From 3ccfccc82aa1cf374af1b822087fb6c4fd41b3c3 Mon Sep 17 00:00:00 2001 From: Alberto Ruiz Date: Sat, 18 Apr 2009 18:46:57 +0000 Subject: Algorithms module reorganized to fix documentation structure --- lib/Numeric/LinearAlgebra/Algorithms.hs | 205 +++++++++++++++++++------------- 1 file changed, 124 insertions(+), 81 deletions(-) (limited to 'lib/Numeric/LinearAlgebra/Algorithms.hs') diff --git a/lib/Numeric/LinearAlgebra/Algorithms.hs b/lib/Numeric/LinearAlgebra/Algorithms.hs index d978fa4..e22246f 100644 --- a/lib/Numeric/LinearAlgebra/Algorithms.hs +++ b/lib/Numeric/LinearAlgebra/Algorithms.hs @@ -3,7 +3,7 @@ ----------------------------------------------------------------------------- {- | Module : Numeric.LinearAlgebra.Algorithms -Copyright : (c) Alberto Ruiz 2006-7 +Copyright : (c) Alberto Ruiz 2006-9 License : GPL-style Maintainer : Alberto Ruiz (aruiz at um dot es) @@ -20,27 +20,33 @@ imported from "Numeric.LinearAlgebra.LAPACK". ----------------------------------------------------------------------------- module Numeric.LinearAlgebra.Algorithms ( --- * Linear Systems +-- * Supported types + Field(), +-- * Products multiply, dot, + outer, kronecker, +-- * Linear Systems linearSolve, + luSolve, + linearSolveSVD, inv, pinv, - pinvTol, det, rank, rcond, + det, rank, rcond, -- * Matrix factorizations -- ** Singular value decomposition svd, full, economy, --thin, -- ** Eigensystems - eig, eigSH, + eig, eigSH, eigSH', -- ** QR qr, -- ** Cholesky - chol, + chol, cholSH, -- ** Hessenberg hess, -- ** Schur schur, -- ** LU - lu, luPacked, luSolve, + lu, luPacked, -- * Matrix functions expm, sqrtm, @@ -53,11 +59,10 @@ module Numeric.LinearAlgebra.Algorithms ( -- * Misc ctrans, eps, i, - outer, kronecker, -- * Util haussholder, unpackQR, unpackHess, - Field(linearSolveSVD,eigSH',cholSH) + pinvTol ) where @@ -71,82 +76,120 @@ import Data.List(foldl1') import Data.Array - -- | Auxiliary typeclass used to define generic computations for both real and complex matrices. class (Normed (Matrix t), Linear Vector t, Linear Matrix t) => Field t where - -- | Singular value decomposition using lapack's dgesvd or zgesvd. - svd :: Matrix t -> (Matrix t, Vector Double, Matrix t) - -- | Obtains the LU decomposition of a matrix in a compact data structure suitable for 'luSolve'. - luPacked :: Matrix t -> (Matrix t, [Int]) - -- | Solution of a linear system (for several right hand sides) from the precomputed LU factorization - -- obtained by 'luPacked'. - luSolve :: (Matrix t, [Int]) -> Matrix t -> Matrix t - -- | Solution of a general linear system (for several right-hand sides) using lapacks' dgesv or zgesv. - -- It is similar to 'luSolve' . 'luPacked', but @linearSolve@ raises an error if called on a singular system. - -- See also other versions of linearSolve in "Numeric.LinearAlgebra.LAPACK". - linearSolve :: Matrix t -> Matrix t -> Matrix t - linearSolveSVD :: Matrix t -> Matrix t -> Matrix t - -- | Eigenvalues and eigenvectors of a general square matrix using lapack's dgeev or zgeev. - -- - -- If @(s,v) = eig m@ then @m \<> v == v \<> diag s@ - eig :: Matrix t -> (Vector (Complex Double), Matrix (Complex Double)) - -- | Similar to eigSH without checking that the input matrix is hermitian or symmetric. - eigSH' :: Matrix t -> (Vector Double, Matrix t) - -- | Similar to chol without checking that the input matrix is hermitian or symmetric. - cholSH :: Matrix t -> Matrix t - -- | QR factorization using lapack's dgeqr2 or zgeqr2. - -- - -- If @(q,r) = qr m@ then @m == q \<> r@, where q is unitary and r is upper triangular. - qr :: Matrix t -> (Matrix t, Matrix t) - -- | Hessenberg factorization using lapack's dgehrd or zgehrd. - -- - -- If @(p,h) = hess m@ then @m == p \<> h \<> ctrans p@, where p is unitary - -- and h is in upper Hessenberg form. - hess :: Matrix t -> (Matrix t, Matrix t) - -- | Schur factorization using lapack's dgees or zgees. - -- - -- If @(u,s) = schur m@ then @m == u \<> s \<> ctrans 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) - schur :: Matrix t -> (Matrix t, Matrix t) - -- | Conjugate transpose. - ctrans :: Matrix t -> Matrix t - -- | Matrix product. - multiply :: Matrix t -> Matrix t -> Matrix t + svd' :: Matrix t -> (Matrix t, Vector Double, Matrix t) + luPacked' :: Matrix t -> (Matrix t, [Int]) + luSolve' :: (Matrix t, [Int]) -> Matrix t -> Matrix t + linearSolve' :: Matrix t -> Matrix t -> Matrix t + linearSolveSVD' :: Matrix t -> Matrix t -> Matrix t + eig' :: Matrix t -> (Vector (Complex Double), Matrix (Complex Double)) + eigSH'' :: Matrix t -> (Vector Double, Matrix t) + cholSH' :: Matrix t -> Matrix t + qr' :: Matrix t -> (Matrix t, Matrix t) + hess' :: Matrix t -> (Matrix t, Matrix t) + schur' :: Matrix t -> (Matrix t, Matrix t) + ctrans' :: Matrix t -> Matrix t + multiply' :: Matrix t -> Matrix t -> Matrix t + + +-- | Singular value decomposition using lapack's dgesvd or zgesvd. +svd :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t) +svd = svd' + +-- | Obtains the LU decomposition of a matrix in a compact data structure suitable for 'luSolve'. +luPacked :: Field t => Matrix t -> (Matrix t, [Int]) +luPacked = luPacked' + +-- | Solution of a linear system (for several right hand sides) from the precomputed LU factorization +-- obtained by 'luPacked'. +luSolve :: Field t => (Matrix t, [Int]) -> Matrix t -> Matrix t +luSolve = luSolve' + +-- | Solution of a general linear system (for several right-hand sides) using lapacks' dgesv or zgesv. +-- It is similar to 'luSolve' . 'luPacked', but @linearSolve@ raises an error if called on a singular system. +-- See also other versions of linearSolve in "Numeric.LinearAlgebra.LAPACK". +linearSolve :: Field t => Matrix t -> Matrix t -> Matrix t +linearSolve = linearSolve' +linearSolveSVD :: Field t => Matrix t -> Matrix t -> Matrix t +linearSolveSVD = linearSolveSVD' + +-- | Eigenvalues and eigenvectors of a general square matrix using lapack's dgeev or zgeev. +-- +-- If @(s,v) = eig m@ then @m \<> v == v \<> diag s@ +eig :: Field t => Matrix t -> (Vector (Complex Double), Matrix (Complex Double)) +eig = eig' + +-- | Similar to 'eigSH' without checking that the input matrix is hermitian or symmetric. +eigSH' :: Field t => Matrix t -> (Vector Double, Matrix t) +eigSH' = eigSH'' + +-- | Similar to 'chol' without checking that the input matrix is hermitian or symmetric. +cholSH :: Field t => Matrix t -> Matrix t +cholSH = cholSH' + +-- | QR factorization using lapack's dgeqr2 or zgeqr2. +-- +-- If @(q,r) = qr m@ then @m == q \<> r@, where q is unitary and r is upper triangular. +qr :: Field t => Matrix t -> (Matrix t, Matrix t) +qr = qr' + +-- | Hessenberg factorization using lapack's dgehrd or zgehrd. +-- +-- If @(p,h) = hess m@ then @m == p \<> h \<> ctrans p@, where p is unitary +-- and h is in upper Hessenberg form. +hess :: Field t => Matrix t -> (Matrix t, Matrix t) +hess = hess' + +-- | Schur factorization using lapack's dgees or zgees. +-- +-- If @(u,s) = schur m@ then @m == u \<> s \<> ctrans 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) +schur :: Field t => Matrix t -> (Matrix t, Matrix t) +schur = schur' + +-- | Generic conjugate transpose. +ctrans :: Field t => Matrix t -> Matrix t +ctrans = ctrans' + +-- | Matrix product. +multiply :: Field t => Matrix t -> Matrix t -> Matrix t +multiply = multiply' instance Field Double where - svd = svdR - luPacked = luR - luSolve (l_u,perm) = lusR l_u perm - linearSolve = linearSolveR -- (luSolve . luPacked) ?? - linearSolveSVD = linearSolveSVDR Nothing - ctrans = trans - eig = eigR - eigSH' = eigS - cholSH = cholS - qr = unpackQR . qrR - hess = unpackHess hessR - schur = schurR - multiply = multiplyR + svd' = svdR + luPacked' = luR + luSolve' (l_u,perm) = lusR l_u perm + linearSolve' = linearSolveR -- (luSolve . luPacked) ?? + linearSolveSVD' = linearSolveSVDR Nothing + ctrans' = trans + eig' = eigR + eigSH'' = eigS + cholSH' = cholS + qr' = unpackQR . qrR + hess' = unpackHess hessR + schur' = schurR + multiply' = multiplyR instance Field (Complex Double) where - svd = svdC - luPacked = luC - luSolve (l_u,perm) = lusC l_u perm - linearSolve = linearSolveC - linearSolveSVD = linearSolveSVDC Nothing - ctrans = conj . trans - eig = eigC - eigSH' = eigH - cholSH = cholH - qr = unpackQR . qrC - hess = unpackHess hessC - schur = schurC - multiply = multiplyC + svd' = svdC + luPacked' = luC + luSolve' (l_u,perm) = lusC l_u perm + linearSolve' = linearSolveC + linearSolveSVD' = linearSolveSVDC Nothing + ctrans' = conj . trans + eig' = eigC + eigSH'' = eigH + cholSH' = cholH + qr' = unpackQR . qrC + hess' = unpackHess hessC + schur' = schurC + multiply' = multiplyC -- | Eigenvalues and Eigenvectors of a complex hermitian or real symmetric matrix using lapack's dsyev or zheev. @@ -193,8 +236,8 @@ pinv m = linearSolveSVD m (ident (rows m)) -- If @(u,d,v) = full svd m@ then @m == u \<> d \<> trans v@. full :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Matrix Double, Matrix t) -full svd' m = (u, d ,v) where - (u,s,v) = svd' m +full svdFun m = (u, d ,v) where + (u,s,v) = svdFun m d = diagRect s r c r = rows m c = cols m @@ -204,8 +247,8 @@ full svd' m = (u, d ,v) where -- If @(u,s,v) = economy svd m@ then @m == u \<> diag s \<> trans v@. economy :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Vector Double, Matrix t) -economy svd' m = (u', subVector 0 d s, v') where - (u,s,v) = svd' m +economy svdFun m = (u', subVector 0 d s, v') where + (u,s,v) = svdFun m sl@(g:_) = toList s s' = fromList . filter (>tol) $ sl t = 1 -- cgit v1.2.3