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Mapping Lapack to Matlab, Mathematica and Ada 2005 functions (DRAFT version, may contain errors)

sometime in 2011   Compiled on May 21, 2020 at 2:35am

This table lists some of the Lapack functions (only the Single Precision REAL Routines are shown), and Matlab, Mathematica and Ada calls which closely provide that functionality.

Table

 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. OrR=chol(A)x=R$$R’\B) see above. OrR=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 usex=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$$ and provides an estimate of the condition number, and error bounds on the solution A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] Use transpose or conjuagte on A ﬁrst, then call solve(). But missing condition number function. SGBSVX Solves a general banded system of linear equations $$Ax=b$$, $$A^Tx=b$$, or $$A^Hx=b$$,and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] Use transpose or conjuagte on A ﬁrst, then call solve(). But missing condition number function. SGTSVX Solves a general tridiagonal system of linear equations $$Ax=b$$, $$A^Tx=b$$, or $$A^Hx=b$$, and provides an estimate of the condition ,number and error bounds on the solution A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] Use transpose or conjuagte on A ﬁrst, then call solve(). But missing condition number function. SPOSVX Solves a symmetric positive deﬁnite system of linear equations $$Ax=b$$, and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SPPSVX Solves a symmetric positive deﬁnite system of linear equations $$Ax=b$$, where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SPBSVX Solves a symmetric positive deﬁnite banded system of linear equations $$Ax=b$$, where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SPTSVX Solves a symmetric positive deﬁnite tridiagonal system of linear equations $$Ax=b$$, where $$A$$ is held in packed storage, and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SSYSVX Solves a real symmetric indeﬁnite system of linear equations $$Ax=b$$, and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SSPSVX Solves a real symmetric indeﬁnite system of linear equations $$Ax=b$$, where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution. A\bcond(A) LinearSolve[A,b]LinearAlgebra‘MatrixConditionNumber[A] x:solve(A,b). But missing condition number function. SGELSY Computes the minimum norm least squares solution to an over-or under-determined system of linear equations $$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