----------------------------------------------------------------------------- {- | Module : Numeric.LinearAlgebra.HMatrix Copyright : (c) Alberto Ruiz 2006-14 License : BSD3 Maintainer : Alberto Ruiz Stability : provisional -} ----------------------------------------------------------------------------- module Numeric.LinearAlgebra.HMatrix ( -- * 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, peps, relativeError, haussholder, optimiseMult, udot, nullspaceSVD, orthSVD, ranksv, ℝ,ℂ,iC, -- * Auxiliary classes Element, Container, Product, Numeric, LSDiv, Complexable, RealElement, RealOf, ComplexOf, SingleOf, DoubleOf, IndexOf, Field, -- Normed, Transposable, CGState(..), Testable(..) ) where import Numeric.LinearAlgebra.Data import Numeric.Matrix() import Numeric.Vector() import Data.Packed.Numeric hiding ((<>), mul) import Numeric.LinearAlgebra.Algorithms hiding (linearSolve,Normed,orth) import qualified Numeric.LinearAlgebra.Algorithms as A import Numeric.LinearAlgebra.Util import Numeric.LinearAlgebra.Random import Numeric.Sparse((!#>)) import Numeric.LinearAlgebra.Util.CG {- | 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)