lapack

description

Matlab

Mathematica

Ada

SGESV

Solves a general system of linear equations \(Ax=b\)

A\b

f=factorize(A)
x=f\b

S=inverse(A)
x=S*b

pinv(A)* b

LinearSolve[A,B]

PsedudoInverse[A].B

x:=solve(A,b)

SGBSV

Solves a general banded system of linear equations \(Ax=b\)

A\b

LinearSolve[A,B]

x:=solve(A,b)

SGTSV

Solves a general tridiagonal system of linear equations \(Ax=b\)

A\b

LinearSolve[A,B]

x:=solve(A,b)

SPOSV

Solves a symmetric positive definite system of linear \(Ax=b\)

A\b

LinearSolve[A,B]

x:=solve(A,b)

SPPSV

Solves a symmetric positive definite system of linear equations \(Ax=b\), where \(A\) is held in packed storage

A\b

LinearSolve[A,B]

x:=solve(A,b)

SPBSV

Solves a symmetric positive definite banded system \(Ax=b\)

see above. Or
R=chol(A)
x=R\(R’\B)

see above. Or
R=CholeskyDecomposition[A]
LinearSolve[Transpose[R],B]
LinearSolve[R,%]

x:=solve(A,b)

SPTSV

Solves a symmetric positive definite tridiagonal system \(Ax=b\)

A\b

LinearSolve[A,B]

x:=solve(A,b)

SSYSV

Solves a real symmetric indefinite system of linear equations \(Ax=b\)

A\b

LinearSolve[A,B]

x:=solve(A,b)

SSPSV

Solves a real symmetric indefinite system of linear equations \(Ax=b\) where \(A\) is held in packed storage

A\b

LinearSolve[A,B]

x:=solve(A,b)

SGELS

Computes the least squares solution to an overdetermined system of linear equations, \(Ax=b\) or \(A^{T}x=b\), or the minimum norm solution of an underdetermined system, where A is a general rectangular matrix of full rank, using a QR or LQ factorization of A

for overdetermined:
A\b

for underdetermined:
pinv(A)*b

or lsqlin(A,b)

for overdetermined:
LinearSolve[A,b]

for underdetermined:

PseudoInverse[A].b

or LeastSquares[A,b]

x:=solve(A,b)

SGELSD

Computes the least squares solution to an overdetermined system of linear equations, \(Ax=b\) or \(A^{T}x=b\), or the minimum norm solution of an underdetermined system, where \(A\) is a general rectangular matrix of full rank, using singular value decomposition (SVD)

Can also use
x=A\b

[u,s,v]=svd(A)
x=v*inv(s)*v’*b

x=LinearSolve[A,b]

u,w,v=SingularValueDecomposition[A]
invS=DiagonalMatrix[1/Diagonal[s]]
x=v.invS.Transpose[v].b

No SVD. Can use x:=solve(A,b)

SGGLSE

Solves the LSE (Constrained Linear Least Squares Problem) using the Generalized RQ factorization

lsqlin()

FindMinimum[]

Missing?

SGGGLM

Solves the GLM (Generalized Linear Regression Model) using the GQR (Generalized QR) factorization

glmfit()
requires statistics toolbox

see GeneralizedLinearModelFit[]
and LinearModelFit[]

Missing?

SSYEV

Computes all eigenvalues and optionally, eigenvectors of a real symmetric matrix

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSYEVD

Computes all eigenvalues and optionally, eigenvectors of a real symmetric matrix If eigenvectors are desired, it uses a divide and conquer algorithm

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSPEV

Computes all eigenvalues and optionally, eigenvectors of a real symmetric matrix in packed storage

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSPEVD

Computes all eigenvalues and optionally, eigenvectors of a real symmetric matrix in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSBEV

Computes all eigenvalues and optionally, eigenvectors of a real symmetric band matrix

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSBEVD

Computes all eigenvalues and optionally, eigenvectors of a real symmetric band matrix. If eigenvectors are desired, it uses a divide and conquer algorithm

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSTEV

Computes all eigenvalues and optionally, eigenvectors of a real symmetric tridiagonal matrix

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SSTEVD

Computes all eigenvalues and optionally, eigenvectors of a real symmetric tridiagonal matrix. If eigenvectors are desired, it uses a divide and conquer algorithm

eig() or eigs()

Eigensystem[]
Eigenvalues[]
Eigenvectors[]

eigenvalues()
eigensystem()

SGEES

Computes all eigenvalues and Schur factorization of a general matrix and orders the factorization so that selected eigenvalues are at the top left of the Schur form

schur()

SchurDecomposition[]

missing?

SGEEV

Computes the eigenvalues and left and right eigenvectors of a general matrix

For right eigenvectors use [V,D] = eig(A)
For left eigenvectors of A use [W,D] = eig(A.’)
W=conj(W)

For right eigenvectors use D,V=Eigensystem[A]
v=Transpose[v]

For left eigenvectors of A D,W=Eigensystem[Transpose[A]]
W=ConjugateTranspose[W]

For right eigenvectors use eigensystem(A,values,vectors) and for left eigenvectors, use transpose() on A and call eigensystem() again then call conjugate(). See annex G for the exact calls.

SGESVD

Computes the singular value decomposition (SVD) a general matrix

svd()

SingularValueDecomposition[]

missing?

SGESDD

Computes the singular value decomposition (SVD) a general matrix using divide-and-conquer

svd()

SingularValueDecomposition[]

missing?

SSYGV

Computes all eigenvalues and the eigenvectors of a generalized symmetric-definite generalized eigenproblem

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SSYGVD

Computes all eigenvalues and the eigenvectors of a generalized symmetric-definite generalized eigenproblem \(A x=\lambda B x\), \(A B x=\lambda x\), \(B A x=\lambda x\) If eigenvectors are desired, it uses a divide and conquer algorithm

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SSPGV

Computes all eigenvalues and the eigenvectors of a generalized symmetric-definite generalized eigenproblem \(Ax=\lambda Bx\) , \(ABx=\lambda x\), \(BAx=\lambda x\) where A and B are in packed storage

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SSPGVD

Computes all eigenvalues and the eigenvectors of a generalized symmetric-definite generalized eigenproblem \(Ax=\lambda Bx\) , \(ABx=\lambda x\), \(BAx=\lambda x\), where A and B are in packed storage. If eigenvectors are desired, it  uses a divide and conquer algorithm

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SSBGV

Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric of the form the form \(Ax=\lambda Bx\) A and B are assumed to be symmetric and banded, and B is also positive definite

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SSBGVD

Computes all eigenvalues and optionally, the eigenvectors of a real generalized symmetric definite banded eigenproblem of the form \(Ax=\lambda Bx\) A and B are assumed to be symmetric and banded, and B is also positive definite. If eigenvectors are desired, it  uses a divide and conquer algorithm

[V,D]=eig(A,B,’chol’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SGGES

Computes the generalized eigenvalues, Schur form, and left and/or right Schur vectors for a pair of nonsymmetric matrices

schur()

SchurDecomposition[]

missing?

SGGEV

Computes the generalized eigenvalues, and left and/or right generalized eigenvectors for a pair of nonsymmetric matrices

[V,D]=eig(A,B,’qz’)

D,V=Eigensystem[A,B]
D,V=Eigensystem[A,B,k]

missing?

SGGSVD

Computes the Generalized Singular Value Decomposition

gsvd()

SingularValueList[]

missing?

SGESVX

Solve a general system of linear equations, \(Ax=b\), \(A^Tx=b\), or \(A^Hx=b\) and provides an estimate of the condition number, and error bounds on the solution

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

Use transpose or conjuagte on A first, then call solve(). But missing condition number function.

SGBSVX

Solves a general banded system of linear equations \(Ax=b\), \(A^Tx=b\), or \(A^Hx=b\),and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

Use transpose or conjuagte on A first, then call solve(). But missing condition number function.

SGTSVX

Solves a general tridiagonal system of linear equations \(Ax=b\), \(A^Tx=b\), or \(A^Hx=b\), and provides an estimate of the condition ,number and error bounds on the solution

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

Use transpose or conjuagte on A first, then call solve(). But missing condition number function.

SPOSVX

Solves a symmetric positive definite system of linear equations \(Ax=b\), and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SPPSVX

Solves a symmetric positive definite system of linear equations \(Ax=b\), where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SPBSVX

Solves a symmetric positive definite banded system of linear equations \(Ax=b\), where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SPTSVX

Solves a symmetric positive definite tridiagonal system of linear equations \(Ax=b\), where \(A\) is held in packed storage, and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SSYSVX

Solves a real symmetric indefinite system of linear equations \(Ax=b\), and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SSPSVX

Solves a real symmetric indefinite system of linear equations \(Ax=b\), where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution.

A\b
cond(A)

LinearSolve[A,b]
LinearAlgebra‘MatrixConditionNumber[A]

x:solve(A,b). But missing condition number function.

SGELSY

Computes the minimum norm least squares solution to an over-or under-determined system of linear equations \(Ax=b\), using a complete orthogonal factorization of A

for overdetermined:
A\b

for underdetermined:
pinv(A)*b

or lsqlin(A,b)

for overdetermined:
LinearSolve[A,b]

for underdetermined:

PseudoInverse[A].b

or LeastSquares[A,b]

x:=solve(A,b)

SGELSS

Computes the minimum norm least squares solution to an over- or under-determined system of linear equations \(Ax=b\), using the singular value decomposition of A.

for overdetermined:
A\b

for underdetermined:
pinv(A)*b

or lsqlin(A,b)

for overdetermined:
LinearSolve[A,b]

for underdetermined:

PseudoInverse[A].b

or LeastSquares[A,b]

x:=solve(A,b)

SSYEVX

Computes selected eigenvalues and eigenvectors of a symmetric matrix.

use eig() then user selects

Eigenvalues[] then user selects

eigenvalues(A) then user selects

SSYEVR

Computes selected eigenvalues, and optionally, eigenvectors of a real, symmetric matrix. Eigenvalues are computed by the dqds algorithm, and eigenvectors are computed from various "good" \(LDL^T\), representations (also known as Relatively Robust Representations).

No direct support, but can use eig() then user selects

No direct support, but can use Eigensystem() then user selects

No direct support, but can use eigensystem() then user selects

SSYGVX

Computes selected eigenvalues and and optionally, the eigenvectors of a generalized symmetric-definite generalized eigenproblem \(A x=\lambda B x\), \(A B x=\lambda x\), \(B A x=\lambda x\)

No direct support, [V,D]=eig(A,B,’chol’) then user selects

No direct support, but can use D,V=Eigensystem[A,B] or D,V=Eigensystem[A,B,k] then user selects

missing?

SSPEVX

Computes selected eigenvalues and eigenvectors of a symmetric matrix in packed storage.

No direct support, but can use eig() then user selects

No direct support, but can use Eigensystem() then user selects

No direct support, but can use eigensystem() then user selects

SSPGVX

Computes selected eigenvalues and and optionally, the eigenvectors of a generalized symmetric-definite generalized eigenproblem \(A x=\lambda B x\), \(ABx=\lambda x\), \(BAx=\lambda x\) where A and B are in packed storage.

No direct support, [V,D]=eig(A,B,’chol’) then user selects

No direct support, but can use D,V=Eigensystem[A,B] or D,V=Eigensystem[A,B,k] then user selects

missing?

SSBEVX

Computes selected eigenvalues and eigenvectors of a symmetric band matrix.

No direct support, but can use eig() then user selects

No direct support, but can use Eigensystem() then user selects

No direct support, but can use eigensystem() then user selects

SSBGVX

Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. A and B are assumed to be symmetric and banded, and B is also positive definite.

No direct support, [V,D]=eig(A,B,’chol’) then user selects

No direct support, but can use D,V=Eigensystem[A,B] or D,V=Eigensystem[A,B,k] then user selects

missing?

SSTEVX

Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix.

No direct support, but can use eig() then user selects

No direct support, but can use Eigensystem() then user selects

No direct support, but can use eigensystem() then user selects

SSTEVR

Computes selected eigenvalues, and optionally, eigenvectors of a real symmetric tridiagonal matrix. Eigenvalues are computed by the dqds algorithm, and eigenvectors are computed from various "good" \(LDL^T\) representations (also known as Relatively Robust Representations).

No direct support, but can use eig() then user selects

No direct support, but can use Eigensystem() then user selects

No direct support, but can use eigensystem() then user selects

SGEESX

Computes the eigenvalues and Schur factorization of a general matrix, orders the factorization so that selected eigenvalues, are at the top left of the Schur form, and computes reciprocal condition numbers for the average of the selected eigenvalues and for the associated right invariant subspace.

No direct support, but can use eig(), shur(), then user selects

No direct support, but can use Eigensystem(), SchurDecomposition[], then user selects

No direct support, but can use eigensystem() then user selects

SGGESX

Computes the generalized eigenvalues, the real Schur form, and optionally, the left and/or right matrices of Schur vectors.

No direct support, but can use eig(), shur(), then user selects

No direct support, but can use Eigensystem[], SchurDecomposition[], then user selects

No support for generalized eigenvalues. No shur decomposition

SGEEVX

Computes the eigenvalues and left and right eigenvectors of a general matrix, with preliminary balancing of the matrix, and computes reciprocal condition numbers for the eigenvalues and right eigenvectors.

No direct support, but can use eig() and cond()

No direct support, but can use Eigensystem[], and LinearAlgebra‘MatrixConditionNumber[A]

No support but can use eigensystem(), no condition number.

SGGEVX

Computes the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

[V,D]=eig(A,B,’chol’)

[D,V]=Eigensystem[A,B]

No support for generalized eigenvalues