IndexOf Vector scalar' :: Mod m Z -> Vector (Mod m Z) scale' :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) addConstant :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) add' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) sub :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) mul :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) equal :: Vector (Mod m Z) -> Vector (Mod m Z) -> Bool cmap' :: Element b => (Mod m Z -> b) -> Vector (Mod m Z) -> Vector b konst' :: Mod m Z -> IndexOf Vector -> Vector (Mod m Z) build' :: IndexOf Vector -> ArgOf Vector (Mod m Z) -> Vector (Mod m Z) atIndex' :: Vector (Mod m Z) -> IndexOf Vector -> Mod m Z minIndex' :: Vector (Mod m Z) -> IndexOf Vector maxIndex' :: Vector (Mod m Z) -> IndexOf Vector minElement' :: Vector (Mod m Z) -> Mod m Z maxElement' :: Vector (Mod m Z) -> Mod m Z sumElements' :: Vector (Mod m Z) -> Mod m Z prodElements' :: Vector (Mod m Z) -> Mod m Z step' :: Vector (Mod m Z) -> Vector (Mod m Z) ccompare' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector I cselect' :: Vector I -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) find' :: (Mod m Z -> Bool) -> Vector (Mod m Z) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Mod m Z -> [(IndexOf Vector, Mod m Z)] -> Vector (Mod m Z) accum' :: Vector (Mod m Z) -> (Mod m Z -> Mod m Z -> Mod m Z) -> [(IndexOf Vector, Mod m Z)] -> Vector (Mod m Z) scaleRecip :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) divide :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) arctan2' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) cmod' :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) fromInt' :: Vector I -> Vector (Mod m Z) toInt' :: Vector (Mod m Z) -> Vector I fromZ' :: Vector Z -> Vector (Mod m Z) toZ' :: Vector (Mod m Z) -> Vector Z
KnownNat m => Container Vector (Mod m I) Source #
Instance details Defined in Internal.Modular Methods conj' :: Vector (Mod m I) -> Vector (Mod m I) size' :: Vector (Mod m I) -> IndexOf Vector scalar' :: Mod m I -> Vector (Mod m I) scale' :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) addConstant :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) add' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) sub :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) mul :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) equal :: Vector (Mod m I) -> Vector (Mod m I) -> Bool cmap' :: Element b => (Mod m I -> b) -> Vector (Mod m I) -> Vector b konst' :: Mod m I -> IndexOf Vector -> Vector (Mod m I) build' :: IndexOf Vector -> ArgOf Vector (Mod m I) -> Vector (Mod m I) atIndex' :: Vector (Mod m I) -> IndexOf Vector -> Mod m I minIndex' :: Vector (Mod m I) -> IndexOf Vector maxIndex' :: Vector (Mod m I) -> IndexOf Vector minElement' :: Vector (Mod m I) -> Mod m I maxElement' :: Vector (Mod m I) -> Mod m I sumElements' :: Vector (Mod m I) -> Mod m I prodElements' :: Vector (Mod m I) -> Mod m I step' :: Vector (Mod m I) -> Vector (Mod m I) ccompare' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector I cselect' :: Vector I -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) find' :: (Mod m I -> Bool) -> Vector (Mod m I) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Mod m I -> [(IndexOf Vector, Mod m I)] -> Vector (Mod m I) accum' :: Vector (Mod m I) -> (Mod m I -> Mod m I -> Mod m I) -> [(IndexOf Vector, Mod m I)] -> Vector (Mod m I) scaleRecip :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) divide :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) arctan2' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) cmod' :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) fromInt' :: Vector I -> Vector (Mod m I) toInt' :: Vector (Mod m I) -> Vector I fromZ' :: Vector Z -> Vector (Mod m I) toZ' :: Vector (Mod m I) -> Vector Z

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 #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix Double -> Matrix Double -> Matrix Double absSum :: Vector Double -> RealOf Double norm1 :: Vector Double -> RealOf Double norm2 :: Vector Double -> RealOf Double normInf :: Vector Double -> RealOf Double
Product Float Source #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix Float -> Matrix Float -> Matrix Float absSum :: Vector Float -> RealOf Float norm1 :: Vector Float -> RealOf Float norm2 :: Vector Float -> RealOf Float normInf :: Vector Float -> RealOf Float
Product Z Source #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix Z -> Matrix Z -> Matrix Z absSum :: Vector Z -> RealOf Z norm1 :: Vector Z -> RealOf Z norm2 :: Vector Z -> RealOf Z normInf :: Vector Z -> RealOf Z
Product I Source #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix I -> Matrix I -> Matrix I absSum :: Vector I -> RealOf I norm1 :: Vector I -> RealOf I norm2 :: Vector I -> RealOf I normInf :: Vector I -> RealOf I
Product (Complex Double) Source #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) absSum :: Vector (Complex Double) -> RealOf (Complex Double) norm1 :: Vector (Complex Double) -> RealOf (Complex Double) norm2 :: Vector (Complex Double) -> RealOf (Complex Double) normInf :: Vector (Complex Double) -> RealOf (Complex Double)
Product (Complex Float) Source #
Instance details Defined in Internal.Numeric Methods multiply :: Matrix (Complex Float) -> Matrix (Complex Float) -> Matrix (Complex Float) absSum :: Vector (Complex Float) -> RealOf (Complex Float) norm1 :: Vector (Complex Float) -> RealOf (Complex Float) norm2 :: Vector (Complex Float) -> RealOf (Complex Float) normInf :: Vector (Complex Float) -> RealOf (Complex Float)
KnownNat m => Product (Mod m Z) Source #
Instance details Defined in Internal.Modular Methods multiply :: Matrix (Mod m Z) -> Matrix (Mod m Z) -> Matrix (Mod m Z) absSum :: Vector (Mod m Z) -> RealOf (Mod m Z) norm1 :: Vector (Mod m Z) -> RealOf (Mod m Z) norm2 :: Vector (Mod m Z) -> RealOf (Mod m Z) normInf :: Vector (Mod m Z) -> RealOf (Mod m Z)
KnownNat m => Product (Mod m I) Source #
Instance details Defined in Internal.Modular Methods multiply :: Matrix (Mod m I) -> Matrix (Mod m I) -> Matrix (Mod m I) absSum :: Vector (Mod m I) -> RealOf (Mod m I) norm1 :: Vector (Mod m I) -> RealOf (Mod m I) norm2 :: Vector (Mod m I) -> RealOf (Mod m I) normInf :: Vector (Mod m I) -> RealOf (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 Source #

Instances

Numeric Double Source #
Instance details Defined in Internal.Numeric
Numeric Float Source #
Instance details Defined in Internal.Numeric
Numeric Z Source #
Instance details Defined in Internal.Numeric
Numeric I Source #
Instance details Defined in Internal.Numeric
Numeric (Complex Double) Source #
Instance details Defined in Internal.Numeric
Numeric (Complex Float) Source #
Instance details Defined in Internal.Numeric
KnownNat m => Numeric (Mod m Z) Source #
Instance details Defined in Internal.Modular
KnownNat m => Numeric (Mod m I) Source #
Instance details Defined in Internal.Modular

class LSDiv c Source #

Minimal complete definition

linSolve

Instances

LSDiv Vector Source #
Instance details Defined in Internal.Container Methods linSolve :: Field t => Matrix t -> Vector t -> Vector t
LSDiv Matrix Source #
Instance details Defined in Internal.Container Methods linSolve :: Field t => Matrix t -> Matrix t -> Matrix t

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 #
Instance details Defined in Internal.Algorithms Methods scale :: t -> Herm t -> Herm t Source #
(Show t, Element t) => Show (Herm t) Source #
Instance details Defined in Internal.Algorithms Methods showsPrec :: Int -> Herm t -> ShowS # show :: Herm t -> String # showList :: [Herm t] -> ShowS #
(NFData t, Numeric t) => NFData (Herm t) Source #
Instance details Defined in Internal.Algorithms Methods rnf :: Herm t -> () #
Field t => Additive (Herm t) Source #
Instance details Defined in Internal.Algorithms Methods add :: Herm t -> Herm t -> 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 #
Instance details Defined in Internal.Conversion Methods toComplex' :: RealElement e => (Vector e, Vector e) -> Vector (Complex e) fromComplex' :: RealElement e => Vector (Complex e) -> (Vector e, Vector e) comp' :: RealElement e => Vector e -> Vector (Complex e) single' :: Precision a b => Vector b -> Vector a double' :: Precision a b => Vector a -> Vector b
Complexable Matrix Source #
Instance details Defined in Internal.Conversion Methods toComplex' :: RealElement e => (Matrix e, Matrix e) -> Matrix (Complex e) fromComplex' :: RealElement e => Matrix (Complex e) -> (Matrix e, Matrix e) comp' :: RealElement e => Matrix e -> Matrix (Complex e) single' :: Precision a b => Matrix b -> Matrix a double' :: Precision a b => Matrix a -> Matrix b

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

Supported real types

Instances

RealElement Double Source #
Instance details Defined in Internal.Conversion
RealElement Float Source #
Instance details Defined in Internal.Conversion

type family RealOf x Source #

Instances

type RealOf Double Source #
Instance details Defined in Internal.Numeric type RealOf Double = Double
type RealOf Float Source #
Instance details Defined in Internal.Numeric type RealOf Float = Float
type RealOf Z Source #
Instance details Defined in Internal.Numeric type RealOf Z = Z
type RealOf I Source #
Instance details Defined in Internal.Numeric type RealOf I = I
type RealOf (Complex Double) Source #
Instance details Defined in Internal.Numeric type RealOf (Complex Double) = Double
type RealOf (Complex Float) Source #
Instance details Defined in Internal.Numeric type RealOf (Complex Float) = Float
type RealOf (Mod n Z) Source #
Instance details Defined in Internal.Modular type RealOf (Mod n Z) = Z
type RealOf (Mod n I) Source #
Instance details Defined in Internal.Modular type RealOf (Mod n I) = I

type ComplexOf x = Complex (RealOf x) Source #

type family SingleOf x Source #

Instances

type SingleOf Double Source #
Instance details Defined in Internal.Numeric type SingleOf Double = Float
type SingleOf Float Source #
Instance details Defined in Internal.Numeric type SingleOf Float = Float
type SingleOf (Complex a) Source #
Instance details Defined in Internal.Numeric type SingleOf (Complex a) = Complex (SingleOf a)

type family DoubleOf x Source #

Instances

type DoubleOf Double Source #
Instance details Defined in Internal.Numeric type DoubleOf Double = Double
type DoubleOf Float Source #
Instance details Defined in Internal.Numeric type DoubleOf Float = Double
type DoubleOf (Complex a) Source #
Instance details Defined in Internal.Numeric type DoubleOf (Complex a) = Complex (DoubleOf a)

type family IndexOf (c :: * -> *) Source #

Instances

type IndexOf Vector Source #
Instance details Defined in Internal.Numeric type IndexOf Vector = Int
type IndexOf Matrix Source #
Instance details Defined in Internal.Numeric 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 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', triSolve', triDiagSolve', ldlPacked', ldlSolve', linearSolveSVD', linearSolveLS', eig', geig', eigSH'', eigOnly, geigOnly, eigOnlySH, cholSH', mbCholSH', qr', qrgr', hess', schur'

Instances

Field Double Source #
Instance details Defined in Internal.Algorithms Methods svd' :: Matrix Double -> (Matrix Double, Vector Double, Matrix Double) thinSVD' :: Matrix Double -> (Matrix Double, Vector Double, Matrix Double) sv' :: Matrix Double -> Vector Double luPacked' :: Matrix Double -> (Matrix Double, [Int]) luSolve' :: (Matrix Double, [Int]) -> Matrix Double -> Matrix Double mbLinearSolve' :: Matrix Double -> Matrix Double -> Maybe (Matrix Double) linearSolve' :: Matrix Double -> Matrix Double -> Matrix Double cholSolve' :: Matrix Double -> Matrix Double -> Matrix Double triSolve' :: UpLo -> Matrix Double -> Matrix Double -> Matrix Double triDiagSolve' :: Vector Double -> Vector Double -> Vector Double -> Matrix Double -> Matrix Double ldlPacked' :: Matrix Double -> (Matrix Double, [Int]) ldlSolve' :: (Matrix Double, [Int]) -> Matrix Double -> Matrix Double linearSolveSVD' :: Matrix Double -> Matrix Double -> Matrix Double linearSolveLS' :: Matrix Double -> Matrix Double -> Matrix Double eig' :: Matrix Double -> (Vector (Complex Double), Matrix (Complex Double)) geig' :: Matrix Double -> Matrix Double -> (Vector (Complex Double), Vector Double, Matrix (Complex Double)) eigSH'' :: Matrix Double -> (Vector Double, Matrix Double) eigOnly :: Matrix Double -> Vector (Complex Double) geigOnly :: Matrix Double -> Matrix Double -> (Vector (Complex Double), Vector Double) eigOnlySH :: Matrix Double -> Vector Double cholSH' :: Matrix Double -> Matrix Double mbCholSH' :: Matrix Double -> Maybe (Matrix Double) qr' :: Matrix Double -> (Matrix Double, Vector Double) qrgr' :: Int -> (Matrix Double, Vector Double) -> Matrix Double hess' :: Matrix Double -> (Matrix Double, Matrix Double) schur' :: Matrix Double -> (Matrix Double, Matrix Double)
Field (Complex Double) Source #
Instance details Defined in Internal.Algorithms Methods svd' :: Matrix (Complex Double) -> (Matrix (Complex Double), Vector Double, Matrix (Complex Double)) thinSVD' :: Matrix (Complex Double) -> (Matrix (Complex Double), Vector Double, Matrix (Complex Double)) sv' :: Matrix (Complex Double) -> Vector Double luPacked' :: Matrix (Complex Double) -> (Matrix (Complex Double), [Int]) luSolve' :: (Matrix (Complex Double), [Int]) -> Matrix (Complex Double) -> Matrix (Complex Double) mbLinearSolve' :: Matrix (Complex Double) -> Matrix (Complex Double) -> Maybe (Matrix (Complex Double)) linearSolve' :: Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) cholSolve' :: Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) triSolve' :: UpLo -> Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) triDiagSolve' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) ldlPacked' :: Matrix (Complex Double) -> (Matrix (Complex Double), [Int]) ldlSolve' :: (Matrix (Complex Double), [Int]) -> Matrix (Complex Double) -> Matrix (Complex Double) linearSolveSVD' :: Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) linearSolveLS' :: Matrix (Complex Double) -> Matrix (Complex Double) -> Matrix (Complex Double) eig' :: Matrix (Complex Double) -> (Vector (Complex Double), Matrix (Complex Double)) geig' :: Matrix (Complex Double) -> Matrix (Complex Double) -> (Vector (Complex Double), Vector (Complex Double), Matrix (Complex Double)) eigSH'' :: Matrix (Complex Double) -> (Vector Double, Matrix (Complex Double)) eigOnly :: Matrix (Complex Double) -> Vector (Complex Double) geigOnly :: Matrix (Complex Double) -> Matrix (Complex Double) -> (Vector (Complex Double), Vector (Complex Double)) eigOnlySH :: Matrix (Complex Double) -> Vector Double cholSH' :: Matrix (Complex Double) -> Matrix (Complex Double) mbCholSH' :: Matrix (Complex Double) -> Maybe (Matrix (Complex Double)) qr' :: Matrix (Complex Double) -> (Matrix (Complex Double), Vector (Complex Double)) qrgr' :: Int -> (Matrix (Complex Double), Vector (Complex Double)) -> Matrix (Complex Double) hess' :: Matrix (Complex Double) -> (Matrix (Complex Double), Matrix (Complex Double)) schur' :: Matrix (Complex Double) -> (Matrix (Complex Double), Matrix (Complex Double))

class Linear t c Source #

Minimal complete definition

scale

Instances

Container Matrix t => Linear t Matrix Source #
Instance details Defined in Internal.Numeric Methods scale :: t -> Matrix t -> Matrix t Source #
Container Vector t => Linear t Vector Source #
Instance details Defined in Internal.Numeric Methods scale :: t -> Vector t -> Vector t Source #
Field t => Linear t Herm Source #
Instance details Defined in Internal.Algorithms Methods scale :: t -> Herm t -> Herm t Source #

class Additive c Source #

Minimal complete definition

add

Instances

Container Vector t => Additive (Vector t) Source #
Instance details Defined in Internal.Numeric Methods add :: Vector t -> Vector t -> Vector t Source #
Container Matrix t => Additive (Matrix t) Source #
Instance details Defined in Internal.Numeric Methods add :: Matrix t -> Matrix t -> Matrix t Source #
Field t => Additive (Herm t) Source #
Instance details Defined in Internal.Algorithms Methods add :: Herm t -> Herm t -> Herm t Source #
Additive (C n) Source #
Instance details Defined in Internal.Static Methods add :: C n -> C n -> C n Source #
Additive (R n) Source #
Instance details Defined in Internal.Static Methods add :: R n -> R n -> R n Source #
KnownNat n => Additive (Sym n) Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods add :: Sym n -> Sym n -> Sym n Source #
(KnownNat m, KnownNat n) => Additive (M m n) Source #
Instance details Defined in Internal.Static Methods add :: M m n -> M m n -> M m n Source #
(KnownNat m, KnownNat n) => Additive (L m n) Source #
Instance details Defined in Internal.Static Methods add :: L m n -> L m n -> L m n Source #

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

Minimal complete definition

tr, tr'

Instances

Transposable GMatrix GMatrix Source #
Instance details Defined in Internal.Sparse Methods tr :: GMatrix -> GMatrix Source # tr' :: GMatrix -> GMatrix Source #
(CTrans t, Container Vector t) => Transposable (Matrix t) (Matrix t) Source #
Instance details Defined in Internal.Numeric Methods tr :: Matrix t -> Matrix t Source # tr' :: Matrix t -> Matrix t Source #
KnownNat n => Transposable (Her n) (Her n) Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods tr :: Her n -> Her n Source # tr' :: Her n -> Her n Source #
KnownNat n => Transposable (Sym n) (Sym n) Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods tr :: Sym n -> Sym n Source # tr' :: Sym n -> Sym n Source #
(KnownNat n, KnownNat m) => Transposable (M m n) (M n m) Source #
Instance details Defined in Internal.Static Methods tr :: M m n -> M n m Source # tr' :: M m n -> M n m Source #
(KnownNat n, KnownNat m) => Transposable (L m n) (L n m) Source #
Instance details Defined in Internal.Static Methods tr :: L m n -> L n m Source # tr' :: 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 #
Instance details Defined in Internal.Algorithms Methods showsPrec :: Int -> LU t -> ShowS # show :: LU t -> String # showList :: [LU t] -> ShowS #
(NFData t, Numeric t) => NFData (LU t) Source #
Instance details Defined in Internal.Algorithms Methods rnf :: LU t -> () #

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 #
Instance details Defined in Internal.Algorithms Methods showsPrec :: Int -> LDL t -> ShowS # show :: LDL t -> String # showList :: [LDL t] -> ShowS #
(NFData t, Numeric t) => NFData (LDL t) Source #
Instance details Defined in Internal.Algorithms Methods rnf :: LDL t -> () #

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 #
Instance details Defined in Internal.Algorithms Methods rnf :: QR t -> () #

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

Testable GMatrix Source #
Instance details Defined in Internal.CG Methods checkT :: GMatrix -> (Bool, IO ()) Source # ioCheckT :: GMatrix -> IO (Bool, IO ()) Source #
Testable (Matrix I) Source #
Instance details Defined in Internal.Util Methods checkT :: Matrix I -> (Bool, IO ()) Source # ioCheckT :: Matrix I -> IO (Bool, IO ()) Source #
KnownNat m => Testable (Matrix (Mod m I)) Source #
Instance details Defined in Internal.Modular Methods checkT :: Matrix (Mod m I) -> (Bool, IO ()) Source # ioCheckT :: Matrix (Mod m I) -> IO (Bool, IO ()) Source #
(KnownNat n', KnownNat m') => Testable (L n' m') Source #
Instance details Defined in Numeric.LinearAlgebra.Static Methods checkT :: L n' m' -> (Bool, IO ()) Source # ioCheckT :: L n' m' -> IO (Bool, IO ()) Source #