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

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1 Mapping Lapack to Matlab, Mathematica and Ada 2005 functions (DRAFT version, may contain errors) Nasser M. Abbasi sometime in 2011 page compiled on July 1, 2015 at 7:19pm 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 equations Ax = b f=factorize(a) x=f\b LinearSolve[A,B] PsedudoInverse[A].B x:=solve(a,b) S=inverse(A) x=s*b SGBSV SGTSV SPOSV Solves a general banded system of linear equations Ax = b Solves a general tridiagonal system of linear equations Ax = b Ax = b pinv(a)* b LinearSolve[A,B] x:=solve(a,b) LinearSolve[A,B] x:=solve(a,b) LinearSolve[A,B] x:=solve(a,b) 1

2 SPPSV SPBSV SPTSV SSYSV SSPSV SGELS equations Ax = b, where A is held in packed storage banded system Ax = b tridiagonal system Ax = b Solves a real symmetric indefinite equations Ax = b Solves a real symmetric indefinite equations Ax = b where A is held in packed storage least squares solution to an overdetermined 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 LinearSolve[A,B] x:=solve(a,b) see above. Or R=chol(A) x=r\(r \B) see above. Or x:=solve(a,b) R=CholeskyDecomposition[A] LinearSolve[Transpose[R],B] LinearSolve[R,%] LinearSolve[A,B] x:=solve(a,b) LinearSolve[A,B] x:=solve(a,b) LinearSolve[A,B] x:=solve(a,b) for overdetermined: 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) 2

3 SGELSD SGGLSE SGGGLM SSYEV SSYEVD SSPEV least squares solution to an overdetermined 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) Solves the LSE (Constrained Linear Least Squares Problem) using the Generalized RQ factorization Solves the GLM (Generalized Linear Regression Model) using the GQR (Generalized QR) factorization symmetric matrix symmetric matrix If desired, it uses a divide and conquer algorithm symmetric matrix in packed storage Can also use x= x=linearsolve[a,b] No SVD. Can use x:=solve(a,b) u,w,v=singularvaluedecomposition[a] [u,s,v]=svd(a) x=v*inv(s)*v *b invs=diagonalmatrix[1/diagonal[s]] x=v.invs.transpose[v].b lsqlin() FindMinimum[] Missing? glmfit() requires statistics toolbox see GeneralizedLinearModelFit[] and LinearModelFit[] Missing? 3

4 SSPEVD symmetric matrix in packed storage. If desired, it uses a divide and conquer algorithm SSBEV symmetric band matrix SSBEVD symmetric band matrix. If desired, it uses a divide and conquer algorithm SSTEV symmetric tridiagonal matrix SSTEVD symmetric tridiagonal matrix. If desired, it uses a divide and conquer algorithm 4

5 SGEES 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[] SGEEV left and right 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 again then call conjugate(). See annex G for the exact calls. SGESVD SGESDD SSYGV singular value decomposition (SVD) a general matrix singular value decomposition (SVD) a general matrix using divide-and-conquer the symmetric-definite eigenproblem svd() svd() SingularValueDecomposition[] SingularValueDecomposition[] [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] 5

6 SSYGVD SSPGV SSPGVD the symmetric-definite eigenproblem Ax = λbx, ABx = λx, BAx = λx If desired, it uses a divide and conquer algorithm the symmetric-definite eigenproblem Ax = λbx, ABx = λx, BAx = λx where A and B are in packed storage the symmetric-definite eigenproblem Ax = λbx, ABx = λx, BAx = λx, where A and B are in packed storage. If desired, it uses a divide and conquer algorithm [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] 6

7 SSBGV SSBGVD SGGES SGGEV the eigenvalues, and the symmetric of the form the form Ax = λbx A and B are assumed to be symmetric and banded, and B is also the symmetric definite banded eigenproblem of the form Ax = λbx A and B are assumed to be symmetric and banded, and B is also. If desired, it uses a divide and conquer algorithm eigenvalues, Schur form, and left and/or right Schur vectors for a pair of nonsymmetric matrices eigenvalues, and left and/or right eigenvectors for a pair of nonsymmetric matrices [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] schur() SchurDecomposition[] [V,D]=eig(A,B, qz ) D,V=Eigensystem[A,B] 7

8 SGGSVD SGESVX SGBSVX SGTSVX SPOSVX Generalized Singular Value Decomposition Solve a general equations, Ax = b, A T x = b, or A H x = b and provides an estimate of the condition number, and error bounds on the solution Solves a general banded system of linear equations Ax = b, A T x = b, or A H x = b,and provides an estimate of the condition number and error bounds on the solution. Solves a general tridiagonal system of linear equations Ax = b, A T x = b, or A H x = b, and provides an estimate of the condition,number and error bounds on the solution equations Ax = b, and provides an estimate of the condition number and error bounds on the solution. gsvd() SingularValueList[] LinearSolve[A,b] Use transpose or conjuagte A first, then call solve(). But LinearSolve[A,b] Use transpose or conjuagte A first, then call solve(). But LinearSolve[A,b] Use transpose or conjuagte A first, then call solve(). But LinearSolve[A,b] x:solve(a,b). But 8

9 SPPSVX equations Ax = b, where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution LinearSolve[A,b] x:solve(a,b). But SPBSVX 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. LinearSolve[A,b] x:solve(a,b). But SPTSVX 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. LinearSolve[A,b] x:solve(a,b). But SSYSVX Solves a real symmetric indefinite equations Ax = b, and provides an estimate of the condition number and error bounds on the solution. LinearSolve[A,b] x:solve(a,b). But 9

10 SSPSVX Solves a real symmetric indefinite equations Ax = b, where A is held in packed storage, and provides an estimate of the condition number and error bounds on the solution. LinearSolve[A,b] x:solve(a,b). But SGELSY minimum norm least squares solution to an over-or under-determined equations Ax = b, using a complete orthogonal factorization of A for overdetermined: 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 minimum norm least squares solution to an over- or under-determined equations Ax = b, using the singular value decomposition of A. for overdetermined: 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 symmetric matrix. use eig() then then eigenvalues(a) then 10

11 SSYEVR eigenvalues, and, symmetric matrix. Eigenvalues are computed by the dqds algorithm, and computed from various good LDL T, representations (also known as Relatively Robust Representations). can use eig() Eigensystem() then then SSYGVX and the symmetric-definite eigenproblem Ax = λbx, ABx = λx, BAx = λx No direct support, [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] or SSPEVX symmetric matrix in packed storage. can use eig() Eigensystem() then then SSPGVX and the symmetric-definite eigenproblem Ax = λbx, ABx = λx, BAx = λx where A and B are in packed storage. No direct support, [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] or 11

12 SSBEVX symmetric band matrix. can use eig() Eigensystem() then then SSBGVX eigenvalues, and the 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. No direct support, [V,D]=eig(A,B, chol ) D,V=Eigensystem[A,B] or SSTEVX symmetric tridiagonal matrix. can use eig() Eigensystem() then then SSTEVR eigenvalues, and symmetric tridiagonal matrix. Eigenvalues are computed by the dqds algorithm, and computed from various good LDL T representations (also known as Relatively Robust Representations). can use eig() Eigensystem() then then 12

13 SGEESX 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. can use eig(), shur(), then Eigensystem(), Schur- Decomposition[], then then SGGESX eigenvalues, the real Schur form, and the left and/or right matrices of Schur vectors. can use eig(), shur(), then, Schur- Decomposition[], then No support for eigenvalues. No shur decomposition SGEEVX left and right general matrix, with preliminary balancing of the matrix, and computes reciprocal condition numbers for the eigenvalues and right eigenvectors. can use eig() and cond(), and No support but can use, no condition number. SGGEVX eigenvalues, and the left and/or right eigenvectors. [V,D]=eig(A,B, chol ) [D,V]=Eigensystem[A,B] No support for eigenvalues references 13

14 Ada 2005 reference manual Mathematica and Matlab help 14