Oracle Developer Studio Performance Library Routines

This appendix lists the Oracle Developer Studio Performance Library routines by library, routine name, and function.

For a description of the function and a listing of the Fortran and C interfaces, refer to the section 3P man pages for the individual routines. For example, to display the man page for the SBDSQR routine, type man -s 3P sbdsqr. The man page routine names use lowercase letters.

For many routines, separate routines exist that operate on different data types. Rather than list each routine separately, a lowercase x is used in a routine name to denote single, double, complex, and double complex data types. For example, the routine xBDSQR is available as four routines that operate with the following data types:

• SBDSQR – Single data type

• DBDSQR – Double data type

• CBDSQR – Complex data type

• ZBDSQR – Double complex data type

If a routine name is not available for S, D, C, and Z, the x prefix will not be used and each routine name will be listed. Also available (but not listed) in 64-bit enable operating environments are the corresponding routines in 64-bit. Their names are denoted by the _64 suffix. For example, the 64-bit versions of xBDSQR are the following:

• SBDSQR_64

• DBDSQR_64

• CBDSQR_64

• ZBDSQR_64

LAPACK Routines

The following set of tables lists the Oracle Developer Studio Performance Library LAPACK routines. (P) denotes routines that are parallelized.

• Table 25

• Table 26

• Table 27

• Table 28

• Table 29

• Table 30

• Table 31

• Table 32

• Table 33

• Table 34

• Table 35

• Table 36

• Table 37

• Table 38

• Table 39

• Table 40

• Table 41

• Table 42

• Table 43

• Table 44

• Table 45

• Table 46

• Table 47

• Table 48

• Table 49

• Table 50

• Table 51

• Table 52

• Table 53

• Table 54

• Table 55

Table 25  Bidiagonal Matrix Routines

Routine Function SBDSDC (P) or DBDSDC (P) Computes the singular value decomposition (SVD) of a bidiagonal matrix, using a divide and conquer method. SBDSVDX or DBDSVDX Computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix (driver). xBDSQR Computes SVD of a real upper or lower bidiagonal matrix, using the implicit zero-shift QR algorithm. SLARTGS or DLARTGS Generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem. Used by SBBCSD or DBBCSD.

Table 26  Common or Calculating Routines

Routine Function CHLA_TRANSTYPE Translates from a BLAST-specified integer constant to the character string specifying a transposition operation. CLA_HERPVGRW (P) or ZLA_HERPVGRW (P) Computes the reciprocal pivot growth factor norm(A)/norm(U) for a complex Hermitian matrix. ILADIAG Translates from a character string specifying, if a matrix has the unit diagonal or not, to the relevant BLAST-specified integer constant. ILAPREC Translates from a character string specifying an intermediate precision to the relevant BLAST-specified integer constant. ILATRANS Translates from a character string specifying a transposition operation to the relevant BLAST-specified integer constant. ILAENV Is called from the LAPACK routines to choose problem-dependent parameters for the local environment. ILAUPLO Translates from a character string specifying an upper or lower triangular matrix to the relevant BLAST-specified integer constant. ILAVER Returns the LAPACK version. xLA_GBRPVGRW Computes the reciprocal pivot growth factor norm(A)/norm(U) for a real or complex general banded matrix. xLA_GERPVGRW (P) Computes the reciprocal pivot growth factor norm(A)/norm(U) for a general indefinite matrix. xLA_PORPVGRW (P) Computes the reciprocal pivot growth factor norm(A)/norm(U) for a real symmetric or Hermitian positive definite matrix. xLA_SYRPVGRW (P) Computes the reciprocal pivot growth factor norm(A)/norm(U) for a real or complex symmetric indefinite matrix. SLAMRG (P) or DLAMRG (P) Creates a permutation list to merge the entries of two independently sorted sets into a single set sorted in ascending order. CLANHF (P) or ZLANHF (P) Returns a value of the one-norm, Frobenius norm, infinity norm, or the element of largest absolute value of a Hermitian matrix in the RFP format. SLANSF (P) or DLANSF (P) Returns a value of the one-norm, Frobenius norm, infinity norm, or the element of largest absolute value of a real symmetric matrix in the RFP format. xLARSCL2 (P) Performs a reciprocal diagonal scaling on a vector. xLASCL2 (P) Performs a diagonal scaling on a vector. SLASQ1 or DLASQ1 Computes the singular values of a real square bidiagonal matrix. Used by SBDSQR or DBDSQR. SLASQ2 or DLASQ2 Computes all the eigenvalues of a real symmetric positive definite tridiagonal matrix (high relative accuracy). Used by SBDSQR and SSTEGR or DBDSQR and DSTEGR. SLASQ3 or DLASQ3 Checks for deflation, computes a shift and calls the DQDS algorithm. Used by SBDSQR or DBDSQR. SLASQ4 or DLASQ4 Computes an approximation to the smallest eigenvalue using values from the previous transform. Used by SBDSQR or DBDSQR. SLASQ5 or DLASQ5 Computes one DQDS transform in the ping-pong form. Used by SBDSQR and SSTEGR or DBDSQR and DSTEGR. SLASQ6 or DLASQ6 Computes one DQD transform (shift equal to zero) in ping-pong form, with protection against underflow and overflow. Used by SBDSQR and SSTEGR or DBDSQR and DSTEGR. SLASRT or DLASRT Sorts numbers in a vector in increasing or decreasing order. xLATRZ (P) Factors a real or complex upper trapezoidal matrix by means of orthogonal transformations. CROT, ZROT Apply Givens plane rotation Note that SROT/DROT are included in level 1 BLAS.

Table 27  Cosine-Sine (CS) Decomposition Routines

Routine Function xBBCSD (P) Computes the CS decomposition of an unitary or orthogonal matrix in a bidiagonal-block form. SORCSD (P) or DORCSD (P) Computes the CS decomposition of a real partitioned orthogonal matrix. SORCSD2BY1 or DORCSD2BY1 (P) Computes the CS decomposition of an M-by-Q matrix X with orthonormal columns that has been partitioned into a 2-by-1 block structure. CUNCSD (P) or ZUNCSD (P) Computes the CS decomposition of an M-by-M partitioned unitary matrix.

Table 28  Diagonal Matrix Routines

Routine Function SDISNA (P) or DDISNA (P) Computes the reciprocal of the condition numbers for eigenvectors of a real symmetric or complex Hermitian matrix.

Table 29  General Band Matrix Routines

Routine Function CGBBRD or ZGBBRD Reduces a complex general band matrix to an upper bidiagonal form by the orthogonal transformation. SGBBRD (P) or DGBBRD (P) Reduces a real general band matrix to an upper bidiagonal form by the orthogonal transformation. xGBCON Estimates the reciprocal of the condition number of a general band matrix using LU factorization. xGBEQU (P) Computes row and column scalings to equilibrate a general band matrix and reduce its condition number. xGBEQUB (P) Computes row and column scalings intended to equilibrate a general band matrix and reduce its condition number. Differs from CGEEQU by restricting the scaling factors to a power of the radix. xGBRFS (P) Improves the computed solution to a system of linear equations when the coefficient matrix is banded, and provides error bounds and backward error estimates for the solution. xGBRFSX (P) Improves the computed solution to a banded system of linear equations and provides error bounds and backward error estimates. In addition to normwise error bound, the code provides maximum componentwise error bound if possible. xGBSV Solves a general banded system of linear equations (simple driver). xGBSVX (P) Solves a general banded system of linear equations (expert driver). xGBSVXX (P) Solves a general banded system of linear equations (expert driver, extra precision). If requested, both normwise and maximum componentwise error bounds are returned. xGBTF2 (P) Computes the LU factorization of a real or complex general band matrix using partial pivoting with row interchanges (unblocked algorithm). xGBTRF (P) Computes the LU factorization of a general band matrix using partial pivoting with row interchanges. xGBTRS Solves a general banded system of linear equations, using the factorization computed by xGBTRF. xLA_GBAMV Performs a matrix-vector operation to calculate error bounds for a real or complex band matrix. xLA_GBRFSX_EXTENDED Improves the computed solution to a system of linear equations for a real or complex general banded matrix by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Table 30  General Matrix (Unsymmetric or Rectangular) Routines

Routine Function xGEJSV (P) Computes the singular value decomposition (SVD) of a real or complex general matrix. DSGESV Computes the solution to a real system of linear equations with a general matrices (mixed precision with iterative refinement). xGESVJ (P) Computes the singular value decomposition (SVD) of a real or complex general matrix. Implements a preconditioned Jacobi SVD algorithm. Uses xGEQP3, xGEQR, xGELQF or xGEQP3 as a preprocessor. ZCGESV Computes the solution to a complex system of linear equations with a general matrices (mixed precision with iterative refinement). ZCPOSV Computes the solution to a complex system of linear equations with a positive definite matrix (mixed precision with iterative refinement). xGEBAK Forms the right or left eigenvectors of a general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by xGEBAL. xGEBAL (P) Balances a real or complex general matrix. xGEBD2 Reduces a general matrix to bidiagonal form (unblocked algorithm). xGEBRD Reduces a general matrix to upper or lower bidiagonal form by an unitary or orthogonal transformation (blocked algorithm). xGECON Estimates the reciprocal of the condition number of a general matrix, using the factorization computed by xGETRF. xGEEQU (P) Computes row and column scalings intended to equilibrate a general rectangular matrix and reduce its condition number. xGEEQUB (P) Computes row and column scalings intended to equilibrate a general rectangular matrix and reduce its condition number. Differs from xGETRF by restricting the scaling factors to a power of the radix. xGEES Computes the eigenvalues and Schur factorization of a general matrix (simple driver). xGEESX Computes the eigenvalues and Schur factorization of a general matrix (expert driver). xGEEV (P) Computes the eigenvalues and left and right eigenvectors of a general matrix (simple driver). xGEEVX (P) Computes the eigenvalues and left and right eigenvectors of a general matrix (expert driver). xGEGS Deprecated routine replaced by xGGES. xGEGV (P) Deprecated routine replaced by xGGEV. xGEHD2 Reduces a general square matrix to an upper Hessenberg form by the unitary or orthogonal similarity transformation (unblocked algorithm). xGEHRD (P) Reduces a general matrix to upper Hessenberg form by an orthogonal similarity transformation. xGELQ2 Computes the LQ factorization of a real or complex general rectangular matrix (unblocked algorithm). xGELQF Computes the LQ factorization of a general rectangular matrix. xGELS (P) Computes the least squares solution to an over-determined system of linear equations using a QR or LQ factorization of A. xGELSD Computes the least squares solution to an over-determined system of linear equations using a divide and conquer method and a QR or LQ factorization of A. xGELSS Computes the minimum-norm solution to a linear least squares problem by using the SVD of a general rectangular matrix (simple driver). xGELSX (P) Deprecated routine replaced by xSELSY. xGELSY (P) Computes the minimum-norm solution to a linear least squares problem using a complete orthogonal factorization. xGEMQRT Overwrites a general matrix with the result of its transformation by an orthogonal matrix, defined as the product of elementary reflectors generated using the compact WY representation as returned by xGEQRT. xGEQL2 Computes the QL factorization of a real or complex general rectangular matrix (unblocked algorithm). xGEQLF Computes the QL factorization of a real or complex general rectangular matrix. xGEQP3 Computes the QR factorization of general rectangular matrix using Level 3 BLAS. xGEQPF Deprecated routine replaced by xGEQP3. xGEQR2 Computes the QR factorization of a real or complex general rectangular matrix (unblocked algorithm). xGEQR2P Computes the QR factorization of a real or complex general rectangular matrix with non-negative diagonal elements (unblocked algorithm). xGEQRFP Computes the QR factorization of a real or complex general rectangular matrix. xGEQRT Computes a blocked QR factorization of a general real or complex matrix using the compact WY representation of Q. xGEQRT2 Computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. xGEQRT3 (P) Recursively computes a QR factorization of a general real or complex matrix using the compact WY representation of Q. xGERFS (P) Refines the solution to a system of linear equations. xGERFSX (P) Improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution (extra precision). xGERQ2 Computes the RQ factorization of a real or complex general rectangular matrix using an unblocked algorithm. xGERQF Computes the RQ factorization of a real or complex general rectangular matrix. xGESDD Computes the singular value decomposition (SVD) of a real or complex general rectangular matrix using a divide and conquer method (driver). xGESV Solves a general system of linear equations (simple driver). xGESVD Computes the singular value decomposition (SVD) for a real or complex general matrix (driver). xGESVDX Computes the singular value decomposition (SVD) for a real or complex general matrix, allows the computation of a subset of singular values and vectors (driver). xGESVJ Computes the singular value decomposition (SVD) of a real or complex general rectangular matrix. xGESVX (P) Solves a general system of linear equations (expert driver). xGESVXX (P) Computes the solution to a system of linear equations for general matrices (extra precision). xGETF2 Computes the LU factorization of a real or complex general matrix using partial pivoting with row interchanges (unblocked algorithm). xGETRF (P) Computes the LU factorization of a real or complex general rectangular matrix using partial pivoting with row interchanges. xGETRF2 (P) Computes the LU factorization of a real or complex general rectangular matrix using partial pivoting with row interchanges (recursive algorithm). xGETRI Computes the inverse of a general matrix using the factorization computed by xGETRF. xGETRS Solves a general system of linear equations using the factorization computed by xGETRF. xGSVJ0 (P) Preprocessor for xGESVJ. Applies Jacobi rotations targeting only particular pivots. xGSVJ1 (P) Preprocessor for xGESVJ. Applies Jacobi rotations in the same way as xGESVJ does, but it does not check convergence (stopping criterion). xLA_GEAMV (P) Performs a matrix-vector operation to calculate error bounds for a real or complex general matrix. CLA_GERCOND_C (P) or ZLA_GERCOND_C (P) Computes the infinity norm condition number of op(A)*inv(diag(c)) for a complex general matrix. C is a REAL vector. CLA_GERCOND_X (P) or ZLA_GERCOND_X (P) Computes the infinity norm condition number of op(A)*inv(diag(x)) for a complex general matrix. X is a COMPLEX vector. SLA_GERCOND(P) or DLA_GERCOND (P) Estimates the Skeel condition number for a real general matrix. xLA_GERFSX_EXTENDED (P) Improves the computed solution to a system of linear equations for a real or complex general matrix by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. xLA_GERFSX_GBRPVGRW Computes the reciprocal pivot growth factor norm(A)/norm(U) for a real or complex general matrix. xLALS0 (P) Applies back multiplying factors in solving the least squares problem using the divide and conquer SVD approach. Used by xLALSA. CLALSA (P) or ZLALSA (P) Computes the SVD of a complex matrix in compact form. Used by SGELSD. SLALSA or DLALSA Computes the SVD of a real matrix in compact form. Used by SGELSD or DGELSD. xLALSD (P) Solves the least squares problem using the SVD. Used by xGELSD.

Table 31  General Matrix-Generalized Problem (Pair of General Matrices) Routines

Routine Function xGGBAK Forms the right or left eigenvectors of a generalized eigenvalue problem based on the output by xGGBAL. xGGBAL (P) Balances a pair of general matrices for the generalized eigenvalue problem. xGGES Computes the generalized eigenvalues, Schur form, and, optionally, left and/or right Schur vectors for two nonsymmetric matrices (simple driver). xGGES3 Computes the generalized eigenvalues, Schur form, and, optionally, left and/or right Schur vectors for two nonsymmetric matrices using a blocked algorithm (simple driver). xGGESX Computes the generalized eigenvalues, Schur form, and, optionally, left and/or right Schur vectors (expert driver). xGGEV (P) Computes the generalized eigenvalues and the left and/or right generalized eigenvectors for two nonsymmetric matrices (simple driver). xGGEV3 Computes the generalized eigenvalues and the left and/or right generalized eigenvectors for two nonsymmetric matrices using a blocked algorithm (expert driver). xGGEVX (P) Computes the generalized eigenvalues and the left and/or right generalized eigenvectors for two nonsymmetric matrices (expert driver). xGGGLM (P) Solves the general Gauss-Markov linear model (GLM) problem. xGGHD3 (P) Reduces two matrices to the generalized upper Hessenberg form using orthogonal transformations. This is a blocked variant of xGGHRD used to enhance performance. xGGHRD (P) Reduces two matrices to the generalized upper Hessenberg form using orthogonal transformations. xGGLSE Solves the LSE (Constrained Linear Least Squares Problem) using the GRQ (Generalized RQ) factorization. xGGQRF Computes the generalized QR factorization of two matrices. xGGRQF Computes the generalized RQ factorization of two matrices. xGGSVD Computes the generalized singular value decomposition (driver). xGGSVD3 Computes the generalized singular value decomposition (driver). xGGSVP (P) Computes an orthogonal or unitary matrix as a preprocessing step for calculating the generalized singular value decomposition using xGGSVD. xGGSVP3 (P) Computes an orthogonal or unitary matrix as a preprocessing step for calculating the generalized singular value decomposition using xGGSVD3.

Table 32  General Tridiagonal Matrix Routines

Routine Function xGTCON Estimates the reciprocal of the condition number of a tridiagonal matrix, using the LU factorization as computed by xGTTRF. xGTRFS (P) Refines the solution to a general tridiagonal system of linear equations. xGTSV (P) Solves a general tridiagonal system of linear equations (simple driver). xGTSVX Solves a general tridiagonal system of linear equations (expert driver). xGTTRF (P) Computes an LU factorization of a general tridiagonal matrix using partial pivoting and row exchanges. xGTTRS Solves general tridiagonal system of linear equations using the factorization computed by xGTTRF. xGTTS2 (P) Solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by xGTTRF.

Table 33  Hermitian Band Matrix Routines

Routine Function CHBEV or ZHBEV Computes all the eigenvalues and eigenvectors of a Hermitian band matrix. Replacement with newer version CHBEVD or ZHBEVD suggested. CHBEVD or ZHBEVD Computes all the eigenvalues and eigenvectors of a Hermitian band matrix and uses a divide and conquer method to calculate eigenvectors (driver). CHBEVX (P) or ZHBEVX (P) Computes selected eigenvalues and eigenvectors of a Hermitian band matrix. CHBGST (P) or ZHBGST (P) Reduces Hermitian-definite banded generalized eigenproblem to a standard form. CHBGV or ZHBGV Computes all the eigenvalues and eigenvectors of a generalized Hermitian-definite banded eigenproblem. Replacement with newer version CHBGVD or ZHBGVD suggested. CHBGVD or ZHBGVD Computes all the eigenvalues and eigenvectors of a generalized Hermitian-definite banded eigenproblem and uses a divide and conquer method to calculate eigenvectors (driver). CHBGVX (P) or ZHBGVX (P) Computes selected eigenvalues and eigenvectors of a generalized Hermitian-definite banded eigenproblem. CHBTRD (P) or ZHBTRD (P) Reduces a Hermitian band matrix to a real symmetric tridiagonal form by using a unitary similarity transformation.

Table 34  Hermitian Matrix Routines

Routine Function CHECON or ZHECON Estimates the reciprocal of the condition number of a Hermitian matrix using the factorization computed by CHETRF or ZHETRF. CHECON_ROOK or ZHECON_ROOK Estimates the reciprocal of the condition number of a Hermitian matrix using the factorization computed by CHETRF_ROOK or ZHETRF_ROOK. CHEEQUB (P) or ZHEEQUB (P) Computes row and column scalings intended to equilibrate a Hermitian matrix and reduce its condition number with respect to the two-norm. CHEEV or ZHEEV Computes all the eigenvalues and eigenvectors of a Hermitian matrix (simple driver). Replacement with newer version CHEEVR or ZHEEVR suggested. CHEEVD or ZHEEVD Computes all the eigenvalues and eigenvectors of a Hermitian matrix and uses a divide and conquer method to calculate eigenvectors (driver). Replacement with newer version CHEEVR or ZHEEVR suggested. CHEEVR or ZHEEVR Computes selected eigenvalues and the eigenvectors of a complex Hermitian matrix. CHEEVX (P) or ZHEEVX (P) Computes selected eigenvalues and eigenvectors of a Hermitian matrix (expert driver). CHEGST or ZHEGST Reduces a Hermitian-definite generalized eigenproblem to a standard form using the factorization computed by CPOTRF or ZPOTRF. CHEGV or ZHEGV Computes all the eigenvalues and eigenvectors of a complex generalized Hermitian-definite eigenproblem. Replacement with newer version CHEGVD or ZHEGVD suggested. CHEGVD or ZHEGVD Computes all the eigenvalues and eigenvectors of a complex generalized Hermitian-definite eigenproblem and uses a divide and conquer method to calculate eigenvectors (driver). CHEGVX or ZHEGVX Computes selected eigenvalues and eigenvectors of a complex generalized Hermitian-definite eigenproblem. CHERFS (P) or ZHERFS (P) Improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite. CHERFSX (P) or ZHERFSX (P) Improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite (extra precision). CHESV or ZHESV Solves a complex Hermitian-indefinite system of linear equations (simple driver). CHETRF is called to compute the factorization of a complex Hermitian matrix CHESV_ROOK or ZHESV_ROOK Solves a complex Hermitian-indefinite system of linear equations (simple driver). CHETRF_ROOK is called to compute the factorization of a complex Hermitian matrix. CHESVX or ZHESVX Solves a complex Hermitian-indefinite system of linear equations (expert driver). CHESVXX (P) or ZHESVXX (P) Computes the solution to a complex system of linear equations with a square symmetric matrix using the diagonal pivoting factorization (extra precision). CHETD2 or ZHETD2 Reduces a complex Hermitian matrix to a real symmetric tridiagonal form by an unitary similarity transformation (an unblocked algorithm). CHETF2 (P) or ZHETF2 (P) Computes the factorization of a complex Hermitian matrix using the diagonal pivoting method (unblocked algorithm). CHETF2_ROOK (P) or ZHETF2_ROOK (P) Computes the factorization of a complex Hermitian matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (unblocked algorithm). CHETRD or ZHETRD Reduces a Hermitian matrix to a real symmetric tridiagonal form by using a unitary similarity transformation. CHETRF (P) or ZHERTF (P) Computes the factorization of a complex Hermitian-indefinite matrix using the diagonal pivoting method. CHETRF_ROOK (P) or ZHERTF_ROOK (P) Computes the factorization of a complex Hermitian-indefinite matrix using the Bunch-Kaufman ("rook") diagonal pivoting method. CHETRI (P) or ZHETRI (P) Computes the inverse of a complex Hermitian indefinite matrix using the factorization computed by CHETRF or ZHETRF. CHETRI_ROOK (P) or ZHETRI_ROOK (P) Computes the inverse of a complex Hermitian indefinite matrix using the factorization computed by CHETRF_ROOK or ZHETRF_ROOK. CHETRI2 or ZHETRI2 Computes the inverse of a complex Hermitian-indefinite matrix using the factorization computed by CHETRF or ZHETRS.Sets the leading dimension of the workspace before calling CHETRI2X or ZHETRI2X that actually computes the inverse (extra precision). CHETRI2X (P) or ZHETRI2X (P) Computes the inverse of a complex Hermitian-indefinite matrix using the factorization computed by CHETRF or ZHETRS (extra precision). CHETRS (P) or ZHETRS (P) Solves a complex Hermitian-indefinite matrix using the factorization computed by CHETRF or ZHETRF. CHETRS_ROOK (P) or ZHETRS_ROOK (P) Solves a complex Hermitian-indefinite matrix using the factorization computed by CHETRF_ROOK or ZHETRF_ROOK. CHETRS2 (P) or ZHETRS2 (P) Solves a system of linear equations with a complex Hermitian matrix using the factorization computed by CHETRF or ZHERTF and converted by CSYCONV or ZSYCONV. CHFRK (P) or ZHFRK (P) Performs a Hermitian rank-k operation for a matrix in the RFP format. CLA_HEAMV or ZLA_HEAMV Performs a matrix-vector operation to calculate error bounds for a complex Hermitian-indefinite matrix. CLA_HERCOND_C (P) or ZLA_HERCOND_C (P) Computes the infinity norm condition number of op(A)*inv(diag(c)) for a complex Hermitian-indefinite matrix. C is a REAL vector. CLA_HERCOND_X (P) or ZLA_HERCOND_X (P) Computes the infinity norm condition number of op(A)*inv(diag(x)) for a complex Hermitian-indefinite matrix. X is a COMPLEX vector. CLA_HERFSX_EXTENDED (P) or ZLA_HERFSX_EXTENDED (P) Improves the computed solution to a system of linear equations for a complex Hermitian-indefinite matrix by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. CLAHEF (P) or ZLAHEF (P) Computes a partial factorization of a complex Hermitian-indefinite matrix, using the diagonal pivoting method. Used by CHETRF or CHETRF. CLAHEF_ROOK (P) or ZLAHEF_ROOK (P) Computes a partial factorization of a complex Hermitian-indefinite matrix, using the Bunch-Kaufman ("rook") diagonal pivoting method. Used by CHETRF_ROOK or CHETRF_ROOK.

Table 35  Hermitian Matrix in Packed Storage Routines

Routine Function CHPCON or ZHPCON Estimates the reciprocal of the condition number of a Hermitian-indefinite matrix in packed storage using the factorization computed by CHPTRF or ZHPTRF. CHPEV or ZHPEV Computes all the eigenvalues and eigenvectors of a Hermitian matrix in packed storage (simple driver). Replacement with newer version CHPEVD or ZHPEVD suggested. CHPEVX (P) or ZHPEVX (P) Computes selected eigenvalues and eigenvectors of a Hermitian matrix in packed storage (expert driver). CHPEVD or ZHPEVD Computes all the eigenvalues and eigenvectors of a Hermitian matrix in packed storage, and uses a divide and conquer method to calculate eigenvectors (driver). CHPGST or ZHPGST Reduces a Hermitian-definite generalized eigenproblem to standard form, where the coefficient matrices are in packed storage, and uses the factorization computed by CPPTRF or ZPPTRF. CHPGV or ZHPGV Computes all the eigenvalues and eigenvectors of a generalized Hermitian-definite eigenproblem where the coefficient matrices are in packed storage (simple driver). Replacement with newer version CHPGVD or ZHPGVD suggested. CHPGVD or ZHPGVD Computes all the eigenvalues and eigenvectors of a generalized Hermitian-definite eigenproblem where the coefficient matrices are in packed storage, and uses a divide and conquer method to calculate eigenvectors (driver). CHPGVX or ZHPGVX Computes selected eigenvalues and eigenvectors of a complex Hermitian-definite eigenproblem, where the coefficient matrices are in packed storage (expert driver). CHPRFS (P) or ZHPRFS (P) Improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite in packed storage. CHPSV or ZHPSV Computes the solution to a complex system of linear equations where the coefficient matrix is a Hermitian matrix stored in the packed format (simple driver). CHPSVX or ZHPSVX Uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations where the coefficient matrix is a Hermitian matrix stored in the packed format (expert driver). CHPTRD or ZHPTRD Reduces a complex Hermitian matrix stored in the packed form to a real symmetric tridiagonal form by the unitary similarity transformation. CHPTRF or ZHPTRF Computes the factorization of a complex Hermitian packed matrix using the Bunch-Kaufman diagonal pivoting method. CHPTRI or ZHPTRI Computes the inverse of a complex Hermitian-indefinite matrix in packed storage using the factorization computed by CHPTRF or ZHPTRF. CHPTRS (P) or ZHPTRS (P) Solves a complex Hermitian-indefinite matrix stored in the packed format using the factorization computed by CHPTRF or ZHPTRF.

Table 36  Upper Hessenberg Matrix Routines

Routine Function xHSEIN (P) Computes the specified right and/or left eigenvectors of an upper Hessenberg matrix using inverse iteration. CHSEQR or ZHSEQR Computes the eigenvalues of a complex upper Hessenberg matrix and the Shur factorization using the multishift QR algorithm. SHSEQR (P) or DHSEQR (P) Computes the eigenvalues of a real upper Hessenberg matrix and the Shur factorization using the multishift QR algorithm.

Table 37  Upper Hessenberg Matrix-Generalized Problem (Hessenberg and Triangular Matrix) Routines

Routine Function xHGEQZ (P) Computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is an upper triangular, using the single/double-shift QZ method. Matrix pairs of this type are produced by xGGHRD.

Table 38  Real Orthogonal Matrix in Packed Storage Routines

Routine Function SOPGTR (P) or DOPGTR (P) Generates an orthogonal transformation matrix from a real tridiagonal matrix determined by SSPTRD or DSPTRD. SOPMTR or DOPMTR Multiplies a real general matrix by the orthogonal transformation matrix reduced to the tridiagonal form by SSPTRD or DSPTRD.

Table 39  Real Orthogonal Matrix Routines

Routine Function SORBDB or DORBDB Simultaneously bidiagonalizes the blocks of a real partitioned orthogonal matrix. SORBDB1 or DORBDB1 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 1). SORBDB2 or DORBDB2 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 2). SORBDB3 or DORBDB3 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 3). SORBDB4 or DORBDB4 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 4). SORBDB5 or DORBDB5 Orthogonalizes the column vector X with respect to the orthonormal columns of Q. SORBDB6 or DORBDB6 Orthogonalizes the column vector X with respect to the orthonormal columns of Q. Used by SORBDB4 or DORBDB5. SORG2L (P) or DORG2L (P) Generates all or part of a real orthogonal matrix Q from a QL factorization, as determined by SGEQLF or DGEQLF (unblocked algorithm). SORG2R (P) or DORG2R (P) Generates all or part of a real orthogonal matrix Q from a QR factorization, as determined by SGEQRF or DGEQRF (unblocked algorithm). SORGBR (P) or DORGBR Generates the real orthogonal transformation matrices from reduction to the bidiagonal form, as determined by SGEBRD or DGEBRD. SORGHR (P) or DORGHR (P) Generates the real orthogonal transformation matrix reduced to the Hessenberg form, as determined by SGEHRD or DGEHRD. SORGL2 (P) or DORGL2 (P) Generates a real rectangular matrix with orthonormal rows, as returned by SGELQF or DGELQF. SORGLQ (P) or DORGLQ (P) Generates a real orthogonal matrix Q from an LQ factorization, as returned by SGELQF or DGELQF. SORGQL (P) or DORGQL (P) Generates a real orthogonal matrix Q from a QL factorization, as returned by SGEQLF or DGEQLF. SORGQR (P) or DORGQR (P) Generates a real orthogonal matrix Q from a QR factorization, as returned by SGEQRF or DGEQRF. SORGR2 (P) or DORGR2 (P) Generates all or part of a real orthogonal matrix Q from an RQ factorization determined SGEQRF or DGEQRF (unblocked algorithm). SORGRQ (P) or DORGRQ (P) Generates a real orthogonal matrix Q from an RQ factorization, as returned by SGERQF or DGERQF. SORGTR (P) or DORGTR (P) Generates a real orthogonal matrix reduced to tridiagonal form by SSYTRD or DSYTRD. SORM22 or DORM22 Multiplies a real general matrix by the orthogonal matrix. SORM2L or DORM2L Multiplies a real general matrix by the orthogonal matrix from a QL factorization determined by SGEQLF or DGEQLF (unblocked algorithm). SORM2R or DORM2R Multiplies a real general matrix by the orthogonal matrix from a QR factorization determined by SGEQRF or DGEQRF (unblocked algorithm). SORMBR or DORMBR Multiplies a real general matrix with the orthogonal matrix reduced to the bidiagonal form, as determined by SGEBRD or DGEBRD. SORMHR or DORMHR Multiplies a real general matrix by the orthogonal matrix reduced to the Hessenberg form by SGEHRD or DGEHRD. SORML2 or DORML2 Multiplies a real general matrix by the orthogonal matrix from an LQ factorization determined by SGELQF (unblocked algorithm). SORMLQ or DORMLQ Multiplies a real general matrix by the orthogonal matrix from an LQ factorization, as returned by SGELQF or DGELQF. SORMQL or DORMQL Multiplies a real general matrix by the orthogonal matrix from a QL factorization, as returned by SGEQLF or DGEQLF. SORMQR or DORMQR Multiplies a real general matrix by the orthogonal matrix from a QR factorization, as returned by SGEQRF or DGEQRF. SORMR2 or DORMR2 Multiplies a real general matrix by the orthogonal matrix from an RQ factorization determined by STZRZF or DTZRZF (unblocked algorithm). SORMR3 or DORMR3 Multiplies a real general matrix by the orthogonal matrix from an RZ factorization determined by STZRZF or DTZRZF (unblocked algorithm). SORMRQ or DORMRQ Multiplies a real general matrix by the orthogonal matrix from an RQ factorization returned by SGERQF or DGERQF. SORMRZ or DORMRZ Multiplies a real general matrix by the orthogonal matrix from an RZ factorization, as returned by STZRZF or DTZRZF. SORMTR or DORMTR Multiplies a real general matrix by the orthogonal transformation matrix reduced to tridiagonal form by SSYTRD or DSYTRD.

Table 40  Symmetric or Hermitian Positive Definite Band Matrix Routines

Routine Function xPBCON Estimates the reciprocal of the condition number of a symmetric or Hermitian positive definite band matrix using the Cholesky factorization returned by xPBTRF. xPBEQU (P) Computes equilibration scale factors for a symmetric or Hermitian positive definite band matrix. xPBRFS (P) Refines solution to a symmetric or Hermitian positive definite banded system of linear equations. xPBSTF Computes a split Cholesky factorization of a real symmetric positive definite band matrix. xPBSV Solves a symmetric or Hermitian positive definite banded system of linear equations (simple driver). xPBSVX (P) Solves a symmetric or Hermitian positive definite banded system of linear equations (expert driver). xPBTF2 Computes the Cholesky factorization of a real symmetric or complex Hermitian positive definite band matrix (unblocked algorithm). xPBTRF Computes the Cholesky factorization of a symmetric or Hermitian positive definite band matrix. xPBTRS Solves a system of linear equations with a real symmetric or complex Hermitian positive definite banded matrix using the Cholesky factorization computed by xPBTRF.

Table 41  Symmetric or Hermitian Positive Definite Matrix Routines

Routine Function CLA_PORCOND_C (P) or ZLA_PORCOND_C (P) Computes the infinity norm condition number of op(A)*inv(diag(c)) for a complex Hermitian positive definite matrix. C is a REAL vector. CLA_PORCOND_X (P) or ZLA_PORCOND_X (P) Computes the infinity norm condition number of op(A)*inv(diag(x)) for a complex Hermitian positive definite matrix. X is a COMPLEX vector. SLA_PORCOND (P) or DLA_PORCOND(P) Estimates the Skeel condition number for a real symmetric positive definite matrix. xLA_LIN_BERR (P) Computes a component-wise relative backward error. xLA_PORFSX_EXTENDED (P) Improves the computed solution to a system of linear equations for a real symmetric or complex Hermitian positive definite matrix by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. xLA_WWADDW Adds a vector W into a doubled-single vector (X, Y). This works for all extant IBM's hex and binary floating point arithmetics, but not for decimal. xPFTRF Computes the Cholesky factorization of a real symmetric or Hermitian positive definite band matrix. xPFTRI Computes the inverse of a real symmetric or Hermitian positive definite matrix, using the Cholesky factorization computed by xPFTRF. xPFTRS Solves a system of linear equations with a symmetric or Hermitian positive definite matrix, using the Cholesky factorization computed by xPFTRF. xPOCON Estimates the reciprocal of the condition number of a symmetric or Hermitian positive definite matrix, using the Cholesky factorization returned by xPOTRF. xPOEQU (P) Computes equilibration scale factors for a symmetric or Hermitian positive definite matrix. xPOEQUB (P) Computes row and column scalings intended to equilibrate a symmetric or Hermitian positive definite matrix and reduce its condition number with respect to the two-norm. xPORFS (P) Refines the solution to a linear system in a Cholesky-factored symmetric or Hermitian positive definite matrix. xPORFSX (P) Improves the computed solution to a system of linear equations, when the coefficient matrix is a real symmetric or Hermitian positive definite, and provides the error bounds and backward-error estimates for the solution (extra precision). xPOSV Solves a symmetric or Hermitian positive definite system of linear equations (simple driver). xPOSVX (P) Solves a symmetric or Hermitian positive definite system of linear equations (expert driver). xPOSVXX (P) Solves a real symmetric or Hermitian positive definite system of linear equations (expert driver, extra precision). If requested, both normwise and maximum component-wise error bounds are returned. xPOTRF Computes the Cholesky factorization of a real symmetric or Hermitian positive definite matrix. xPOTRF2 Computes the Cholesky factorization of a real symmetric or Hermitian positive definite matrix using the recursive algorithm. xPOTRI Computes the inverse of a real symmetric or Hermitian positive definite matrix using the Cholesky-factorization computed by xPOTRF. xPOTRS Solves a real symmetric or Hermitian positive definite system of linear equations, using the Cholesky factorization computed by xPOTRF. ZCPOSV Computes the solution to a complex system of linear equations with a positive definite matrix (mixed precision with iterative refinement).

Table 42  Symmetric or Hermitian Positive Definite Matrix in Packed Storage Routines

Routine Function xPPCON Estimates the reciprocal of the condition number of a Cholesky-factored symmetric positive definite matrix in packed storage. xPPEQU (P) Computes equilibration scale factors for a symmetric or Hermitian positive definite matrix in packed storage. xPPRFS (P) Refines the solution to a linear system of equations in a Cholesky-factored symmetric or Hermitian positive definite matrix in packed storage. xPPSV Solves a linear system in a symmetric or Hermitian positive definite matrix in packed storage (simple driver). xPPSVX (P) Solves a linear system in a symmetric or Hermitian positive definite matrix in packed storage (expert driver). xPPTRF Computes the Cholesky factorization of a real symmetric or Hermitian positive definite matrix stored in the packed format. xPPTRI Computes the inverse of a real symmetric or Hermitian positive definite matrix in packed storage using the Cholesky factorization returned by xPPTRF. xPPTRS Solves a real symmetric or Hermitian positive definite system of linear equations where the coefficient matrix is in packed storage, using the Cholesky factorization returned by xPPTRF. xPSTF2 (P) Computes the Cholesky factorization with complete pivoting of a real symmetric or Hermitian positive-semi-definite matrix. This version of the algorithm calls level 2 BLAS. xPSTRF (P) Computes the Cholesky factorization with complete pivoting of a real symmetric or Hermitian positive-semi-definite matrix. This version of the algorithm calls level 3 BLAS.

Table 43  Symmetric or Hermitian Positive Definite Tridiagonal Matrix Routines

Routine Function xPTCON Estimates the reciprocal of the condition number of a real symmetric or Hermitian positive definite tridiagonal matrix using the Cholesky factorization computed by xPTTRF. xPTEQR (P) Computes all the eigenvectors and, optionally, the eigenvalues of a real symmetric or Hermitian positive definite matrix. xPTRFS (P) Refines the solution to a symmetric or Hermitian positive definite tridiagonal system of linear equations. xPTSV Solves a real symmetric or Hermitian positive definite tridiagonal system of linear equations (simple driver). xPTSVX Solves a real symmetric or Hermitian positive definite tridiagonal system of linear equations (expert driver). xPTTRF Computes the LDLH or LDLT factorization of a real symmetric or Hermitian positive definite tridiagonal matrix. xPTTRS Solves a real symmetric or Hermitian positive definite tridiagonal system of linear equations using the LDLH or LDLT factorization returned by xPTTRF. xPTTS2 (P) Solves a tridiagonal system of linear equations using the LDLH or LDLT factorization computed by xPTTRF. Used by xPTTRS.

Table 44  Real Symmetric Band Matrix Routines

Routine Function SSBEV or DSBEV Computes all the eigenvalues and, optionally, the left and/or right eigenvectors of a real symmetric band matrix (simple driver). Replacement with newer version SSBEVD or DSBEVD suggested. SSBEVD or DSBEVD Computes all the eigenvalues and, optionally, the eigenvectors of a real symmetric band matrix. If eigenvectors are desired, it uses a divide and conquer algorithm. (driver) SSBEVX (P) or DSBEVX (P) Computes selected eigenvalues and, optionally, the left and/or right eigenvectors of a symmetric band matrix (expert driver). SSBGST (P) or DSBGST (P) Reduces a symmetric-definite banded generalized eigenproblem to a standard form. SSBGV or DSBGV Computes all the eigenvalues and, optionally, the eigenvectors of a generalized symmetric-definite banded eigenproblem (simple driver). Replacement with newer version SSBGVD or DSBGVD suggested. SSBGVD or DSBGVD Computes all the eigenvalues and, optionally, the eigenvectors of generalized symmetric-definite banded eigenproblem and uses a divide and conquer method to calculate eigenvectors (simple driver). SSBGVX (P) or DSBGVX (P) Computes selected eigenvalues and eigenvectors of a generalized symmetric-definite banded eigenproblem (expert driver). SSBTRD (P) or DSBTRD (P) Reduces a symmetric band matrix to real symmetric tridiagonal form by using an orthogonal similarity transformation.

Table 45  Symmetric Matrix in Packed Storage Routines

Routine Function xSPCON Estimates the reciprocal of the condition number of a real or complex symmetric packed matrix using the factorization computed by xSPTRF. SSFRK (P) or DSFRK (P) Performs a symmetric rank-k operation for a real matrix in RFP format. SSPEV or DSPEV Computes all the eigenvalues and eigenvectors of a symmetric matrix in packed storage (simple driver). Replacement with newer version SSPEVD or DSPEVD suggested. SSPEVD or DSPEVD Computes all the eigenvalues and, optionally, the light and/or right eigenvectors of a symmetric matrix in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm (simple driver). SSPEVX (P) or DSPEVX (P) Computes selected eigenvalues and eigenvectors of a symmetric matrix in packed storage (expert driver). SSPGST or DSPGST Reduces a real symmetric-definite generalized eigenproblem to a standard form where the coefficient matrices are in packed storage and uses the factorization computed by SPPTRF or DPPTRF. Replacement with newer version SSPGVD or DSPGVD suggested. SSPGV or DSPGV Computes all the eigenvalues and eigenvectors of a real generalized symmetric-definite eigenproblem where the coefficient matrices are in packed storage (simple driver). Replacement with newer version SSPGVD or DSPGVD suggested. SSPGVD or DSPGVD Computes all the eigenvalues and eigenvectors of a real generalized symmetric-definite eigenproblem where the coefficient matrices are in packed storage, and uses a divide and conquer method to calculate eigenvectors (driver). SSPGVX or DSPGVX Computes selected eigenvalues and eigenvectors of a real generalized symmetric-definite eigenproblem where the coefficient matrices are in packed storage (expert driver). DSPOSV Computes the solution to a real system of linear equations with a real symmetric positive definite matrix: first attempts to factorize the matrix in single precision, then, if necessary - with double precision. xSPRFS (P) Improves the computed solution to a real or complex system of linear equations when the coefficient matrix is symmetric indefinite in packed storage. xSPSV Computes the solution to a real or complex system of linear equations where the coefficient matrix is a symmetric matrix in packed storage (simple driver). xSPSVX Uses the diagonal pivoting factorization to compute the solution to a system of linear equations where the coefficient matrix is a symmetric matrix in packed storage (expert driver). SSPTRD or DSPTRD Reduces a real symmetric matrix stored in the packed form to a real symmetric tridiagonal form using an orthogonal similarity transformation. xSPTRF Computes the factorization of a symmetric packed matrix using the Bunch-Kaufman diagonal pivoting method. xSPTRI Computes the inverse of a symmetric indefinite matrix in packed storage using the factorization computed by xSPTRF. xSPTRS (P) Solves a system of linear equations with a real or complex symmetric matrix in packed storage using the factorization computed by xSPTRF.

Table 46  Real Symmetric Tridiagonal Matrix Routines

Routine Function xLAED0 (P) Computes all the eigenvalues and corresponding eigenvectors of a real or complex unreduced symmetric tridiagonal matrix using the divide and conquer method. Used by xSTEDC. SLAED1 (P) or DLAED1 (P) Computes the updated eigensystem of a real diagonal matrix after modification by a rank-one symmetric matrix. Used by SSTEDC or DSTEDC, when the original matrix is tridiagonal. SLAED2 (P) or DLAED2 (P) Merges the two sets of eigenvalues together into a single sorted set and tries to deflate the size of the problem. Used by SSTEDC or DSTEDC. SLAED3 (P) or DLAED3 Finds the roots of the secular equation and updates the eigenvectors. Used by SSTEDC or DSTEDC, when the original matrix is tridiagonal. SLAED4 (P) or DLAED4 (P) Finds a single root of the secular equation. Used by SSTEDC or DSTEDC. SLAED5 or DLAED5 Solves a 2-by-2 secular equation. Used by SSTEDC or DSTEDC. SLAED6 or DLAED6 Computes the positive or negative root closest to the origin (one Newton step in solution of the secular equation). xLAED7 (P) Computes the updated eigensystem of a diagonal matrix after modification by a rank-one symmetric matrix. Used by xSTEDC, when the original matrix is dense. xLAED8 (P) Merges the two sets of eigenvalues into a single sorted set and deflates the secular equation. Used by xSTEDC, when the original matrix is dense. SLAED9 (P) or DLAED9 (P) Finds the roots of the secular equation and updates the eigenvectors. Used by SSTEDC or DSTEDC, when the original matrix is dense. SLAEDA (P) or DLAEDA (P) Computes a vector determining the rank-one modification of the diagonal matrix. Used by SSTEDC or DSTEDC, when the original matrix is dense. SLAGTF or DLAGTF (P) Computes an LU factorization of a matrix T - (lambda * I), where T is a general tridiagonal matrix, and lambda is a scalar, using partial pivoting with row interchanges. Used by SSTEIN or DSTEIN. SSTEBZ or DSTEBZ Computes the eigenvalues of a real symmetric tridiagonal matrix. CSTEDC (P) or ZSTEDC (P) Computes all the eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band complex Hermitian matrix can also be found if CHETRD/ZHETRD, CHPTRD/ZHPTRD, or CHBTRD/ZHBTRD has been used to reduce this matrix to tridiagonal form. SSTEDC or DSTEDC Computes all the eigenvalues and eigenvectors of a complex symmetric tridiagonal matrix using the divide and conquer method. The eigenvectors of a full or band real symmetric matrix can also be found if SSYTRD, SSPTRD, or SSBTRD; or DSYTRD, DSPTRD, or DSBTRD has been used to reduce this matrix to tridiagonal form. xSTEGR Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using Relatively Robust Representations, xSTEGR is a compatibility wrapper around the improved xSTEMR routine. xSTEIN (P) Computes selected eigenvectors of a real symmetric tridiagonal matrix using inverse iteration. xSTEMR (P) Computes the selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix using Relatively Robust Representations. xSTEQR (P) Computes all the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of a QL or QR algorithm. SSTERF (P) or DSTERF (P) Computes all the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using a root-free QL or QR algorithm variant. SSTEV or DSTEV Computes all the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix (simple driver). Replacement with newer version SSTEVR or DSTEVR suggested. SSTEVD or DSTEVD Computes all the eigenvalues and eigenvectors of a real symmetric tridiagonal matrix (simple driver). Replacement with newer version SSTEVR or DSTEVR suggested. SSTEVR or DSTEVR Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix using Relatively Robust Representations. SSTEVX (P) or DSTEVX (P) Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix (expert driver). xSTSV Computes the solution to a system of linear equations where the coefficient matrix is a symmetric tridiagonal matrix (unblocked algorithm). xSTTRF (P) Computes the factorization of a real or complex symmetric tridiagonal matrix using the Bunch-Kaufman diagonal pivoting method (unblocked algorithm).

Table 47  Symmetric Matrix Routines

Routine Function xLA_SYAMV Performs a matrix-vector operation to calculate error bounds for a real or complex symmetric indefinite matrix. CLA_SYRCOND_C (P) or ZLA_SYRCOND_C (P) Computes the infinity norm condition number of op(A)*inv(diag(c)) for a real or complex symmetric indefinite matrix. C is a REAL vector. CLA_SYRCOND_X (P) or ZLA_SYRCOND_X (P) Computes the infinity norm condition number of op(A)*inv(diag(x)) for a real or complex symmetric indefinite matrix. X is a COMPLEX vector. SLA_SYRCOND (P) or DLA_SYRCOND(P) Estimates the Skeel condition number for a real symmetric indefinite matrix. xLA_SYRFSX_EXTENDED(P) Improves the computed solution to a system of linear equations of a real or complex symmetric indefinite matrix by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. xLASYF Computes a partial factorization of a real or complex symmetric matrix, using the diagonal pivoting method. Used by xSYTRF. xLASYF_ROOK Computes a partial factorization of a real or complex symmetric matrix, using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. Used by xSYTRF_ROOK. xSYCON Estimates the reciprocal of the condition number of a real or complex symmetric matrix using the factorization computed by xSYTRF. xSYCON_ROOK Estimates the reciprocal of the condition number of a real or complex symmetric matrix using the factorization computed by xSYTRF_ROOK. xSYCONV (P) Converts the matrix computed by SSYTRF or DSYTRF into lower and upper triangular matrices and vice-versa. xSYEQUB (P) Computes row and column scalings intended to equilibrate a real or complex symmetric matrix and reduce its condition number with respect to the two-norm. SSYEV or DSYEV Computes all eigenvalues and eigenvectors of a symmetric matrix (simple driver). Replacement with newer version SSYEVR or DSYEVR suggested. SSYEVD or DSYEVD Computes all eigenvalues and eigenvectors of a symmetric matrix and uses a divide and conquer method to calculate eigenvectors (expert driver). Replacement with newer version SSYEVR or DSYEVR suggested. SSYEVR or DSYEVR Computes selected eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. SSYEVX (P) or DSYEVX (P) Computes eigenvalues and eigenvectors of a real symmetric matrix (expert driver). SSYGS2 or DSYGS2 Reduces a real symmetric-definite generalized eigenproblem to a standard form using the factorization results obtained from SPOTRF or DPOTRF (unblocked algorithm). SSYGST or DSYGST Reduces a symmetric-definite generalized eigenproblem to standard form using the factorization computed by SPOTRF or DPOTRF. SSYGV or DSYGV Computes all the eigenvalues and eigenvectors of a generalized symmetric-definite eigenproblem. Replacement with newer version SSYGVD or DSYGVD suggested. SSYGVD or DSYGVD Computes all the eigenvalues and eigenvectors of a generalized symmetric-definite eigenproblem and uses a divide and conquer method to calculate eigenvectors (driver). SSYGVX or DSYGVX Computes selected eigenvalues and eigenvectors of a generalized symmetric-definite eigenproblem (expert driver). xSYRFS (P) Improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite. xSYRFSX (P) Improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and provides error bounds and backward error estimates for the solution (extra precision). xSYSV Solves a real or complex symmetric indefinite system of linear equations (simple driver). xSYTRF is called to compute the factorization of a complex symmetric matrix using the diagonal pivoting method. xSYSV_ROOK Solves a real or complex symmetric indefinite system of linear equations (simple driver). xSYTRF_ROOK is called to compute the factorization of a complex symmetric matrix using the bounded Bunch-Kauffman ("rook") diagonal pivoting method. xSYSVX Solves a real or complex symmetric indefinite system of linear equations (expert driver). xSYSVXX (P) Solves a real or complex symmetric indefinite system of linear equations (expert driver, extra precision). If requested, both normwise and maximum component-wise error bounds are returned. SSYTD2 or DSYTD2 Reduces a real symmetric matrix to a real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). xSYTF2 Computes the factorization of a real or complex symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm). xSYTF2_ROOK Computes the factorization of a real or complex symmetric indefinite matrix, using the bounded Bunch-Kauffman ("rook") diagonal pivoting method (unblocked algorithm). SSYTRD or DSYTRD Reduces a real symmetric matrix to a real symmetric tridiagonal form by using an orthogonal similarity transformation. xSYTRF (P) Computes the factorization of a real or complex symmetric indefinite matrix using the Bunch-Kaufman diagonal pivoting method (blocked algorithm). xSYTRI Computes the inverse of a real or complex symmetric indefinite matrix using the factorization computed by xSYTRF. xSYTRI_ROOK Computes the inverse of a real or complex symmetric indefinite matrix using the factorization computed by xSYTRF_ROOK. xSYTRI2 Computes the inverse of a real or complex symmetric indefinite matrix using the factorization computed by xSYTRF. Sets the leading dimension of the workspace before calling xSYTRF2X that actually computes the inverse. xSYTRI2X (P) Computes the inverse of a real or complex symmetric indefinite matrix using the factorization computed by xSYTRF. Used by xSYTRI2. xSYTRS (P) Solves a system of linear equations with a real or complex symmetric matrix using the factorization computed by xSYTRF. xSYTRS_ROOK (P) Solves a system of linear equations with a real or complex symmetric matrix using the factorization computed by xSYTRF_ROOK. xSYTRS2 (P) Solves a system of linear equations with a real or complex symmetric matrix using the factorization computed by xSYTRF and converted by xSYCONV.

Table 48  Triangular Band Matrix Routines

Routine Function xTBCON Estimates the reciprocal of the condition number of a triangular band matrix. xTBRFS (P) Determines error bounds and estimates for solving a triangular banded system of linear equations. xTBTRS Solves a triangular banded system of linear equations.

Table 49  Triangular Matrix-Generalized Problem (Pair of Triangular Matrices) Routines

Routine Function xTGEVC (P) Computes some or all of the right and/or left eigenvectors of a pair of real or complex triangular matrices, computed by xGGHRD and xHGEQZ. xTGEXC Reorders the generalized Schur decomposition of a real or complex matrix pair using an orthogonal or unitary equivalence transformation. xTGSEN (P) Reorders the generalized Schur decomposition of a real or complex matrix pair and computes the generalized eigenvalues. xTGSJA (P) Computes the generalized singular value decomposition (SVD) from two real or complex triangular or trapezoidal matrices obtained from xGGSVP. CTGSNA (P) or ZTGSNA (P) Estimates the reciprocal of the condition numbers for specified eigenvalues and eigenvectors of two matrices in generalized Schur canonical form. STGSNA or DTGSNA Estimates the reciprocal of the condition numbers for specified eigenvalues and eigenvectors of two matrices in generalized real Schur canonical form. xTGSYL Solves the generalized Sylvester equation.

Table 50  Triangular Matrix in Packed Storage Routines

Routine Function xTPCON Estimates the reciprocal or the condition number of a triangular matrix in packed storage. xTPMQRT Applies a real or complex orthogonal matrix obtained from a “triangular-pentagonal” block reflector to a general matrix, which consists of two blocks. xTPQRT Computes a blocked QR factorization of a real or complex “triangular-pentagonal” matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation. xTPQRT2 Computes a QR factorization of a real or complex “triangular-pentagonal” matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation. xTPRFS (P) Provides error bounds and backward error estimates for the solution to a real or complex system of linear equations with a triangular packed coefficient matrix. The solution should be preliminary obtained by xTPTRS or some other means. xTPTRI Computes the inverse of a real or complex triangular matrix in packed storage. xTPTRS Solves a real or complex triangular system of linear equations where the coefficient matrix is in packed storage. xTPTTF Copies a real or complex triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF). xTPTTR Copies a real or complex triangular matrix from the standard packed format (TP) to the standard full-packed format (TR).

Table 51  Triangular Matrix in Rectangular Full-Packed (RFP) Format and Standard Packed Format Routines

Routine Function xTFSM (P) Solves a matrix equation with real or complex matrices. One operand is a triangular matrix in the RFP format. xTFTRI Computes the inverse of a real or complex triangular matrix stored in RFP format. xTFTTP Copies a real or complex triangular matrix from the rectangular full-packed format (TF) to the standard packed format (TP). xTFTTR Copies a real or complex triangular matrix from the rectangular full-packed format (TF) to the standard full format (TR). xTPTTF Copies a real or complex triangular matrix from the standard packed format (TP) to the rectangular full packed format (TF). xTPTTR Copies a real or complex triangular matrix from the standard packed format (TP) to the standard full-packed format (TR). xTRTTF Copies a real or complex triangular matrix from the standard full format (TR) to the rectangular full-packed format (TF). xTRTTP Copies a real or complex triangular matrix from the standard full format (TR) to the standard packed format (TP).

Table 52  Triangular Matrix Routines

Routine Function xTRCON Estimates the reciprocal or the condition number of a real or complex triangular matrix. xTREVC (P) Computes right and/or left eigenvectors of a real or complex upper triangular matrix. xTREVC3 (P) Computes some or all right and/or left eigenvectors of a real or complex upper quasi-triangular matrix. xTREXC Reorders the Schur factorization of a real or complex matrix using an orthogonal or unitary similarity transformation. xTRRFS (P) Provides error bounds and estimates for a triangular system of linear equations with a real or complex triangular matrix. CTRSEN (P) or ZTRSEN (P) Reorders the Schur factorization of a complex matrix A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in the leading positions in the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. STRSEN or DTRSEN Reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in the leading positions in the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. xTRSNA (P) Estimates the reciprocal condition numbers of selected eigenvalues and eigenvectors of an upper quasi-triangular matrix. xTRSYL Solves a Sylvester matrix equation. xTRTRI Computes the inverse of a real or complex triangular matrix (unblocked algorithm). xTRTRS Solves a triangular system of linear equations.

Table 53  Trapezoidal Matrix Routines

Routine Function xLARZ Applies an elementary reflector (as returned by xTZRZF) to a real or complex general matrix. xLARZB (P) Applies a block reflector or its transpose to a real general matrix or applies a block reflector or its conjugate-transpose to a complex general matrix. xLARZT Forms the triangular factor T of a real or complex block reflector H, which is defined as a product of k elementary reflectors. xLATZM Deprecated routine replaced by xORMZ. Applies a Householder matrix generated by xTZRQF to a real or complex matrix. xTZRQF (P) Deprecated routine replaced by routine xTZRZF. xTZRZF (P) Reduces a rectangular upper trapezoidal matrix to an upper triangular form by means of orthogonal transformations.

Table 54  Unitary Matrix Routines

Routine Function CUNBDB or ZUNBDB Simultaneously bidiagonalizes the blocks of an M-by-M partitioned unitary matrix. CUNBDB1 or ZUNBDB1 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 1). CUNBDB2 or ZUNBDB2 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 2). CUNBDB3 or ZUNBDB3 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 3). CUNBDB4 or ZUNBDB4 Simultaneously bidiagonalizes the blocks of a tall and skinny matrix with orthonormal columns (variant 4). CUNBDB5 or ZUNBDB5 Orthogonalizes the column vector X with respect to the orthonormal columns of Q. CUNBDB5 or ZUNBDB5 Orthogonalizes the column vector X with respect to the orthonormal columns of Q. Used by CUNBDB5 or ZUNBDB5. CUNCSD2BY1 or ZUNCSD2BY1 Computes the CS decomposition of an M-by-Q matrix X with orthonormal columns that has been partitioned into a 2-by-1 block structure. CUNG2L (P) or ZUNG2L (P) Generates an M-by-N complex matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M, as returned by CGEQLF or ZGEQLF. CUNG2R (P) or ZUNG2R (P) Generates an M-by-N complex matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M, as returned by CGEQRF or ZGEQRF. CUNGBR (P) or ZUNGBR (P) Generates an unitary transformation matrix from reduction to bidiagonal form, as determined by CGEBRD or ZGEBRD. CUNGHR (P) or ZUNGHR (P) Generates an orthogonal transformation matrix reduced to Hessenberg form, as determined by CGEHRD or ZGEHRD. CUNGL2 (P) or ZUNGL2 (P) Generates all or part of an unitary matrix Q from an LQ factorization determined by CGELQF or ZGELQF (unblocked algorithm). CUNGLQ (P) or ZUNGLQ (P) Generates an unitary matrix Q from an LQ factorization, as returned by CGELQF or ZGELQF. CUNGQL (P) or ZUNGQL (P) Generates an unitary matrix Q from a QL factorization, as returned by CGEQLF or ZGEQLF. CUNGQR (P) or ZUNGQR (P) Generates an unitary matrix Q from a QR factorization, as returned by CGEQRF or ZGEQRF. CUNGR2 (P) or ZUNGR2 (P) Generates all or part of an unitary matrix Q from an RQ factorization determined by CGERQF or ZGERQF (unblocked algorithm). CUNGRQ (P) or ZUNGRQ (P) Generates an unitary matrix Q from an RQ factorization, as returned by CGERQF or ZGERQF. CUNGTR (P) or ZUNGTR (P) Generates an unitary matrix reduced to a tridiagonal form, by CHETRD or ZHETRD. CUNM22 or ZUNM22 Multiplies a general matrix by a banded unitary matrix. CUNM2L or ZUNM2L Multiplies a general matrix by the unitary matrix from a QL factorization determined by CGEQLF or ZGEQLF (unblocked algorithm). CUNM2R or ZUNM2R Multiplies a general matrix by an unitary matrix from a QR factorization determined by CGEQRF or ZGERLF (unblocked algorithm). CUNMBR or ZUNMBR Multiplies a general matrix with an unitary transformation matrix reduced to a bidiagonal form, as determined by CGEBRD or ZGEBRD. CUNMHR or ZUNMHR Multiplies a general matrix by an unitary matrix reduced to the Hessenberg form by CGEHRD or ZGEHRD. CUNML2 or ZUNML2 Multiplies a general matrix by an unitary matrix from an LQ factorization determined by CGELQF or ZGELQF (unblocked algorithm). CUNMLQ or ZUNMLQ Multiplies a general matrix by an unitary matrix from an LQ factorization, as returned by CGELQF or ZGELQF. CUNMQL or ZUNMQL Multiplies a general matrix by an unitary matrix from a QL factorization, as returned by CGEQLF or ZGEQLF. CUNMQR or ZUNMQR Multiplies a general matrix by an unitary matrix from a QR factorization, as returned by CGEQRF or ZGEQRF. CUNMR2 or ZUNMR2 Multiplies a general matrix by an unitary matrix from an RQ factorization determined by CGERQF or ZGERQF (unblocked algorithm). CUNMR3 or ZUNMR3 Multiplies a general matrix by an unitary matrix from an RZ factorization determined by CTZRZF or ZTZRZF (unblocked algorithm). CUNMRQ or ZUNMRQ Multiplies a general matrix by an unitary matrix from an RQ factorization, as returned by CGERQF or ZGERQF. CUNMRZ or ZUNMRZ Multiplies a general matrix by an unitary matrix from an RZ factorization, as returned by CTZRZF or ZTZRZF. CUNMTR or ZUNMTR Multiplies a general matrix by an unitary transformation matrix reduced to tridiagonal form by CHETRD or ZHETRD.

Table 55  Unitary Matrix in Packed Storage Routines

Routine Function CUPGTR (P) or ZUPGTR (P) Generates an unitary transformation matrix from a tridiagonal matrix determined by CHPTRD or ZHPTRD. CUPMTR or ZUPMTR Multiplies a general matrix by an unitary transformation matrix reduced to tridiagonal form by CHPTRD or ZHPTRD.