LAPACK (Linear Algebra Package) is a new library of dense linear and eigenproblem solvers that supercedes LINPACK and EISPACK, offering better performance and accuracy. CXML includes a compiled and optimized version of LAPACK. LAPACK includes subroutines for solving the most common problems in numerical linear algebra: • Solving systems of simultaneous liner equations • Finding least squares solutions of overdetermined systems of equations • Solving eigenvalue problems • Solving singular value problems The extensive functionality provided by LAPACK includes routines for the following matrix factorizations: • LU • Cholesky • QR • SVD • Schur • Generalized Schur Where appropriate, these functions are provided for the following matrices: • General • General band • General tridiagonal • Symmetric • Symmetric band • Symmetric tridiagonal • Symmetric, packed storage • Symmetric positive definite • Symmetric positive definite band • Symmetric positive definite, tridiagonal • Triangular • Triangular band • Triangular, packed storage LAPACK extends the functionality of LINPACK and EISPACK by including equilibration, iterative refinement, error bounds, and driver routines for linear systems, routines for computing and re-ordering the Schur factorization, and condition estimation routines for eigenvalue problems. LAPACK improves on the accuracy of the standard algorithms in EISPACK by including high accuracy algorithms for finding singular values and eigenvalues of bidiagonal and tridiagonal matrices respectively that arise in SVD and symmetric eigenvalue problems. The performance of the public-domain LAPACK routines on Alpha platforms is improved through the use of the optimized BLAS subprograms. EQUIVALENCE BETWEEN LAPACK AND LINPACK/EISPACK ROUTINES: The LAPACK equivalence utility provides the names and parameter lists of LAPACK routines that are equivalent to the LINPACK and EISPACK routines you specify. The utility command is as follows: /usr/share/equivalence_lapack routine_name [routine_name...] where you replace routine_name with the LINPACK and/or EISPACK routine names. For example: /usr/share/equivalence_lapack dgesl imtql1 return: DGESL: SUBROUTINE SGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO ) SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO ) IMTQL1: SUBROUTINE SSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) SUBROUTINE DSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) The LINPACK or EISPACK routine names are to the left of the colons. The equivalent LAPACK routines and calling sequences are to the right of the colons. This utility helps you to convert LINPACK and EISPACK routine calls to equivalent LAPACK routine calls. The utility has limitations in that the argument lists of the LAPACK routines are generally different from those of the corresponding LINPACK and EISPACK routines, and the workspace requirements are often different as well. NAMING SCHEME: The name of each LAPACK routine is a coded specification of its function (within the very tight limits of standard Fortran 77 6-character names). All driver and computational routines have names of the form XYYZZZ, where for some driver routines the 6th character is blank. The first letter, X, indicates the data type as follows: S REAL D DOUBLE PRECISION C COMPLEX Z COMPLEX*16 or DOUBLE COMPLEX The next two letters, YY, indicate the type of matrix (or of the most significant matrix). Most of these two-letter codes apply to both real and complex matrices; a few apply specifically to one or the other. The last three letters ZZZ indicate the computation performed. For example, SGEBRD is a single precision routine that performs a bidiagonal reduction (BRD) of a real general matrix. LIST OF ROUTINES: LAPACK includes both computational routines that perform a distinct algorithmic task (such as performing an LU factorization) as well as driver routines that solve a complete problem (such as solving a system of linear equations). The driver routines (simple and expert) are listed first, followed by the computational routines. Auxiliary routines from LAPACK are not listed. The Subprogram Name is the name of the manual page containing documentation on the subprogram. Available SIMPLE DRIVER routines: Subprogram Name Operation sgesv Solves a general system of linear equations dgesv AX=B. cgesv zgesv sgbsv Solves a general banded system of linear dgbsv equations AX=B. cgbsv zgbsv sgtsv Solves a general tridiagonal system of linear dgtsv equations AX=B. cgtsv zgtsv sposv Solves a symmetric/Hermitian positive definite dposv system of linear equations AX=B. cposv zposv sppsv Solves a symmetric/Hermitian positive definite dppsv system of linear equations AX=B, where A is held cppsv in packed storage. zppsv spbsv Solves a symmetric/Hermitian positive definite dpbsv banded system of linear equations AX=B. cpbsv zpbsv sptsv Solves a symmetric/Hermitian positive definite dptsv tridiagonal system of linear equations AX=B. cptsv zptsv ssysv Solves a real/complex/complex symmetric/symmetric/ dsysv Hermitian indefinite system of linear equations csysv AX=B. zsysv chesv zhesv sspsv Solves a real/complex/complex symmetric/symmetric/ dspsv Hermitian indefinite system of linear equations cspsv AX=B, where A is held in packed storage. zspsv chpsv zhpsv sgels Computes the least squares solution to an over- dgels determined system of linear equations, A X=B or cgels A**H X=B, or the minimum norm solution of an zgels under-determined system, where A is a general rectangular matrix of full rank, using a QR or LQ factorization of A. sgelss Computes the minimum norm least squares solution dgelss to an over- or under-determined system of linear cgelss equations A X=B, using the singular value zgelss decomposition of A. sgglse Solves the LSE (Constrained Linear Least Squares dgglse Problem) using the GRQ (Generalized RQ) cgglse factorization zgglse sggglm Solves the GLM (Generalized Linear Regression dggglm Model) using the GQR (Generalized QR) cggglm factorization zggglm ssyev Computes all eigenvalues and eigenvectors of a dsyev symmetric/Hermitian matrix. cheev zheev ssyevd Computes all eigenvalues and eigenvectors of a dsyevd symmetric/Hermitian matrix, using a divide and cheevd conquer algorithm. zheevd sspev Computes all eigenvalues and eigenvectors of a dspev symmetric/Hermitian matrix in packed storage. chpev zhpev sspevd Computes all eigenvalues and eigenvectors of a dspevd symmetric/Hermitian matrix in packed storage, chpevd using a divide and conquer algorithm. zhpevd ssbev Computes all eigenvalues and eigenvectors of a dsbev symmetric/Hermitian band matrix. chbev zhbev ssbevd Computes all eigenvalues and eigenvectors of a dsbevd symmetric/Hermitian band matrix, using a divide chbevd and conquer algorithm. zhbevd ssbgv Computes all eigenvalues and eigenvectors of a dsbgv symmetric/Hermitian-definite band matrix. chbgv zhbgv sstev Computes all eigenvalues and eigenvectors of a dstev real symmetric tridiagonal matrix. sstevd Computes all eigenvalues and eigenvectors of a dstevd real symmetric tridiagonal matrix, using a devide and conquer algorithm. sgees Computes the eigenvalues and Schur factorization dgees of a general matrix, and orders the factorization cgees so that selected eigenvalues are at the top left zgees of the Schur form. sgeev Computes the eigenvalues and left and right dgeev eigenvectors of a general matrix cgeev zgeev sgesvd Computes the singular value decomposition (SVD) dgesvd of a general rectangular matrix. cgesvd zgesvd ssygv Computes all eigenvalues and the eigenvectors dsygv of a generalized symmetric/Hermitian-definite chegv generalized eigenproblem, Ax= lambda Bx, ABx= zhegv lambda x, or BAx= lambda x. sspgv Computes all eigenvalues and eigenvectors of a dspgv generalized symmetric/Hermitian-definite generalized chpgv eigenproblem, Ax = lambda Bx, ABx= lambda x, or zhpgv BAx= lambda x, where A and B are in packed storage. sgegs Computes the generalized eigenvalues, Schur form, dgegs and left and/or right Schur vectors for a pair of cgegs nonsymmetric matrices zgegs sgegv Computes the generalized eigenvalues, and left dgegv and/or right generalized eigenvectors for a pair of cgegv nonsymmetric matrices zgegv sggsvd Computes the Generalized Singular Value dggsvd Decomposition cggsvd zggsvd Available EXPERT DRIVER routines: Subprogram Name Operation sgesvx Solves a general system of linear equations AX=B, dgesvx A**T X=B or A**H X=B, and provides an estimate of cgesvx the condition number and error bounds on the zgesvx solution. sgbsvx Solves a general banded system of linear equations dgbsvx AX=B, A**T X=B or A**H X=B, and provides an cgbsvx estimate of the condition number and error bounds zgbsvx on the solution. sgtsvx Solves a general tridiagonal system of linear dgtsvx equations AX=B, A**T X=B or A**H X=B, and provides cgtsvx an estimate of the condition number and error zgtsvx bounds on the solution. sposvx Solves a symmetric/Hermitian positive definite dposvx system of linear equations AX=B, and provides cposvx an estimate of the condition number and error zposvx bounds on the asolution. sppsvx Solves a symmetric/Hermitian positive definite dppsvx system of linear equations AX=B, where A is held cppsvx in packed storage, and provides an estimate of the zppsvx condition number and error bounds on the solution. spbsvx Solves a symmetric/Hermitian positive definite dpbsvx banded system of linear equations AX=B, and provides cpbsvx an estimate of the condition number and error bounds zpbsvx on the solution. sptsvx Solves a symmetric/Hermitian positive definite dptsvx tridiagonal system of linear equations AX=B, and cptsvx provides an estimate of the condition number and zptsvx error bounds on the solution. ssysvx Solves a real/complex/complex symmetric/symmetric/ dsysvx Hermitian indefinite system of linear equations csysvx AX=B, and provides an estimate of the condition zsysvx number and error bounds on the solution. chesvx zhesvx sspsvx Solves a real/complex/complex symmetric/symmetric/ dspsvx Hermitian indefinite system of linear equations AX=B, cspsvx where A is held in packed storage, and provides an zspsvx estimate of the condition number and error bounds on chpsvx the solution. zhpsvx sgelsx Computes the minimum norm least squares solution dgelsx to an over- or under-determined system of linear cgelsx equations A X=B, using a complete orthogonal zgelsx factorization of A. ssyevx Computes selected eigenvalues and eigenvectors of a dsyevx symmetric/Hermitian matrix. cheevx zheevx sspevx Computes selected eigenvalues and eigenvectors of a dspevx symmetric/Hermitian matrix in packed storage. chpevx zhpevx ssbevx Computes selected eigenvalues and eigenvectors of a dsbevx symmetric/Hermitian band matrix. chbevx zhbevx sstevx Computes selected eigenvalues and eigenvectors of a dstevx real symmetric tridiagonal matrix. sgeesx Computes the eigenvalues and Schur factorization of dgeesx a general matrix, orders the factorization so that cgeesx selected eigenvalues are at the top left of the zgeesx Schur form, and computes reciprocal condition numbers for the average of the selected eigenvalues, and for the associated right invariant subspace. sgeevx Computes the eigenvalues and left and right eigen- dgeevx vectors of a general matrix, with preliminary cgeevx balancing of the matrix, and computes reciprocal zgeevx condition numbers for the eigenvalues and right eigenvectors. Available COMPUTATIONAL routines: Subprogram Name Operation sbdsqr Computes the singular value decomposition dbdsqr (SVD) of a real bidiagonal matrix, using cbdsqr the bidiagonal QR algorithm. zbdsqr sgbcon Estimates the reciprocal of the condition dgbcon number of a general band matrix, in either cgbcon the 1-norm or the infinity-norm, using zgbcon the LU factorization computed by SGBTRF/CGBTRF. sgbequ Computes row and column scalings to dgbequ equilibrate a general band matrix and reduce cgbequ its condition number. zgbequ sgbrfs Improves the computed solution to a dgbrfs general banded system of linear equations cgbrfs AX=B, A**T X=B or A**H X=B, and provides zgbrfs forward and backward error bounds for the solution. sgbtrf Computes an LU factorization of a general dgbtrf band matrix, using partial pivoting with cgbtrf row interchanges. zgbtrf sgbtrs Solves a general banded system of linear dgbtrs equations AX=B, A**T X=B or A**H X=B, using cgbtrs the LU factorization computed by zgbtrs SGBTRF/CGBTRF. sgebak Transforms eigenvectors of a balanced dgebak matrix to those of the original matrix cgebak supplied to SGEBAL/CGEBAL. zgebak sgebal Balances a general matrix in order to dgebal improve the accuracy of computed cgebal eigenvalues. zgebal sgebrd Reduces a general rectangular matrix to dgebrd real bidiagonal form by an orthogonal/ cgebrd unitary transformation. zgebrd sgbbrd Reduces a general rectangular banded matrix dgbbrd to real bidiagonal form by an orthogonal/ cgbbrd unitary transformation. zgbbrd sgecon Estimates the reciprocal of the condition dgecon number of a general matrix, in either the cgecon 1-norm or the infinity-norm, using the zgecon LU factorization computed by SGETRF/CGETRF. sgeequ Computes row and column scalings to dgeequ equilibrate a general rectangular matrix cgeequ and reduce its condition number. zgeequ sgehrd Reduces a general matrix to upper dgehrd Hessenberg form by an orthogonal/unitary cgehrd similarity transformation. zgehrd sgelqf Computes an LQ factorization of a general dgelqf rectangular matrix. cgelqf zgelqf sgeqlf Computes a QL factorization of a general dgeqlf rectangular matrix. cgeqlf zgeqlf sgeqpf Computes a QR factorization with column dgeqpf pivoting of a general rectangular matrix. cgeqpf zgeqpf sgeqrf Computes a QR factorization of a general dgeqrf rectangular matrix. cgeqrf zgeqrf sgerfs Improves the computed solution to a dgerfs general system of linear equations AX=B, cgerfs A**T X=B or A**H X=B, and provides forward zgerfs and backward error bounds for the solution. sgerqf Computes an RQ factorization of a dgerqf general rectangular matrix. cgerqf zgerqf sgetrf Computes an LU factorization of a dgetrf general matrix, using partial pivoting cgetrf with row interchanges. zgetrf sgetri Computes the inverse of a general matrix, dgetri using the LU factorization computed by cgetri SGETRF/CGETRF. zgetri sgetrs Solves a general system of linear dgetrs equations AX=B, A**T X=B or A**H X=B, cgetrs using the LU factorization computed by zgetrs SGETRF/CGETRF. sggbak Forms the right or left eigenvectors dggbak of the generalized eigenvalue problem cggbak by backward transformation on the zggbak computed eigenvectors of the balanced matrix output by xGGBAL. sggbal Balances a pair of general real/complex dggbal matrices for the generalized eigenvalue cggbal problem A x = lambda B x. zggbal sgghrd Reduces a pair of real/complex matrices dgghrd to generalized upper Hessenberg form cgghrd using orthogonal/unitary similarity zgghrd transformations sggsvp Computes orthogonal/unitary matrices dggsvp as a preprocessing step for computing cggsvp the generalized singular value zggsvp decomposition sgtcon Estimates the reciprocal of the dgtcon condition number of a general tridiagonal cgtcon matrix, in either the 1-norm or the zgtcon infinity-norm, using the LU factorization computed by SGTTRF/CGTTRF. sgtrfs Improves the computed solution to a dgtrfs general tridiagonal system of linear cgtrfs equations AX=B, A**T X=B or A**H X=B, zgtrfs and providesforward and backward error bounds for the solution. sgttrf Computes an LU factorization of a general dgttrf tridiagonal matrix, using partial cgttrf pivoting with row interchanges. zgttrf sgttrs Solves a general tridiagonal system of dgttrs linear equations AX=B, A**T X=B or cgttrs A**H X=B, using the LU factorization zgttrs computed by SGTTRF/CGTTRF. shgeqz Implements a single-/double-shift dhgeqz version of the QZ method for finding chgeqz the generalized eigenvalues of the equation zhgeqz det(A - w(i) B) = 0 shsein Computes specified right and/or left dhsein eigenvectors of an upper Hessenberg chsein matrix by inverse iteration. zhsein shseqr Computes the eigenvalues and Schur dhseqr factorization of an upper Hessenberg chseqr matrix, using the multishift QR algorithm. zhseqr sopgtr Generates the orthogonal/unitary dopgtr transformation matrix from a reduction cupgtr to tridiagonal form determined by zupgtr SSPTRD/CHPTRD. sopmtr Multiplies a general matrix by the dopmtr orthogonal/unitary transformation matrix cupmtr from a reduction to tridiagonal form zupmtr determined by SSPTRD/CHPTRD. sorgbr Generates the orthogonal/unitary dorgbr transformation matrices from a reduction cungbr to bidiagonal form determined by SGEBRD/CGEBRD. zungbr sorghr Generates the orthogonal/unitary dorghr transformation matrix from a reduction cunghr to Hessenberg form determined by SGEHRD/CGEHRD. zunghr sorglq Generates all or part of the orthogonal/ dorglq unitary matrix Q from an LQ factorization cunglq determined by SGELQF/CGELQF. zunglq sorgql Generates all or part of the orthogonal/ dorgql unitary matrix Q from a QL factorization cungql determined by SGEQLF/CGEQLF. zungql sorgqr Generates all or part of the orthogonal/ dorgqr unitary matrix Q from a QR factorization cungqr determined by SGEQRF/CGEQRF. zungqr sorgrq Generates all or part of the dorgrq orthogonal/unitary matrix Q from an RQ cungrq factorization determined by SGERQF/CGERQF. zungrq sorgtr Generates the orthogonal/unitary dorgtr transformation matrix from a reduction cungtr to tridiagonal form determined by zungtr SSYTRD/CHETRD. sormbr Multiplies a general matrix by one of dormbr the orthogonal/unitary transformation cunmbr matrices from a reduction to bidiagonal form zunmbr determined by SGEBRD/CGEBRD. sormhr Multiplies a general matrix by the dormhr orthogonal/unitary transformation matrix cunmhr from a reduction to Hessenberg form zunmhr determined by SGEHRD/CGEHRD. sormlq Multiplies a general matrix by the dormlq orthogonal/unitary matrix from an LQ cunmlq factorization determined by SGELQF/CGELQF. zunmlq sormql Multiplies a general matrix by the dormql orthogonal/unitary matrix from a QL cunmql factorization determined by SGEQLF/CGEQLF. zunmql sormqr Multiplies a general matrix by the dormqr orthogonal/unitary matrix from a QR cunmqr factorization determined by SGEQRF/CGEQRF. zunmqr sormrq Multiplies a general matrix by the dormrq orthogonal/unitary matrix from an RQ cunmrq factorization determined by SGERQF/CGERQF. zunmrq sormtr Multiplies a general matrix by the dormtr orthogonal/unitary transformation matrix cunmtr from a reduction to tridiagonal form zunmtr determined by SSYTRD/CHETRD. spbcon Estimates the reciprocal of the condition dpbcon number of a symmetric/Hermitian positive cpbcon definite band matrix, using the Cholesky zpbcon factorization computed by SPBTRF/CPBTRF. spbequ Computes row and column scalings to dpbequ equilibrate a symmetric/Hermitian positive cpbequ definite band matrix and reduce its condition zpbequ number. spbrfs Improves the computed solution to a dpbrfs symmetric/Hermitian positive definite banded cpbrfs system of linear equations AX=B, and provides zpbrfs forward and backward error bounds for the solution. spbtrf Computes the Cholesky factorization of a dpbtrf symmetric/Hermitian positive definite band cpbtrf matrix. zpbtrf spbtrs Solves a symmetric/Hermitian positive dpbtrs definite banded system of linear equations cpbtrs AX=B, using the Cholesky factorization zpbtrs computed by SPBTRF/CPBTRF. spocon Estimates the reciprocal of the condition dpocon number of a symmetric/Hermitian positive cpocon definite matrix, using the Cholesky zpocon factorization computed by SPOTRF/CPOTRF. spoequ Computes row and column scalings to equilibrate dpoequ a symmetric/Hermitian positive definite matrix cpoequ and reduce its condition number. zpoequ sporfs Improves the computed solution to a dporfs symmetric/Hermitian positive definite system cporfs of linear equations AX=B, and provides forward zporfs and backward error bounds for the solution. spotrf Computes the Cholesky factorization of a dpotrf symmetric/Hermitian positive definite matrix. cpotrf zpotrf spotri Computes the inverse of a symmetric/Hermitian dpotri positive definite matrix, using the Cholesky cpotri factorization computed by SPOTRF/CPOTRF. zpotri spotrs Solves a symmetric/Hermitian positive definite dpotrs system of linear equations AX=B, using the cpotrs Cholesky factorization computed by SPOTRF/CPOTRF. zpotrs sppcon Estimates the reciprocal of the condition dppcon number of a symmetric/Hermitian positive cppcon definite matrix in packed storage, using the zppcon Cholesky factorization computed by SPPTRF/CPPTRF. sppequ computes row and column scalings to dppequ equilibrate a symmetric/hermitian positive cppequ definite matrix in packed storage and reduce zppequ its condition number. spprfs Improves the computed solution to a symmetric/ dpprfs Hermitian positive definite system of linear cpprfs equations AX=B, where A is held in packed storage, zpprfs and provides forward and backward error bounds for the solution. spptrf Computes the Cholesky factorization of a dpptrf symmetric/Hermitian positive definite matrix cpptrf in packed storage. zpptrf spbstf Computes the Cholesky factorization of a dpbstf symmetric/Hermitian positive definite matrix cpbstf in banded storage. zpbstf spptri Computes the inverse of a symmetric/ dpptri Hermitian positive definite matrix in packed cpptri storage, using the Cholesky factorization computed zpptri by SPPTRF/CPPTRF. spptrs Solves a symmetric/Hermitian positive definite dpptrs system of linear equations AX=B, where A is held cpptrs in packed storage, using the Cholesky factorization zpptrs computed by SPPTRF/CPPTRF. sptcon Computes the reciprocal of the condition dptcon number of a symmetric/Hermitian positive cptcon definite tridiagonal matrix, using the LDL**H zptcon factorization computed by SPTTRF/CPTTRF. spteqr Computes all eigenvalues and eigenvectors dpteqr of a real symmetric positive definite cpteqr tridiagonal matrix, by computing the SVD of zpteqr its bidiagonal Cholesky factor. sptrfs Improves the computed solution to a dptrfs symmetric/Hermitian positive definite cptrfs tridiagonal system of linear equations AX=B, zptrfs and provides forward and backward error bounds for the solution. spttrf Computes the LDL**H factorization of a dpttrf symmetric/Hermitian positive definite cpttrf tridiagonal matrix. zpttrf spttrs Solves a symmetric/Hermitian positive definite dpttrs tridiagonal system of linear equations, using cpttrs the LDL**H factorization computed by SPTTRF/CPTTRF. zpttrs ssbtrd Reduces a symmetric/Hermitian band matrix to dsbtrd real symmetric tridiagonal form by an orthogonal/ chbtrd unitary similarity transformation. zhbtrd sspcon Estimates the reciprocal of the condition dspcon number of a real/complex/complex symmetric/ cspcon symmetric/Hermitian indefinite matrix in packed zspcon storage, using the factorization computed by chpcon SSPTRF/CSPTRF/CHPTRF. zhpcon sspgst Reduces a symmetric/Hermitian-definite dspgst generalized eigenproblem Ax= lambda Bx, chpgst ABx= lambda x, or BAx= lambda x, to standard zhpgst form, where A and B are held in packed storage, and B has been factorized by SPPTRF/CPPTRF. ssbgst Reduces a symmetric/Hermitian-definite dsbgst generalized eigenproblem Ax= lambda Bx, chbgst ABx= lambda x, or BAx= lambda x, to standard zhbgst form, where A and B are held in banded storage, and B has been factorized by SPBSTF/CPBSTF. ssprfs Improves the computed solution to a real/ dsprfs complex/complex symmetric/symmetric/Hermitian csprfs indefinite system of linear equations AX=B, zsprfs where A is held in packed storage, and provides chprfs forward and backward error bounds for the solution. zhprfs ssptrd Reduces a symmetric/Hermitian matrix in packed dsptrd storage to real symmetric tridiagonal form by chptrd an orthogonal/unitary similarity transformation. zhptrd ssbtrd Reduces a symmetric/Hermitian matrix in banded dsbtrd storage to real symmetric tridiagonal form by chbtrd an orthogonal/unitary similarity transformation. zhbtrd ssptrf Computes the factorization of a real/complex/ dsptrf complex symmetric/symmetric/Hermitian-indefinite csptrf matrix in packed storage, using the diagonal zsptrf pivoting method. chptrf zhptrf ssptri Computes the inverse of a real symmetric/ dsptri complex symmetric/complex Hermitian indefinite csptri matrix in packed storage, using the factorization zsptri computed by SSPTRF/CSPTRF/CHPTRF. chptri zhptri ssptrs Solves a real/complex/complex symmetric/ dsptrs symmetric/Hermitian indefinite system of linear csptrs equations AX=B, where A is held in packed zsptrs storage, using the factorization computed chptrs by SSPTRF/CSPTRF/CHPTRF. zhptrs sstebz Computes selected eigenvalues of a real symmetric dstebz tridiagonal matrix by bisection. sstein Computes selected eigenvectors of a real dstein symmetric tridiagonal matrix by inverse iteration. cstein zstein ssteqr Computes all eigenvalues and eigenvectors of dsteqr a real symmetric tridiagonal matrix, using csteqr the implicit QL or QR algorithm. zsteqr ssterf Computes all eigenvalues of a real symmetric dsterf tridiagonal matrix, using a root-free variant of the QL or QR algorithm. ssycon Estimates the reciprocal of the condition number dsycon of a real/complex/complex symmetric/symmetric/ csycon Hermitian indefinite matrix, using the factor- zsycon ization computed by SSYTRF/CSYTRF/CHETRF. checon zhecon ssygst Reduces a symmetric/Hermitian-definite generalized dsygst eigenproblem Ax= lambda Bx, ABx= lambda x, or chegst BAx= lambda x, to standard form, where B has been zhegst factorized by SPOTRF/CPOTRF. ssyrfs Improves the computed solution to a real/complex/ dsyrfs complexsymmetric/symmetric/Hermitian indefinite csyrfs system of linear equations AX=B, and provides zsyrfs forward and backward error bounds for the cherfs solution. zherfs ssytrd Reduces a symmetric/Hermitian matrix to real dsytrd symmetric tridiagonal form by an orthogonal/ chetrd unitary similarity transformation. zhetrd ssytrf Computes the factorization of a real symmetric/ dsytrf complex symmetric/complex Hermitian-indefinite csytrf matrix, using the diagonal pivoting method. zsytrf chetrf zhetrf ssytri Computes the inverse of a real/complex/complex dsytri symmetric/symmetric/Hermitian indefinite matrix, csytri using the factorization computed by SSYTRF/CSYTRF/ zsytri CHETRF. chetri zhetri ssytrs Solves a real/complex/complex symmetric/ dsytrs symmetric/Hermitian indefinite system of csytrs linear equations AX=B, using the factorization zsytrs computed by SSPTRF/CSPTRF/CHPTRF. chetrs zhetrs stbcon Estimates the reciprocal of the condition dtbcon number of a triangular band matrix, in either ctbcon the 1-norm or the infinity-norm. ztbcon stbrfs Provides forward and backward error bounds dtbrfs for the solution of a triangular banded system ctbrfs of linear equations AX=B, A**T X=B or A**H X=B. ztbrfs stbtrs Solves a triangular banded system of linear dtbtrs equations AX=B, A**T X=B or A**H X=B. ctbtrs ztbtrs stgevc Computes selected left and/or right dtgevc generalized eigenvectors of a pair of ctgevc real/complex upper triangular matrices. ztgevc stgsja Computes the generalized singular value dtgsja decomposition of two real/complex upper ctgsja "triangular (or trapezoidal)" matrices as ztgsja output by xGGSVP. stpcon Estimates the reciprocal of the condition dtpcon number of a triangular matrix in packed ctpcon storage, in either the 1-norm or the infinity- ztpcon norm. stprfs Provides forward and backward error bounds dtprfs for the solution of a triangular system of ctprfs linear equations AX=B, A**T X=B or A**H X=B, ztprfs where A is held in packed storage. stptri Computes the inverse of a triangular matrix dtptri in packed storage. ctptri ztptri stptrs Solves a triangular system of linear equations dtptrs AX=B, A**T X=B or A**H X=B, where A is held in ctptrs packed storage. ztptrs strcon Estimates the reciprocal of the condition dtrcon number of a triangular matrix, in either the ctrcon 1-norm or the infinity-norm. ztrcon strevc Computes left and right eigenvectors of an dtrevc upper quasi-triangular/triangular matrix. ctrevc ztrevc strexc Reorders the Schur factorization of a matrix dtrexc by a unitary similarity transformation. ctrexc ztrexc strrfs Provides forward and backward error bounds dtrrfs for the solution of a triangular system of ctrrfs linear equations A X=B, A**T X=B or ztrrfs A**H X=B. strsen Reorders the Schur factorization of a matrix dtrsen in order to find an orthonormal basis of a right ctrsen invariant subspace corresponding to selected ztrsen eigenvalues, and returns reciprocal condition numbers (sensitivities) of the average of the cluster of eigenvalues and of the invariant subspace. strsna Estimates the reciprocal condition numbers dtrsna (sensitivities) of selected eigenvalues and ctrsna eigenvectors of an upper quasi-triangular/ ztrsna triangular matrix. strsyl Solves the Sylvester matrix equation dtrsyl A X +/- X B=C where A and B are upper quasi- ctrsyl triangular/triangular, and may be transposed. ztrsyl strtri Computes the inverse of a triangular matrix. dtrtri ctrtri ztrtri strtrs Solves a triangular system of linear equations dtrtrs AX=B, A**T X=B or A**H X=B. ctrtrs ztrtrs stzrqf Computes an RQ factorization of an upper dtzrqf trapezoidal matrix. ctzrqf ztzrqf