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|
-----------------------------------------------------------------------------
{- |
Module : Numeric.LinearAlgebra
Copyright : (c) Alberto Ruiz 2006-15
License : BSD3
Maintainer : Alberto Ruiz
Stability : provisional
-}
-----------------------------------------------------------------------------
module Numeric.LinearAlgebra (
-- * Basic types and data processing
module Numeric.LinearAlgebra.Data,
-- * Arithmetic and numeric classes
-- |
-- The standard numeric classes are defined elementwise:
--
-- >>> vector [1,2,3] * vector [3,0,-2]
-- fromList [3.0,0.0,-6.0]
--
-- >>> matrix 3 [1..9] * ident 3
-- (3><3)
-- [ 1.0, 0.0, 0.0
-- , 0.0, 5.0, 0.0
-- , 0.0, 0.0, 9.0 ]
--
-- In arithmetic operations single-element vectors and matrices
-- (created from numeric literals or using 'scalar') automatically
-- expand to match the dimensions of the other operand:
--
-- >>> 5 + 2*ident 3 :: Matrix Double
-- (3><3)
-- [ 7.0, 5.0, 5.0
-- , 5.0, 7.0, 5.0
-- , 5.0, 5.0, 7.0 ]
--
-- >>> matrix 3 [1..9] + matrix 1 [10,20,30]
-- (3><3)
-- [ 11.0, 12.0, 13.0
-- , 24.0, 25.0, 26.0
-- , 37.0, 38.0, 39.0 ]
--
-- * Products
-- ** dot
dot, (<·>),
-- ** matrix-vector
app, (#>), (<#), (!#>),
-- ** matrix-matrix
mul, (<>),
-- | 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, kronecker, cross,
scale,
sumElements, prodElements,
-- * Linear Systems
(<\>),
linearSolve,
linearSolveLS,
linearSolveSVD,
luSolve,
cholSolve,
cgSolve,
cgSolve',
-- * Inverse and pseudoinverse
inv, pinv, pinvTol,
-- * Determinant and rank
rcond, rank,
det, invlndet,
-- * Norms
Normed(..),
norm_Frob, norm_nuclear,
-- * Nullspace and range
orth,
nullspace, null1, null1sym,
-- * SVD
svd,
thinSVD,
compactSVD,
singularValues,
leftSV, rightSV,
-- * Eigensystems
eig, eigSH, eigSH',
eigenvalues, eigenvaluesSH, eigenvaluesSH',
geigSH',
-- * QR
qr, rq, qrRaw, qrgr,
-- * Cholesky
chol, cholSH, mbCholSH,
-- * Hessenberg
hess,
-- * Schur
schur,
-- * LU
lu, luPacked,
-- * Matrix functions
expm,
sqrtm,
matFunc,
-- * Correlation and convolution
corr, conv, corrMin, corr2, conv2,
-- * Random arrays
Seed, RandDist(..), randomVector, rand, randn, gaussianSample, uniformSample,
-- * Misc
meanCov, rowOuters, pairwiseD2, unitary, peps, relativeError, haussholder, optimiseMult, udot, nullspaceSVD, orthSVD, ranksv, gaussElim,
ℝ,ℂ,iC,
-- * Auxiliary classes
Element, Container, Product, Numeric, LSDiv,
Complexable, RealElement,
RealOf, ComplexOf, SingleOf, DoubleOf,
IndexOf,
Mod,
Field,
-- Normed,
Transposable,
CGState(..),
Testable(..)
) where
import Numeric.LinearAlgebra.Data
import Numeric.Matrix()
import Numeric.Vector()
import Internal.Matrix
import Internal.Container hiding ((<>))
import Internal.Numeric hiding (mul)
import Internal.Algorithms hiding (linearSolve,Normed,orth)
import qualified Internal.Algorithms as A
import Internal.Util
import Internal.Modular
import Internal.Random
import Internal.Sparse((!#>))
import Internal.CG
import Internal.Conversion
{- | infix synonym of 'mul'
>>> let a = (3><5) [1..]
>>> a
(3><5)
[ 1.0, 2.0, 3.0, 4.0, 5.0
, 6.0, 7.0, 8.0, 9.0, 10.0
, 11.0, 12.0, 13.0, 14.0, 15.0 ]
>>> let b = (5><2) [1,3, 0,2, -1,5, 7,7, 6,0]
>>> b
(5><2)
[ 1.0, 3.0
, 0.0, 2.0
, -1.0, 5.0
, 7.0, 7.0
, 6.0, 0.0 ]
>>> a <> b
(3><2)
[ 56.0, 50.0
, 121.0, 135.0
, 186.0, 220.0 ]
-}
(<>) :: Numeric t => Matrix t -> Matrix t -> Matrix t
(<>) = mXm
infixr 8 <>
-- | dense matrix product
mul :: Numeric t => Matrix t -> Matrix t -> Matrix t
mul = mXm
{- | 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 ]
-}
linearSolve m b = A.mbLinearSolve m b
-- | return an orthonormal basis of the null space of a matrix. See also 'nullspaceSVD'.
nullspace m = nullspaceSVD (Left (1*eps)) m (rightSV m)
-- | return an orthonormal basis of the range space of a matrix. See also 'orthSVD'.
orth m = orthSVD (Left (1*eps)) m (leftSV m)
|
