HMath.Matrix.LAPACK

Module index · Numerical conventions

Description

Double-precision optional, dynamically loaded LAPACKE spectral routines. All foreign buffers are private copies; no Haskell storage is exposed or mutated. Inputs must be finite and dimensions must fit 32-bit indices; the adapter widens them for ILP64. The default system liblapacke SONAME has its conventional LP64 ABI. A custom HMATH_LAPACKE_LIBRARY requires an explicit HMATH_LAPACKE_INTEGER_BITS=32/64 declaration before first use. HMATH_BACKEND=pure disables native routines. Missing native routines fail explicitly; pure alternatives live in HMath.Backend. Pure wrappers are safe because the called routines have no observable effects outside those buffers. LAPACKE interfaces: https://www.netlib.org/lapack/lapacke.html

On this page

API	Purpose
SVD	Real singular-value decomposition with U,descending nonnegative singular values,and VT.
SVD	Declaration and type contract below.
svdU	Left singular vectors as columns of the real thin/full SVD factor.
singularValues	Nonnegative real singular values in descending order.
svdVT	Transposed right singular-vector factor in A=Udiag(s)VT.
svd	Compute thin real SVD A=Udiag(s)VT via LAPACK, with descending nonnegative singular values and k=min(m,n).
fullSVD	Full square U and V^T, including null-space completion.
truncatedSVD	Given retained rank k then real matrix, compute thin SVD and retain its first k components.
randomizedSVD	Randomized range finder with an explicit seed, oversampling, and power iterations, followed by a small deterministic SVD.
Eigen	Complex eigenvalues with corresponding right eigenvector columns, including conjugate pairs from real input.
Eigen	Declaration and type contract below.
eigenvalues	Complex eigenvalues matching eigenvector column order.
eigenvectors	Right eigenvectors as complex matrix columns.
eigen	Compute complex eigenvalues and right eigenvectors of a finite real square matrix via LAPACK.
symmetricEigen	Real symmetric eigensolver with orthonormal columns, sorted ascending.
GeneralizedEigen	Projective eigenvalues alpha/beta preserve infinite eigenvalues and indeterminate singular pencils without dividing by a zero beta.
GeneralizedEigen	Declaration and type contract below.
eigenAlpha	Generalized-eigenvalue numerators alpha; retain with beta to represent infinite eigenvalues.
eigenBeta	Generalized-eigenvalue denominators beta; zero beta can represent an infinite eigenvalue.
generalizedEigenvectors	Right eigenvector columns satisfying betaAv=alphaBv.
generalizedEigen	Given equally sized square matrices A,B, compute generalized alpha/beta eigenvalue representation and right eigenvectors.
Schur	Real Schur factors A=QTtranspose(Q), with quasi-triangular T and complex eigenvalues.
Schur	Declaration and type contract below.
schurQ	Orthogonal basis in A=QTtranspose(Q).
schurT	Real quasi-upper-triangular Schur form, including 2-by-2 blocks for conjugate pairs.
schurEigenvalues	Complex eigenvalues decoded from the real Schur form.
schur	Compute real Schur form A=QTtranspose(Q), returning orthogonal Q, quasi-upper-triangular T and complex eigenvalues.
QZ	Generalized real Schur factors for a matrix pair, retaining alpha/beta eigenvalue representation to allow infinite eigenvalues.
QZ	Declaration and type contract below.
qzQ	Left orthogonal QZ transform, with A=QStranspose(Z).
qzZ	Right orthogonal QZ transform for both matrix factors.
qzS	First generalized real Schur factor S.
qzT	Second generalized real Schur factor T.
qzAlpha	Generalized Schur eigenvalue numerators alpha.
qzBeta	Generalized Schur eigenvalue denominators beta; do not divide blindly when zero.
qz	Compute generalized real Schur factors for a finite square matrix pair A,B, retaining Q,Z,S,T and alphabeta eigenvalues.
Hessenberg	Orthogonal reduction A=QHtranspose(Q), with H upper Hessenberg.
Hessenberg	Declaration and type contract below.
hessenbergQ	Orthogonal transform in A=QHtranspose(Q).
hessenbergH	Upper Hessenberg form with zeros below the first subdiagonal.
hessenberg	Reduce finite real square A to upper Hessenberg H with orthogonal Q, satisfying A=QHtranspose(Q).
Balance	D^-1 A D = balancedMatrix.
Balance	Declaration and type contract below.
balancedMatrix	Matrix after LAPACK diagonal balancing.
balanceScales	Diagonal basis scaling factors used by the balancing transformation.
balance	Balance a real square matrix by diagonal scaling, returning the balanced matrix and scales needed to interpret its basis.
ComplexSVD	Complex128 thin SVD: A = U diag(s) V, where complexVT already stores V.
ComplexSVD	Declaration and type contract below.
complexU	Complex left singular vectors as columns.
complexSingularValues	Descending nonnegative real singular values of the complex matrix.
complexVT	Conjugate-transposed right singular vectors in complex SVD.
complexSVD	Compute thin complex SVD with conjugate-transposed right factor VT and real nonnegative singular values.
complexEigen	Compute complex eigenvalues and right eigenvectors of a finite complex square matrix via LAPACK.

Documentation

SVD

data SVD Source #

Real singular-value decomposition with U,descending nonnegative singular values,and VT. Thin/full dimensions follow the selected factorization function.

Constructors

SVD
Fields svdU :: Matrix Double Left singular vectors as columns of the real thin/full SVD factor. singularValues :: [Double] Nonnegative real singular values in descending order. svdVT :: Matrix Double Transposed right singular-vector factor in A=U*diag(s)*VT.

Instances

Instances details

Show SVD Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> SVD -> ShowS # show :: SVD -> String # showList :: [SVD] -> ShowS #
Eq SVD Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: SVD -> SVD -> Bool # (/=) :: SVD -> SVD -> Bool #

svd

svd :: Matrix Double -> Either MathError SVD Source #

Compute thin real SVD A=U*diag(s)*VT via LAPACK, with descending nonnegative singular values and k=min(m,n). Return m-by-k U and k-by-n VT. Invalid input, unavailable backend, allocation bounds or LAPACK failure returns Left MathError.

fullSVD

fullSVD :: Matrix Double -> Either MathError SVD Source #

Full square U and V^T, including null-space completion.

truncatedSVD

truncatedSVD :: Int -> Matrix Double -> Either MathError SVD Source #

Given retained rank k then real matrix, compute thin SVD and retain its first k components. Reject k outside the available rank dimensions; decomposition failures propagate.

randomizedSVD

randomizedSVD :: Int -> Int -> Int -> Integer -> Matrix Double -> Either MathError SVD Source #

Randomized range finder with an explicit seed, oversampling, and power iterations, followed by a small deterministic SVD. The Gaussian test matrix uses Box-Muller on a local LCG, so results are reproducible, not cryptographic.

Eigen

data Eigen Source #

Complex eigenvalues with corresponding right eigenvector columns, including conjugate pairs from real input.

Constructors

Eigen
Fields eigenvalues :: [Complex Double] Complex eigenvalues matching eigenvector column order. eigenvectors :: Matrix (Complex Double) Right eigenvectors as complex matrix columns.

Instances

Instances details

Show Eigen Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> Eigen -> ShowS # show :: Eigen -> String # showList :: [Eigen] -> ShowS #
Eq Eigen Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: Eigen -> Eigen -> Bool # (/=) :: Eigen -> Eigen -> Bool #

eigen

eigen :: Matrix Double -> Either MathError Eigen Source #

Compute complex eigenvalues and right eigenvectors of a finite real square matrix via LAPACK. Complex-conjugate pairs are decoded into complex columns. Invalid shapebackendnumeric failures return Left MathError.

symmetricEigen

symmetricEigen :: Matrix Double -> Either MathError Eigen Source #

Real symmetric eigensolver with orthonormal columns, sorted ascending.

GeneralizedEigen

data GeneralizedEigen Source #

Projective eigenvalues alpha/beta preserve infinite eigenvalues and indeterminate singular pencils without dividing by a zero beta.

Constructors

GeneralizedEigen
Fields eigenAlpha :: [Complex Double] Generalized-eigenvalue numerators alpha; retain with beta to represent infinite eigenvalues. eigenBeta :: [Double] Generalized-eigenvalue denominators beta; zero beta can represent an infinite eigenvalue. generalizedEigenvectors :: Matrix (Complex Double) Right eigenvector columns satisfying beta*A*v=alpha*B*v.

Instances

Instances details

Show GeneralizedEigen Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> GeneralizedEigen -> ShowS # show :: GeneralizedEigen -> String # showList :: [GeneralizedEigen] -> ShowS #
Eq GeneralizedEigen Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: GeneralizedEigen -> GeneralizedEigen -> Bool # (/=) :: GeneralizedEigen -> GeneralizedEigen -> Bool #

generalizedEigen

generalizedEigen :: Matrix Double -> Matrix Double -> Either MathError GeneralizedEigen Source #

Given equally sized square matrices A,B, compute generalized alpha/beta eigenvalue representation and right eigenvectors. Retain alpha/beta to represent infinite eigenvalues safely; invalid input or LAPACK failure propagates.

Schur

data Schur Source #

Real Schur factors A=Q*T*transpose(Q), with quasi-triangular T and complex eigenvalues.

Constructors

Schur
Fields schurQ :: Matrix Double Orthogonal basis in A=Q*T*transpose(Q). schurT :: Matrix Double Real quasi-upper-triangular Schur form, including 2-by-2 blocks for conjugate pairs. schurEigenvalues :: [Complex Double] Complex eigenvalues decoded from the real Schur form.

Instances

Instances details

Show Schur Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> Schur -> ShowS # show :: Schur -> String # showList :: [Schur] -> ShowS #
Eq Schur Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: Schur -> Schur -> Bool # (/=) :: Schur -> Schur -> Bool #

schur

schur :: Matrix Double -> Either MathError Schur Source #

Compute real Schur form A=Q*T*transpose(Q), returning orthogonal Q, quasi-upper-triangular T and complex eigenvalues. Require finite square input and available LAPACK; failures return Left MathError.

QZ

data QZ Source #

Generalized real Schur factors for a matrix pair, retaining alpha/beta eigenvalue representation to allow infinite eigenvalues.

Constructors

QZ

Fields qzQ :: Matrix Double Left orthogonal QZ transform, with A=Q*S*transpose(Z). qzZ :: Matrix Double Right orthogonal QZ transform for both matrix factors. qzS :: Matrix Double First generalized real Schur factor S. qzT :: Matrix Double Second generalized real Schur factor T. qzAlpha :: [Complex Double] Generalized Schur eigenvalue numerators alpha. qzBeta :: [Double] Generalized Schur eigenvalue denominators beta; do not divide blindly when zero.

Instances

Instances details

Show QZ Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> QZ -> ShowS # show :: QZ -> String # showList :: [QZ] -> ShowS #
Eq QZ Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: QZ -> QZ -> Bool # (/=) :: QZ -> QZ -> Bool #

qz

qz :: Matrix Double -> Matrix Double -> Either MathError QZ Source #

Compute generalized real Schur factors for a finite square matrix pair A,B, retaining Q,Z,S,T and alphabeta eigenvalues. Shapebackend/LAPACK errors return Left MathError.

Hessenberg

data Hessenberg Source #

Orthogonal reduction A=Q*H*transpose(Q), with H upper Hessenberg.

Constructors

Hessenberg
Fields hessenbergQ :: Matrix Double Orthogonal transform in A=Q*H*transpose(Q). hessenbergH :: Matrix Double Upper Hessenberg form with zeros below the first subdiagonal.

Instances

Instances details

Show Hessenberg Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> Hessenberg -> ShowS # show :: Hessenberg -> String # showList :: [Hessenberg] -> ShowS #
Eq Hessenberg Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: Hessenberg -> Hessenberg -> Bool # (/=) :: Hessenberg -> Hessenberg -> Bool #

hessenberg

hessenberg :: Matrix Double -> Either MathError Hessenberg Source #

Reduce finite real square A to upper Hessenberg H with orthogonal Q, satisfying A=Q*H*transpose(Q). Invalid shape or backend/decomposition failures propagate.

Balance

data Balance Source #

D^-1 A D = balancedMatrix. Scale-only balancing preserves index order.

Constructors

Balance
Fields balancedMatrix :: Matrix Double Matrix after LAPACK diagonal balancing. balanceScales :: [Double] Diagonal basis scaling factors used by the balancing transformation.

Instances

Instances details

Show Balance Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> Balance -> ShowS # show :: Balance -> String # showList :: [Balance] -> ShowS #
Eq Balance Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: Balance -> Balance -> Bool # (/=) :: Balance -> Balance -> Bool #

balance

balance :: Matrix Double -> Either MathError Balance Source #

Balance a real square matrix by diagonal scaling, returning the balanced matrix and scales needed to interpret its basis. Invalid finite-input/shape checks or LAPACK failures return Left MathError.

ComplexSVD

data ComplexSVD Source #

Complex128 thin SVD: A = U diag(s) V*, where complexVT already stores V*.

Constructors

ComplexSVD
Fields complexU :: Matrix (Complex Double) Complex left singular vectors as columns. complexSingularValues :: [Double] Descending nonnegative real singular values of the complex matrix. complexVT :: Matrix (Complex Double) Conjugate-transposed right singular vectors in complex SVD.

Instances

Instances details

Show ComplexSVD Source #
Instance details Defined in HMath.Matrix.LAPACK Methods showsPrec :: Int -> ComplexSVD -> ShowS # show :: ComplexSVD -> String # showList :: [ComplexSVD] -> ShowS #
Eq ComplexSVD Source #
Instance details Defined in HMath.Matrix.LAPACK Methods (==) :: ComplexSVD -> ComplexSVD -> Bool # (/=) :: ComplexSVD -> ComplexSVD -> Bool #

complexSVD

complexSVD :: Matrix (Complex Double) -> Either MathError ComplexSVD Source #

Compute thin complex SVD with conjugate-transposed right factor VT and real nonnegative singular values. Require finite complex entries and available LAPACK. Invalid storage sizesbackenddecomposition fails.

complexEigen

complexEigen :: Matrix (Complex Double) -> Either MathError Eigen Source #

Compute complex eigenvalues and right eigenvectors of a finite complex square matrix via LAPACK. Invalid shape, backend availability, bounds or decomposition errors return Left MathError.