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| Numeric.LinearAlgebra.Algorithms | | Portability | uses ffi | | Stability | provisional | | Maintainer | Alberto Ruiz (aruiz at um dot es) |
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| Description |
A generic interface for some common functions. Using it we can write higher level algorithms
and testing properties both for real and complex matrices.
In any case, the specific functions for particular base types can also be explicitly
imported from Numeric.LinearAlgebra.LAPACK.
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| Synopsis |
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| class (Normed (Matrix t), Linear Vector t, Linear Matrix t) => Field t | | | multiply :: Field t => Matrix t -> Matrix t -> Matrix t | | | dot :: Field t => Vector t -> Vector t -> t | | | outer :: Field t => Vector t -> Vector t -> Matrix t | | | kronecker :: Field t => Matrix t -> Matrix t -> Matrix t | | | linearSolve :: Field t => Matrix t -> Matrix t -> Matrix t | | | luSolve :: Field t => (Matrix t, [Int]) -> Matrix t -> Matrix t | | | linearSolveSVD :: Field t => Matrix t -> Matrix t -> Matrix t | | | inv :: Field t => Matrix t -> Matrix t | | | pinv :: Field t => Matrix t -> Matrix t | | | det :: Field t => Matrix t -> t | | | rank :: Field t => Matrix t -> Int | | | rcond :: Field t => Matrix t -> Double | | | svd :: Field t => Matrix t -> (Matrix t, Vector Double, Matrix t) | | | full :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Matrix Double, Matrix t) | | | economy :: Element t => (Matrix t -> (Matrix t, Vector Double, Matrix t)) -> Matrix t -> (Matrix t, Vector Double, Matrix t) | | | eig :: Field t => Matrix t -> (Vector (Complex Double), Matrix (Complex Double)) | | | eigSH :: Field t => Matrix t -> (Vector Double, Matrix t) | | | eigSH' :: Field t => Matrix t -> (Vector Double, Matrix t) | | | qr :: Field t => Matrix t -> (Matrix t, Matrix t) | | | chol :: Field t => Matrix t -> Matrix t | | | cholSH :: Field t => Matrix t -> Matrix t | | | hess :: Field t => Matrix t -> (Matrix t, Matrix t) | | | schur :: Field t => Matrix t -> (Matrix t, Matrix t) | | | lu :: Field t => Matrix t -> (Matrix t, Matrix t, Matrix t, t) | | | luPacked :: Field t => Matrix t -> (Matrix t, [Int]) | | | expm :: Field t => Matrix t -> Matrix t | | | sqrtm :: Field t => Matrix t -> Matrix t | | | matFunc :: Field t => (Complex Double -> Complex Double) -> Matrix t -> Matrix (Complex Double) | | | nullspacePrec :: Field t => Double -> Matrix t -> [Vector t] | | | nullVector :: Field t => Matrix t -> Vector t | | | class Normed t where | | | | | | ctrans :: Field t => Matrix t -> Matrix t | | | eps :: Double | | | i :: Complex Double | | | haussholder :: Field a => a -> Vector a -> Matrix a | | | unpackQR :: Field t => (Matrix t, Vector t) -> (Matrix t, Matrix t) | | | unpackHess :: Field t => (Matrix t -> (Matrix t, Vector t)) -> Matrix t -> (Matrix t, Matrix t) | | | pinvTol :: Double -> Matrix Double -> Matrix Double |
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| Supported types
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| Auxiliary typeclass used to define generic computations for both real and complex matrices.
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| Products
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| Matrix product.
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| Euclidean inner product.
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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 ] |
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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 ] |
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| Linear Systems
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| 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.
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| Solution of a linear system (for several right hand sides) from the precomputed LU factorization
obtained by luPacked.
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| Inverse of a square matrix using lapacks' dgesv and zgesv.
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| Pseudoinverse of a general matrix using lapack's dgelss or zgelss.
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| determinant of a square matrix, computed from the LU decomposition.
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| Number of linearly independent rows or columns.
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| Reciprocal of the 2-norm condition number of a matrix, computed from the SVD.
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| Matrix factorizations
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| Singular value decomposition
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| Singular value decomposition using lapack's dgesvd or zgesvd.
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A version of svd which returns an appropriate diagonal matrix with the singular values.
If (u,d,v) = full svd m then m == u <> d <> trans v.
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A version of svd which returns only the nonzero singular values and the corresponding rows and columns of the rotations.
If (u,s,v) = economy svd m then m == u <> diag s <> trans v.
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| Eigensystems
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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
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Eigenvalues and Eigenvectors of a complex hermitian or real symmetric matrix using lapack's dsyev or zheev.
If (s,v) = eigSH m then m == v <> diag s <> ctrans v
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| Similar to eigSH without checking that the input matrix is hermitian or symmetric.
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| QR
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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.
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| Cholesky
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Cholesky factorization of a positive definite hermitian or symmetric matrix using lapack's dpotrf or zportrf.
If c = chol m then m == ctrans c <> c.
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| Similar to chol without checking that the input matrix is hermitian or symmetric.
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| Hessenberg
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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.
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| Schur
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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)
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| LU
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Explicit LU factorization of a general matrix using lapack's dgetrf or zgetrf.
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.
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| Obtains the LU decomposition of a matrix in a compact data structure suitable for luSolve.
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| Matrix functions
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| Matrix exponential. It uses a direct translation of Algorithm 11.3.1 in Golub & Van Loan,
based on a scaled Pade approximation.
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Matrix square root. Currently it uses a simple iterative algorithm described in Wikipedia.
It only works with invertible matrices that have a real solution. For diagonalizable matrices you can try matFunc sqrt.
m = (2><2) [4,9
,0,4] :: Matrix Double>sqrtm m
(2><2)
[ 2.0, 2.25
, 0.0, 2.0 ] |
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Generic matrix functions for diagonalizable matrices. For instance:
logm = matFunc log |
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| Nullspace
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| :: Field t | | | => Double | input matrix
| | -> Matrix t | list of unitary vectors spanning the nullspace
| | -> [Vector t] | | | The nullspace of a matrix from its SVD decomposition.
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| The nullspace of a matrix, assumed to be one-dimensional, with default tolerance (shortcut for last . nullspacePrec 1).
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| Norms
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Objects which have a p-norm.
Using it you can define convenient shortcuts:
norm2 x = pnorm PNorm2 x frobenius m = norm2 . flatten $ m | | | Methods | | | | Instances | |
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| Misc
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| Generic conjugate transpose.
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| The machine precision of a Double: eps = 2.22044604925031e-16 (the value used by GNU-Octave).
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| The imaginary unit: i = 0.0 :+ 1.0
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| Util
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| Produced by Haddock version 2.6.0 |
