NAG Library Manual

## F08 – Least-squares and Eigenvalue Problems (LAPACK)

F08 Chapter Introduction
 RoutineName Mark ofIntroduction Purpose F08AAFExample TextExample Data 21 Solves an overdetermined or underdetermined real linear system F08AEFExample TextExample Data 16 QR factorization of real general rectangular matrix F08AFFExample TextExample Data 16 Form all or part of orthogonal Q from QR factorization determined by F08AEF (DGEQRF) or F08BEF (DGEQPF) F08AGF 16 Apply orthogonal transformation determined by F08AEF (DGEQRF) or F08BEF (DGEQPF) F08AHFExample TextExample Data 16 LQ factorization of real general rectangular matrix F08AJFExample TextExample Data 16 Form all or part of orthogonal Q from LQ factorization determined by F08AHF (DGELQF) F08AKF 16 Apply orthogonal transformation determined by F08AHF (DGELQF) F08ANFExample TextExample Data 21 Solves an overdetermined or underdetermined complex linear system F08ASFExample TextExample Data 16 QR factorization of complex general rectangular matrix F08ATFExample TextExample Data 16 Form all or part of unitary Q from QR factorization determined by F08ASF (ZGEQRF) or F08BSF (ZGEQPF) F08AUF 16 Apply unitary transformation determined by F08ASF (ZGEQRF) or F08BSF (ZGEQPF) F08AVFExample TextExample Data 16 LQ factorization of complex general rectangular matrix F08AWFExample TextExample Data 16 Form all or part of unitary Q from LQ factorization determined by F08AVF (ZGELQF) F08AXF 16 Apply unitary transformation determined by F08AVF (ZGELQF) F08BAFExample TextExample Data 21 Computes the minimum-norm solution to a real linear least-squares problem F08BEFExample TextExample Data 16 QR factorization of real general rectangular matrix with column pivoting F08BFFExample TextExample Data 21 QR factorization of real general rectangular matrix with column pivoting, using BLAS-3 F08BHFExample TextExample Data 21 Reduces a real upper trapezoidal matrix to upper triangular form F08BKF 21 Apply orthogonal transformation determined by F08BHF (DTZRZF) F08BNFExample TextExample Data 21 Computes the minimum-norm solution to a complex linear least-squares problem F08BSFExample TextExample Data 16 QR factorization of complex general rectangular matrix with column pivoting F08BTFExample TextExample Data 21 QR factorization of complex general rectangular matrix with column pivoting, using BLAS-3 F08BVFExample TextExample Data 21 Reduces a complex upper trapezoidal matrix to upper triangular form F08BXF 21 Apply unitary transformation determined by F08BVF (ZTZRZF) F08CEFExample TextExample Data 21 QL factorization of real general rectangular matrix F08CFFExample TextExample Data 21 Form all or part of orthogonal Q from QL factorization determined by F08CEF (DGEQLF) F08CGF 21 Apply orthogonal transformation determined by F08CEF (DGEQLF) F08CHFExample TextExample Data 21 RQ factorization of real general rectangular matrix F08CJFExample TextExample Data 21 Form all or part of orthogonal Q from RQ factorization determined by F08CHF (DGERQF) F08CKF 21 Apply orthogonal transformation determined by F08CHF (DGERQF) F08CSFExample TextExample Data 21 QL factorization of complex general rectangular matrix F08CTFExample TextExample Data 21 Form all or part of orthogonal Q from QL factorization determined by F08CSF (ZGEQLF) F08CUF 21 Apply unitary transformation determined by F08CSF (ZGEQLF) F08CVFExample TextExample Data 21 RQ factorization of complex general rectangular matrix F08CWFExample TextExample Data 21 Form all or part of orthogonal Q from RQ factorization determined by F08CVF (ZGERQF) F08CXF 21 Apply unitary transformation determined by F08CVF (ZGERQF) F08FAFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix F08FBFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix F08FCFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of real symmetric matrix (divide-and-conquer) F08FDFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix (Relatively Robust Representations) F08FEFExample TextExample Data 16 Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form F08FFFExample TextExample Data 16 Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08FEF (DSYTRD) F08FGFExample TextExample Data 16 Apply orthogonal transformation determined by F08FEF (DSYTRD) F08FLF 21 Computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex Hermitian matrix or for the left or right singular vectors of a general matrix F08FNFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix F08FPFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix F08FQFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of complex Hermitian matrix (divide-and-conquer) F08FRFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix (Relatively Robust Representations) F08FSFExample TextExample Data 16 Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form F08FTFExample TextExample Data 16 Generate unitary transformation matrix from reduction to tridiagonal form determined by F08FSF (ZHETRD) F08FUFExample TextExample Data 16 Apply unitary transformation matrix determined by F08FSF (ZHETRD) F08GAFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage F08GBFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix, packed storage F08GCFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of real symmetric matrix, packed storage (divide-and-conquer) F08GEFExample TextExample Data 16 Orthogonal reduction of real symmetric matrix to symmetric tridiagonal form, packed storage F08GFFExample TextExample Data 16 Generate orthogonal transformation matrix from reduction to tridiagonal form determined by F08GEF (DSPTRD) F08GGFExample TextExample Data 16 Apply orthogonal transformation determined by F08GEF (DSPTRD) F08GNFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage F08GPFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix, packed storage F08GQFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of complex Hermitian matrix, packed storage (divide-and-conquer) F08GSFExample TextExample Data 16 Unitary reduction of complex Hermitian matrix to real symmetric tridiagonal form, packed storage F08GTFExample TextExample Data 16 Generate unitary transformation matrix from reduction to tridiagonal form determined by F08GSF (ZHPTRD) F08GUFExample TextExample Data 16 Apply unitary transformation matrix determined by F08GSF (ZHPTRD) F08HAFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix F08HBFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix F08HCFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of real symmetric band matrix (divide-and-conquer) F08HEFExample TextExample Data 16 Orthogonal reduction of real symmetric band matrix to symmetric tridiagonal form F08HNFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix F08HPFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix F08HQFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of complex Hermitian band matrix (divide-and-conquer) F08HSFExample TextExample Data 16 Unitary reduction of complex Hermitian band matrix to real symmetric tridiagonal form F08JAFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix F08JBFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix F08JCFExample TextExample Data 19 All eigenvalues and optionally all eigenvectors of real symmetric tridiagonal matrix (divide-and-conquer) F08JDFExample TextExample Data 21 Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix (Relatively Robust Representations) F08JEFExample TextExample Data 16 All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from real symmetric matrix using implicit QL or QR F08JFFExample TextExample Data 16 All eigenvalues of real symmetric tridiagonal matrix, root-free variant of QL or QR F08JGFExample TextExample Data 16 All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from real symmetric positive-definite matrix F08JHFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a matrix reduced to this form (divide-and-conquer) F08JJF 16 Selected eigenvalues of real symmetric tridiagonal matrix by bisection F08JKF 16 Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in real array F08JLFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a symmetric matrix reduced to this form (Relatively Robust Representations) F08JSF 16 All eigenvalues and eigenvectors of real symmetric tridiagonal matrix, reduced from complex Hermitian matrix, using implicit QL or QR F08JUFExample TextExample Data 16 All eigenvalues and eigenvectors of real symmetric positive-definite tridiagonal matrix, reduced from complex Hermitian positive-definite matrix F08JVFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (divide-and-conquer) F08JXF 16 Selected eigenvectors of real symmetric tridiagonal matrix by inverse iteration, storing eigenvectors in complex array F08JYFExample TextExample Data 21 Computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix or a complex Hermitian matrix reduced to this form (Relatively Robust Representations) F08KAFExample TextExample Data 21 Computes the minimum-norm solution to a real linear least-squares problem using singular value decomposition F08KBFExample TextExample Data 21 Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors F08KCFExample TextExample Data 21 Computes the minimum-norm solution to a real linear least-squares problem using singular value decomposition (divide-and-conquer) F08KDFExample TextExample Data 21 Computes the singular value decomposition of a real matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) F08KEFExample TextExample Data 16 Orthogonal reduction of real general rectangular matrix to bidiagonal form F08KFFExample TextExample Data 16 Generate orthogonal transformation matrices from reduction to bidiagonal form determined by F08KEF (DGEBRD) F08KGFExample TextExample Data 16 Apply orthogonal transformations from reduction to bidiagonal form determined by F08KEF (DGEBRD) F08KNFExample TextExample Data 21 Computes the minimum-norm solution to a complex linear least-squares problem using singular value decomposition F08KPFExample TextExample Data 21 Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors F08KQFExample TextExample Data 21 Computes the minimum-norm solution to a complex linear least-squares problem using singular value decomposition (divide-and-conquer) F08KRFExample TextExample Data 21 Computes the singular value decomposition of a complex matrix, optionally computing the left and/or right singular vectors (divide-and-conquer) F08KSFExample TextExample Data 16 Unitary reduction of complex general rectangular matrix to bidiagonal form F08KTFExample TextExample Data 16 Generate unitary transformation matrices from reduction to bidiagonal form determined by F08KSF (ZGEBRD) F08KUFExample TextExample Data 16 Apply unitary transformations from reduction to bidiagonal form determined by F08KSF (ZGEBRD) F08LEFExample TextExample Data 19 Reduction of real rectangular band matrix to upper bidiagonal form F08LSFExample TextExample Data 19 Reduction of complex rectangular band matrix to upper bidiagonal form F08MDFExample TextExample Data 21 Computes the singular value decomposition of a real bidiagonal matrix, optionally computing the singular vectors (divide-and-conquer) F08MEFExample TextExample Data 16 SVD of real bidiagonal matrix reduced from real general matrix F08MSF 16 SVD of real bidiagonal matrix reduced from complex general matrix F08NAFExample TextExample Data 21 Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix F08NBFExample TextExample Data 21 Computes all eigenvalues and, optionally, left and/or right eigenvectors of a real nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08NEFExample TextExample Data 16 Orthogonal reduction of real general matrix to upper Hessenberg form F08NFFExample TextExample Data 16 Generate orthogonal transformation matrix from reduction to Hessenberg form determined by F08NEF (DGEHRD) F08NGFExample TextExample Data 16 Apply orthogonal transformation matrix from reduction to Hessenberg form determined by F08NEF (DGEHRD) F08NHFExample TextExample Data 16 Balance real general matrix F08NJF 16 Transform eigenvectors of real balanced matrix to those of original matrix supplied to F08NHF (DGEBAL) F08NNFExample TextExample Data 21 Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix F08NPFExample TextExample Data 21 Computes all eigenvalues and, optionally, left and/or right eigenvectors of a complex nonsymmetric matrix; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08NSFExample TextExample Data 16 Unitary reduction of complex general matrix to upper Hessenberg form F08NTFExample TextExample Data 16 Generate unitary transformation matrix from reduction to Hessenberg form determined by F08NSF (ZGEHRD) F08NUFExample TextExample Data 16 Apply unitary transformation matrix from reduction to Hessenberg form determined by F08NSF (ZGEHRD) F08NVFExample TextExample Data 16 Balance complex general matrix F08NWF 16 Transform eigenvectors of complex balanced matrix to those of original matrix supplied to F08NVF (ZGEBAL) F08PAFExample TextExample Data 21 Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors F08PBFExample TextExample Data 21 Computes for real square nonsymmetric matrix, the eigenvalues, the real Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08PEFExample TextExample Data 16 Eigenvalues and Schur factorization of real upper Hessenberg matrix reduced from real general matrix F08PKF 16 Selected right and/or left eigenvectors of real upper Hessenberg matrix by inverse iteration F08PNFExample TextExample Data 21 Computes for complex square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors F08PPFExample TextExample Data 21 Computes for real square nonsymmetric matrix, the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08PSFExample TextExample Data 16 Eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix F08PXF 16 Selected right and/or left eigenvectors of complex upper Hessenberg matrix by inverse iteration F08QFFExample TextExample Data 16 Reorder Schur factorization of real matrix using orthogonal similarity transformation F08QGFExample TextExample Data 16 Reorder Schur factorization of real matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities F08QHFExample TextExample Data 16 Solve real Sylvester matrix equation AX+XB=C, A and B are upper quasi-triangular or transposes F08QKF 16 Left and right eigenvectors of real upper quasi-triangular matrix F08QLFExample TextExample Data 16 Estimates of sensitivities of selected eigenvalues and eigenvectors of real upper quasi-triangular matrix F08QTFExample TextExample Data 16 Reorder Schur factorization of complex matrix using unitary similarity transformation F08QUFExample TextExample Data 16 Reorder Schur factorization of complex matrix, form orthonormal basis of right invariant subspace for selected eigenvalues, with estimates of sensitivities F08QVFExample TextExample Data 16 Solve complex Sylvester matrix equation AX+XB=C, A and B are upper triangular or conjugate-transposes F08QXF 16 Left and right eigenvectors of complex upper triangular matrix F08QYFExample TextExample Data 16 Estimates of sensitivities of selected eigenvalues and eigenvectors of complex upper triangular matrix F08SAFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem F08SBFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem F08SCFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem (divide-and-conquer) F08SEFExample TextExample Data 16 Reduction to standard form of real symmetric-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, B factorized by F07FDF (DPOTRF) F08SNFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem F08SPFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem F08SQFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem (divide-and-conquer) F08SSFExample TextExample Data 16 Reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, B factorized by F07FRF (ZPOTRF) F08TAFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage F08TBFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage F08TCFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, packed storage (divide-and-conquer) F08TEFExample TextExample Data 16 Reduction to standard form of real symmetric-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, packed storage, B factorized by F07GDF (DPPTRF) F08TNFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage F08TPFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage F08TQFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, packed storage (divide-and-conquer) F08TSFExample TextExample Data 16 Reduction to standard form of complex Hermitian-definite generalized eigenproblem Ax=λBx, ABx=λx or BAx=λx, packed storage, B factorized by F07GRF (ZPPTRF) F08UAFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem F08UBFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem F08UCFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a real banded generalized symmetric-definite eigenproblem (divide-and-conquer) F08UEFExample TextExample Data 19 Reduction of real symmetric-definite banded generalized eigenproblem Ax=λBx to standard form Cy=λy, such that C has the same bandwidth as A F08UFF 19 Computes a split Cholesky factorization of real symmetric positive-definite band matrix A F08UNFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem F08UPFExample TextExample Data 21 Computes selected eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem F08UQFExample TextExample Data 21 Computes all the eigenvalues, and optionally, the eigenvectors of a complex banded generalized Hermitian-definite eigenproblem (divide-and-conquer) F08USFExample TextExample Data 19 Reduction of complex Hermitian-definite banded generalized eigenproblem Ax=λBx to standard form Cy=λ y, such that C has the same bandwidth as A F08UTF 19 Computes a split Cholesky factorization of complex Hermitian positive-definite band matrix A F08VAFExample TextExample Data 21 Computes the generalized singular value decomposition of a real matrix pair F08VEFExample TextExample Data 21 Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a real matrix pair F08VNFExample TextExample Data 21 Computes the generalized singular value decomposition of a complex matrix pair F08VSFExample TextExample Data 21 Computes orthogonal matrices as processing steps for computing the generalized singular value decomposition of a complex matrix pair F08WAFExample TextExample Data 21 Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors F08WBFExample TextExample Data 21 Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08WEF 20 Orthogonal reduction of a pair of real general matrices to generalized upper Hessenberg form F08WHF 20 Balance a pair of real general matrices F08WJF 20 Transform eigenvectors of a pair of real balanced matrices to those of original matrix pair supplied to F08WHF (DGGBAL) F08WNFExample TextExample Data 21 Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors F08WPFExample TextExample Data 21 Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors; also, optionally, the balancing transformation, the reciprocal condition numbers for the eigenvalues and for the right eigenvectors F08WSF 20 Unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form F08WVF 20 Balance a pair of complex general matrices F08WWF 20 Transform eigenvectors of a pair of complex balanced matrices to those of original matrix pair supplied to F08WVF (ZGGBAL) F08XAFExample TextExample Data 21 Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors F08XBFExample TextExample Data 21 Computes, for a real nonsymmetric matrix pair, the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08XEFExample TextExample Data 20 Eigenvalues and generalized Schur factorization of real generalized upper Hessenberg form reduced from a pair of real general matrices F08XNFExample TextExample Data 21 Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors F08XPFExample TextExample Data 21 Computes, for a complex nonsymmetric matrix pair, the generalized eigenvalues, the generalized complex Schur form and, optionally, the left and/or right matrices of Schur vectors; also, optionally, computes reciprocal condition numbers for selected eigenvalues F08XSFExample TextExample Data 20 Eigenvalues and generalized Schur factorization of complex generalized upper Hessenberg form reduced from a pair of complex general matrices F08YEFExample TextExample Data 21 Computes the generalized singular value decomposition of a real upper triangular (or trapezoidal) matrix pair F08YFFExample TextExample Data 21 Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation F08YGFExample TextExample Data 21 Reorders the generalized real Schur decomposition of a real matrix pair using an orthogonal equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces F08YHFExample TextExample Data 21 Solves the real-valued generalized Sylvester equation F08YKFExample TextExample Data 20 Left and right eigenvectors of a pair of real upper quasi-triangular matrices F08YLFExample TextExample Data 21 Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a real matrix pair in generalized real Schur canonical form F08YSFExample TextExample Data 21 Computes the generalized singular value decomposition of a complex upper triangular (or trapezoidal) matrix pair F08YTFExample TextExample Data 21 Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation F08YUFExample TextExample Data 21 Reorders the generalized Schur decomposition of a complex matrix pair using an unitary equivalence transformation, computes the generalized eigenvalues of the reordered pair and, optionally, computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces F08YVFExample TextExample Data 21 Solves the complex generalized Sylvester equation F08YXFExample TextExample Data 20 Left and right eigenvectors of a pair of complex upper triangular matrices F08YYFExample TextExample Data 21 Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a complex matrix pair in generalized Schur canonical form F08ZAFExample TextExample Data 21 Solves the real linear equality-constrained least-squares (LSE) problem F08ZBFExample TextExample Data 21 Solves a real general Gauss–Markov linear model (GLM) problem F08ZEFExample TextExample Data 21 Computes a generalized QR factorization of a real matrix pair F08ZFFExample TextExample Data 21 Computes a generalized RQ factorization of a real matrix pair F08ZNFExample TextExample Data 21 Solves the complex linear equality-constrained least-squares (LSE) problem F08ZPFExample TextExample Data 21 Solves a complex general Gauss–Markov linear model (GLM) problem F08ZSFExample TextExample Data 21 Computes a generalized QR factorization of a complex matrix pair F08ZTFExample TextExample Data 21 Computes a generalized RQ factorization of a complex matrix pair