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CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) NNNNAAAAMMMMEEEE complib, complib.sgimath, sgimath - Scientific and Mathematical Library DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN The Silicon Graphics Scientific Mathematical Library, complib.sgimath, is a comprehensive collection of high-performance math libraries providing technical support for mathematical and numerical techniques used in scientific and technical computing. This library is provided by SGI for the convenience of the users. Support is limited to bug fixes at SGI's discretion. The library complib.sgimath contains an extensive collection of industry standard libraries such as Basic Linear Algebra Subprograms (BLAS), the Extended BLAS (Level 2 and Level 3), EISPACK, LINPACK, and LAPACK. Internally developed libraries for calculating Fast Fourier Transforms (FFT's) and Convolutions are also included, as well as select direct sparse matrix solvers. Documentation is available per routine via individual man pages. General man pages for the Blas ( mmmmaaaannnn bbbbllllaaaassss ), fft routines ( mmmmaaaannnn fffffffftttt ), convolution routines ( mmmmaaaannnn ccccoooonnnnvvvv ) and LAPACK ( mmmmaaaannnn llllaaaappppaaaacccckkkk ) are also available. The complib.sgimath library is available on Silicon Graphics Inc. machines via the -l compilation flag, -lcomplib.sgimath (append _mp for multiprocessing libraries) for OS versions 5.1 and higher. The library is available for R3000, R4000 (-mips2) and R8000 architectures (-mips4), and single and multiple processor architectures (-mp). Documentation for LAPACK and LINPACK is available by writing: SIAM Department BKLP93 P.O. Box 7260 Philadelphia, Pennsylvania 19101 Anderson E., et. al. SIAM 1992 "LAPACK Users Guide", $19.50 Dongarra J., et. al. SIAM 1979 "LINPACK Users Guide", $19.50 AAAAVVVVAAAAIIIILLLLAAAABBBBIIIILLLLIIIITTTTYYYY Many of the routines in complib.sgimath are available from: netlib@research.att.com. mail netlib@research.att.com send index The Internet address "netlib@research.att.com" refers to a gateway machine, 192.20.225.2, at AT&T Bell Labs in Murray Hill, New Jersey. This address should be understood on all the major networks. For systems having only uucp connections, use uunet!research!netlib. In this case, someone will be paying for long distance 1200bps phone calls, so keep PPPPaaaaggggeeee 1111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) your requests to a reasonable size! If ftp is more convenient for you than email, you may connect to "research.att.com"; log in as "netlib". (This is for read-only ftp, not telnet.) Filesnames end in ".Z", reflecting the need to have the "uncompress" command applied after you've ftp'd them. "compress" source code for a variety of machines and operating systems can be obtained by anonymous ftp from ftp.uu.net. The files in netlib/crc/res/ have a list of files with modification times, lengths, and checksums to assist people who wish to automatically track changes. For access from Europe, try the duplicate collection in Oslo: Internet: netlib@nac.no EARN/BITNET: netlib%nac.no@norunix.bitnet (now livid.uib.no ?) X.400: s=netlib; o=nac; prmd=uninett; c=no; EUNET/uucp: nuug!netlib For the Pacific, try netlib@draci.cs.uow.edu.au located at the University of Wollongong, NSW, Australia. The contents of netlib (other than toms) is available on CD-ROM from Prime Time Freeware. The price of their two-disc set, which also includes statlib, TeX, Modula3, Interview, Postgres, Tcl/Tk, and more is about $60; for current information contact Prime Time Freeware 370 Altair Way, Suite 150 Tel: +1 408-738-4832 ptf@cfcl.com Sunnyvale, CA 94086 USA Fax: +1 408-738-2050 The following libraries are available from "netlib@research.att.com". These libraries are part of complib.sgimath. The BLAS library, level 1, 2 and 3 and machine constants. The LAPACK library, for the most common problems in numerical linear algebra: linear equations, linear least squares problems, eigenvalue problems, and singular value problems. It has been designed to be efficient on a wide range of modern high-performance computers. The LINPACK library, for linear equations and linear least squares problems, linear systems whose matrices are general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square. In addition, the package computes the QR and singular value decompositions of rectangular matrices and applies them to least squares problems. The EISPACK library, a collection of double precision Fortran subroutines that compute the eigenvalues and eigenvectors of nine classes of matrices. The package can determine the eigensystems of double complex general, double complex Hermitian, double precision general, double precision symmetric, double precision symmetric band, double precision symmetric tridiagonal, special double precision tridiagonal, generalized double precision, and generalized double precision symmetric matrices. In PPPPaaaaggggeeee 2222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) addition, there are two routines which use the singular value decomposition to solve certain least squares problems. IIIINNNNDDDDEEEEXXXX BBBBLLLLAAAASSSS LLLLIIIIBBBBRRRRAAAARRRRYYYY ---- BBBBaaaassssiiiicccc LLLLiiiinnnneeeeaaaarrrr AAAAllllggggeeeebbbbrrrraaaa SSSSuuuubbbbpppprrrrooooggggrrrraaaammmmssss BBBBLLLLAAAASSSS LLLLeeeevvvveeeellll 1111 dnrm2, snrm2, zdnrm2, csnrm2 - BLAS level ONE Euclidean norm functions. dcopy, scopy, zcopy, ccopy - BLAS level ONE copy subroutines drotg, srotg, drot, srot - BLAS level ONE rotation subroutines idamax, isamax, izamax, icamax - BLAS level ONE Maximum index functions ddot, sdot, zdotc, cdotc, zdotu, cdotu - BLAS level ONE, dot product functions dswap, sswap, zswap, cswap - BLAS level ONE swap subroutines dasum, sasum, dzasum, scasum - BLAS level ONE L1 norm functions. dscal, sscal, zscal, cscal, zdscal, csscal - BLAS level ONE scaling subroutines daxpy, saxpy, zaxpy, caxpy - BLAS level ONE axpy subroutines BBBBLLLLAAAASSSS LLLLeeeevvvveeeellll 2222 dgemv, sgemv, zgemv, cgemv - BLAS Level Two Matrix-Vector Product dspr, sspr, zhpr, chpr - BLAS Level Two Symmetric Packed Matrix Rank 1 Update dsyr, ssyr, zher, cher - BLAS Level Two (Symmetric/Hermitian)Matrix Rank 1 Update dtpmv, stpmv, ztpmv, ctpmv - BLAS Level Two Matrix-Vector Product dtpsv, stpsv, ztpsv, ctpsv - BLAS Level Two Solution of Triangular System dger, sger, zgeru, cgeru, zgerc, cgerc - BLAS Level Two Rank 1 Operation dspr2, sspr2, zhpr2, chpr2 - BLAS Level Two Symmetric Packed Matrix Rank 2 Update dsyr2, ssyr2, zher2, cher2 - BLAS Level Two (Symmetric/Hermitian)Matrix Rank 2 Update dsbmv, ssbmv, zhbmv, chbmv - BLAS Level Two (Symmetric/Hermitian) Banded Matrix - Vector Product dtrmv, strmv, ztrmv, ctrmv - BLAS Level Two Matrix-Vector Product dtrsv, strsv, ztrsv, ctrsv - BLAS Level Two Solution of triangular system of equations. dgbmv, sgbmv, zgbmv, cgbmv - BLAS Level Two Matrix-Vector Product dspmv, sspmv, zhpmv, chpmv - BLAS Level Two (Symmetric/Hermitian) Packed Matrix - Vector Product dsymv, ssymv, zhemv, chemv - BLAS Level Two (Symmetric/Hermitian)Matrix - Vector Product dtbmv, stbmv, ztbmv, ctbmv, dtbsv, stbsv, ztbsv, ctbsv - BLAS Level Two Matrix-Vector Product and Solution of System of Equations. BBBBLLLLAAAASSSS LLLLeeeevvvveeeellll 3333 dtrmm, strmm, ztrmm, ctrmm - BLAS level three Matrix Product PPPPaaaaggggeeee 3333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) zhemm, chemm - BLAS level three Hermitian Matrix Product dsyr2k, ssyr2k, zsyr2k, csyr2k - BLAS level three Symetric Rank 2K Update. zher2k and cher2k - BLAS level three Hermitian Rank 2K Update dsymm, ssymm, zsymm, csymm - BLAS level three Symmetric Matrix Product dsyrk, ssyrk, zsyrk, csyrk - BLAS level three Symetric Rank K Update. dtrsm, strsm, ztrsm, ctrsm - BLAS level three Solution of Systems of Equations dgemm, sgemm, zgemm, cgemm - BLAS level three Matrix Product zherk and cherk - BLAS level three Hermitiam Rank K Update EEEEIIIISSSSPPPPAAAACCCCKKKK LLLLIIIIBBBBRRRRAAAARRRRYYYY BAKVEC - This subroutine forms the eigenvectors of a NONSYMMETRIC TRIDIAGONAL matrix by back transforming those of the corresponding symmetric matrix determined by FIGI. BALANC - This subroutine balances a REAL matrix and isolates eigenvalues whenever possible. BALBAK - This subroutine forms the eigenvectors of a REAL GENERAL matrix by back transforming those of the corresponding balanced matrix determined by BALANC. BANDR - This subroutine reduces a REAL SYMMETRIC BAND matrix to a symmetric tridiagonal matrix using and optionally accumulating orthogonal similarity transformations. BANDV - This subroutine finds those eigenvectors of a REAL SYMMETRIC BAND matrix corresponding to specified eigenvalues, using inverse iteration. The subroutine may also be used to solve systems of linear equations with a symmetric or non-symmetric band coefficient matrix. BISECT - This subroutine finds those eigenvalues of a TRIDIAGONAL SYMMETRIC matrix which lie in a specified interval, using bisection. BQR - This subroutine finds the eigenvalue of smallest (usually) magnitude of a REAL SYMMETRIC BAND matrix using the QR algorithm with shifts of origin. Consecutive calls can be made to find further eigenvalues. CBABK2 - This subroutine forms the eigenvectors of a COMPLEX GENERAL matrix by back transforming those of the corresponding balanced matrix PPPPaaaaggggeeee 4444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) determined by CBAL. CBAL - This subroutine balances a COMPLEX matrix and isolates eigenvalues whenever possible. CDIV - COMPLEX DIVISION, (CR,CI) = (AR,AI)/(BR,BI) CG - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a COMPLEX GENERAL matrix. CH - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a COMPLEX HERMITIAN matrix. CINVIT - This subroutine finds those eigenvectors of A COMPLEX UPPER Hessenberg matrix corresponding to specified eigenvalues, using inverse iteration. COMBAK - This subroutine forms the eigenvectors of a COMPLEX GENERAL matrix by back transforming those of the corresponding upper Hessenberg matrix determined by COMHES. COMHES - Given a COMPLEX GENERAL matrix, this subroutine reduces a submatrix situated in rows and columns LOW through IGH to upper Hessenberg form by stabilized elementary similarity transformations. COMLR - This subroutine finds the eigenvalues of a COMPLEX UPPER Hessenberg matrix by the modified LR method. COMLR2 - This subroutine finds the eigenvalues and eigenvectors of a COMPLEX UPPER Hessenberg matrix by the modified LR method. The eigenvectors of a COMPLEX GENERAL matrix can also be found if COMHES has been used to reduce this general matrix to Hessenberg form. COMQR - This subroutine finds the eigenvalues of a COMPLEX upper Hessenberg matrix by the QR method. COMQR2 - This subroutine finds the eigenvalues and eigenvectors of a COMPLEX UPPER Hessenberg matrix by the QR method. The eigenvectors of a COMPLEX GENERAL matrix can also be found if CORTH has been used to PPPPaaaaggggeeee 5555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) reduce this general matrix to Hessenberg form. CORTB - This subroutine forms the eigenvectors of a COMPLEX GENERAL matrix by back transforming those of the corresponding upper Hessenberg matrix determined by CORTH. CORTH - Given a COMPLEX GENERAL matrix, this subroutine reduces a submatrix situated in rows and columns LOW through IGH to upper Hessenberg form by unitary similarity transformations. CSROOT - (YR,YI) = COMPLEX SQRT(XR,XI) BRANCH CHOSEN SO THAT YR .GE. 0.0 AND SIGN(YI) .EQ. SIGN(XI) ELMBAK - This subroutine forms the eigenvectors of a REAL GENERAL matrix by back transforming those of the corresponding upper Hessenberg matrix determined by ELMHES. ELMHES - Given a REAL GENERAL matrix, this subroutine reduces a submatrix situated in rows and columns LOW through IGH to upper Hessenberg form by stabilized elementary similarity transformations. ELTRAN - This subroutine accumulates the stabilized elementary similarity transformations used in the reduction of a REAL GENERAL matrix to upper Hessenberg form by ELMHES. EPSLON - ESTIMATE UNIT ROUNDOFF IN QUANTITIES OF SIZE X. FIGI - Given a NONSYMMETRIC TRIDIAGONAL matrix such that the products of corresponding pairs of off-diagonal elements are all non-negative, this subroutine reduces it to a symmetric tridiagonal matrix with the same eigenvalues. If, further, a zero product only occurs when both factors are zero, the reduced matrix is similar to the original matrix. FIGI2 - Given a NONSYMMETRIC TRIDIAGONAL matrix such that the products of corresponding pairs of off-diagonal elements are all non-negative, and zero only when both factors are zero, this subroutine reduces it to a SYMMETRIC TRIDIAGONAL matrix using and accumulating diagonal similarity transformations. HQR - This subroutine finds the eigenvalues of a REAL UPPER Hessenberg matrix by the QR method. PPPPaaaaggggeeee 6666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) HQR2 - This subroutine finds the eigenvalues and eigenvectors of a REAL UPPER Hessenberg matrix by the QR method. The eigenvectors of a REAL GENERAL matrix can also be found if ELMHES and ELTRAN or ORTHES and ORTRAN have been used to reduce this general matrix to Hessenberg form and to accumulate the similarity transformations. HTRIB3 - This subroutine forms the eigenvectors of a COMPLEX HERMITIAN matrix by back transforming those of the corresponding real symmetric tridiagonal matrix determined by HTRID3. HTRIBK - This subroutine forms the eigenvectors of a COMPLEX HERMITIAN matrix by back transforming those of the corresponding real symmetric tridiagonal matrix determined by HTRIDI. HTRID3 - This subroutine reduces a COMPLEX HERMITIAN matrix, stored as a single square array, to a real symmetric tridiagonal matrix using unitary similarity transformations. HTRIDI - This subroutine reduces a COMPLEX HERMITIAN matrix to a real symmetric tridiagonal matrix using unitary similarity transformations. IMTQL1 - This subroutine finds the eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method. IMTQL2 - This subroutine finds the eigenvalues and eigenvectors of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method. The eigenvectors of a FULL SYMMETRIC matrix can also be found if TRED2 has been used to reduce this full matrix to tridiagonal form. IMTQLV - This subroutine finds the eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method and associates with them their corresponding submatrix indices. INVIT - This subroutine finds those eigenvectors of a REAL UPPER Hessenberg matrix corresponding to specified eigenvalues, using inverse iteration. MINFIT - This subroutine determines, towards the solution of the linear T system AX=B, the singular value decomposition A=USV of a real T M by N rectangular matrix, forming U B rather than U. Householder bidiagonalization and a variant of the QR algorithm are used. PPPPaaaaggggeeee 7777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ORTBAK - This subroutine forms the eigenvectors of a REAL GENERAL matrix by back transforming those of the corresponding upper Hessenberg matrix determined by ORTHES. ORTHES - Given a REAL GENERAL matrix, this subroutine reduces a submatrix situated in rows and columns LOW through IGH to upper Hessenberg form by orthogonal similarity transformations. ORTRAN - This subroutine accumulates the orthogonal similarity transformations used in the reduction of a REAL GENERAL matrix to upper Hessenberg form by ORTHES. PYTHAG - FINDS SQRT(A**2+B**2) WITHOUT OVERFLOW OR DESTRUCTIVE UNDERFLOW QZHES - This subroutine accepts a pair of REAL GENERAL matrices and reduces one of them to upper Hessenberg form and the other to upper triangular form using orthogonal transformations. It is usually followed by QZIT, QZVAL and, possibly, QZVEC. QZIT - This subroutine accepts a pair of REAL matrices, one of them in upper Hessenberg form and the other in upper triangular form. It reduces the Hessenberg matrix to quasi-triangular form using orthogonal transformations while maintaining the triangular form of the other matrix. It is usually preceded by QZHES and followed by QZVAL and, possibly, QZVEC. QZVAL - This subroutine accepts a pair of REAL matrices, one of them in quasi-triangular form and the other in upper triangular form. It reduces the quasi-triangular matrix further, so that any remaining 2-by-2 blocks correspond to pairs of complex eigenvalues, and returns quantities whose ratios give the generalized eigenvalues. It is usually preceded by QZHES and QZIT and may be followed by QZVEC. QZVEC - This subroutine accepts a pair of REAL matrices, one of them in quasi-triangular form (in which each 2-by-2 block corresponds to a pair of complex eigenvalues) and the other in upper triangular form. It computes the eigenvectors of the triangular problem and transforms the results back to the original coordinate system. It is usually preceded by QZHES, QZIT, and QZVAL. RATQR - This subroutine finds the algebraically smallest or largest eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the rational QR method with Newton corrections. PPPPaaaaggggeeee 8888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) REBAK - This subroutine forms the eigenvectors of a generalized SYMMETRIC eigensystem by back transforming those of the derived symmetric matrix determined by REDUC. REBAKB - This subroutine forms the eigenvectors of a generalized SYMMETRIC eigensystem by back transforming those of the derived symmetric matrix determined by REDUC2. REDUC - This subroutine reduces the generalized SYMMETRIC eigenproblem Ax=(LAMBDA)Bx, where B is POSITIVE DEFINITE, to the standard symmetric eigenproblem using the Cholesky factorization of B. REDUC2 - This subroutine reduces the generalized SYMMETRIC eigenproblems ABx=(LAMBDA)x OR BAy=(LAMBDA)y, where B is POSITIVE DEFINITE, to the standard symmetric eigenproblem using the Cholesky factorization of B. RG - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) To find the eigenvalues and eigenvectors (if desired) of a REAL GENERAL matrix. RGG - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) for the REAL GENERAL GENERALIZED eigenproblem Ax = (LAMBDA)Bx. RS - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a REAL SYMMETRIC matrix. RSB - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a REAL SYMMETRIC BAND matrix. RSG - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) To find the eigenvalues and eigenvectors (if desired) for the REAL SYMMETRIC generalized eigenproblem Ax = (LAMBDA)Bx. RSGAB - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) for the REAL SYMMETRIC generalized eigenproblem ABx = (LAMBDA)x. PPPPaaaaggggeeee 9999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) RSGBA - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) for the REAL SYMMETRIC generalized eigenproblem BAx = (LAMBDA)x. RSM - THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK) TO FIND ALL OF THE EIGENVALUES AND SOME OF THE EIGENVECTORS OF A REAL SYMMETRIC MATRIX. RSP - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a REAL SYMMETRIC PACKED matrix. RST - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a REAL SYMMETRIC TRIDIAGONAL matrix. RT - This subroutine calls the recommended sequence of subroutines from the eigensystem subroutine package (EISPACK) to find the eigenvalues and eigenvectors (if desired) of a special REAL TRIDIAGONAL matrix. SVD - This subroutine determines the singular value decomposition T A=USV of a REAL M by N rectangular matrix. Householder bidiagonalization and a variant of the QR algorithm are used. TINVIT - This subroutine finds those eigenvectors of a TRIDIAGONAL SYMMETRIC matrix corresponding to specified eigenvalues, using inverse iteration. TQL1 - This subroutine finds the eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the QL method. TQL2 - This subroutine finds the eigenvalues and eigenvectors of a SYMMETRIC TRIDIAGONAL matrix by the QL method. The eigenvectors of a FULL SYMMETRIC matrix can also be found if TRED2 has been used to reduce this full matrix to tridiagonal form. TQLRAT - This subroutine finds the eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the rational QL method. TRBAK1 - This subroutine forms the eigenvectors of a REAL SYMMETRIC matrix by back transforming those of the corresponding symmetric PPPPaaaaggggeeee 11110000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) tridiagonal matrix determined by TRED1. TRBAK3 - This subroutine forms the eigenvectors of a REAL SYMMETRIC matrix by back transforming those of the corresponding symmetric tridiagonal matrix determined by TRED3. TRED1 - This subroutine reduces a REAL SYMMETRIC matrix to a symmetric tridiagonal matrix using orthogonal similarity transformations. TRED2 - This subroutine reduces a REAL SYMMETRIC matrix to a symmetric tridiagonal matrix using and accumulating orthogonal similarity transformations. TRED3 - This subroutine reduces a REAL SYMMETRIC matrix, stored as a one-dimensional array, to a symmetric tridiagonal matrix using orthogonal similarity transformations. TRIDIB - This subroutine finds those eigenvalues of a TRIDIAGONAL SYMMETRIC matrix between specified boundary indices, using bisection. TSTURM - This subroutine finds those eigenvalues of a TRIDIAGONAL SYMMETRIC matrix which lie in a specified interval and their associated eigenvectors, using bisection and inverse iteration. LLLLIIIINNNNPPPPAAAACCCCKKKK LLLLIIIIBBBBRRRRAAAARRRRYYYY CCHDC - CCHDC computes the Cholesky decomposition of a positive definite matrix. A pivoting option allows the user to estimate the condition of a positive definite matrix or determine the rank of a positive semidefinite matrix. CCHDD - CCHDD downdates an augmented Cholesky decomposition or the triangular factor of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, CCHDD determines a unitary matrix U and a scalar ZETA such that (R Z ) (RR ZZ) U * ( ) = ( ) , (0 ZETA) ( X Y) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) removed. In this PPPPaaaaggggeeee 11111111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) case, if RHO is the norm of the residual vector, then the norm of the residual vector of the downdated problem is SQRT(RHO**2 - ZETA**2). CCHDD will simultaneously downdate several triplets (Z,Y,RHO) along with R. For a less terse description of what CCHDD does and how it may be applied, see the LINPACK Guide. CCHEX - CCHEX updates the Cholesky factorization A = CTRANS(R)*R of a positive definite matrix A of order P under diagonal permutations of the form TRANS(E)*A*E where E is a permutation matrix. Specifically, given an upper triangular matrix R and a permutation matrix E (which is specified by K, L, and JOB), CCHEX determines a unitary matrix U such that U*R*E = RR, where RR is upper triangular. At the users option, the transformation U will be multiplied into the array Z. If A = CTRANS(X)*X, so that R is the triangular part of the QR factorization of X, then RR is the triangular part of the QR factorization of X*E, i.e. X with its columns permuted. For a less terse description of what CCHEX does and how it may be applied, see the LINPACK Guide. CCHUD - CCHUD updates an augmented Cholesky decomposition of the triangular part of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, CCHUD determines a unitary matrix U and a scalar ZETA such that (R Z) (RR ZZ ) U * ( ) = ( ) , (X Y) ( 0 ZETA) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) appended. In this case, if RHO is the norm of the residual vector, then the norm of the residual vector of the updated problem is SQRT(RHO**2 + ZETA**2). CCHUD will simultaneously update several triplets (Z,Y,RHO). CGBCO - CGBCO factors a complex band matrix by Gaussian elimination and estimates the condition of the matrix. CGBDI - CGBDI computes the determinant of a band matrix using the factors computed by CGBCO or CGBFA. If the inverse is needed, use CGBSL N times. PPPPaaaaggggeeee 11112222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CGBFA - CGBFA factors a complex band matrix by elimination. CGBSL - CGBSL solves the complex band system A * X = B or CTRANS(A) * X = B using the factors computed by CGBCO or CGBFA. CGECO - CGECO factors a complex matrix by Gaussian elimination and estimates the condition of the matrix. CGEDI - CGEDI computes the determinant and inverse of a matrix using the factors computed by CGECO or CGEFA. CGEFA - CGEFA factors a complex matrix by Gaussian elimination. CGESL - CGESL solves the complex system A * X = B or CTRANS(A) * X = B using the factors computed by CGECO or CGEFA. CGTSL - CGTSL given a general tridiagonal matrix and a right hand side will find the solution. CHICO - CHICO factors a complex Hermitian matrix by elimination with symmetric pivoting and estimates the condition of the matrix. CHIDI - CHIDI computes the determinant, inertia and inverse of a complex Hermitian matrix using the factors from CHIFA. CHIFA - CHIFA factors a complex Hermitian matrix by elimination with symmetric pivoting. CHISL - CHISL solves the complex Hermitian system A * X = B using the factors computed by CHIFA. CHPCO - CHPCO factors a complex Hermitian matrix stored in packed form by elimination with symmetric pivoting and estimates the condition of the matrix. CHPDI - CHPDI computes the determinant, inertia and inverse of a complex Hermitian matrix using the factors from CHPFA, where the matrix is stored in packed form. CHPFA - CHPFA factors a complex Hermitian matrix stored in packed form by elimination with symmetric pivoting. CHPSL - CHISL solves the complex Hermitian system A * X = B using the factors computed by CHPFA. CPBCO - CPBCO factors a complex Hermitian positive definite matrix stored in band form and estimates the condition of the matrix. CPBDI - CPBDI computes the determinant of a complex Hermitian positive definite band matrix using the factors computed by CPBCO or CPBFA. If the inverse is needed, use CPBSL N times. PPPPaaaaggggeeee 11113333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CPBFA - CPBFA factors a complex Hermitian positive definite matrix stored in band form. CPBSL - CPBSL solves the complex Hermitian positive definite band system A*X = B using the factors computed by CPBCO or CPBFA. CPOCO - CPOCO factors a complex Hermitian positive definite matrix and estimates the condition of the matrix. CPODI - CPODI computes the determinant and inverse of a certain complex Hermitian positive definite matrix (see below) using the factors computed by CPOCO, CPOFA or CQRDC. CPOFA - CPOFA factors a complex Hermitian positive definite matrix. CPOSL - CPOSL solves the COMPLEX Hermitian positive definite system A * X = B using the factors computed by CPOCO or CPOFA. CPPCO - CPPCO factors a complex Hermitian positive definite matrix stored in packed form and estimates the condition of the matrix. CPPDI - CPPDI computes the determinant and inverse of a complex Hermitian positive definite matrix using the factors computed by CPPCO or CPPFA . CPPFA - CPPFA factors a complex Hermitian positive definite matrix stored in packed form. CPPSL - CPPSL solves the complex Hermitian positive definite system A * X = B using the factors computed by CPPCO or CPPFA. CPTSL - CPTSL given a positive definite tridiagonal matrix and a right hand side will find the solution. CQRDC - CQRDC uses Householder transformations to compute the QR factorization of an N by P matrix X. Column pivoting based on the 2- norms of the reduced columns may be performed at the users option. CQRSL - CQRSL applies the output of CQRDC to compute coordinate transformations, projections, and least squares solutions. For K .LE. MIN(N,P), let XK be the matrix XK = (X(JVPT(1)),X(JVPT(2)), ... ,X(JVPT(K))) formed from columnns JVPT(1), ... ,JVPT(K) of the original N x P matrix X that was input to CQRDC (if no pivoting was done, XK consists of the first K columns of X in their original order). CQRDC produces a factored unitary matrix Q and an upper triangular matrix R such that XK = Q * (R) (0) PPPPaaaaggggeeee 11114444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) This information is contained in coded form in the arrays X and QRAUX. CSICO - CSICO factors a complex symmetric matrix by elimination with symmetric pivoting and estimates the condition of the matrix. CSIDI - CSIDI computes the determinant and inverse of a complex symmetric matrix using the factors from CSIFA. CSIFA - CSIFA factors a complex symmetric matrix by elimination with symmetric pivoting. CSISL - CSISL solves the complex symmetric system A * X = B using the factors computed by CSIFA. CSPCO - CSPCO factors a complex symmetric matrix stored in packed form by elimination with symmetric pivoting and estimates the condition of the matrix. CSPDI - CSPDI computes the determinant and inverse of a complex symmetric matrix using the factors from CSPFA, where the matrix is stored in packed form. CSPFA - CSPFA factors a complex symmetric matrix stored in packed form by elimination with symmetric pivoting. CSPSL - CSISL solves the complex symmetric system A * X = B using the factors computed by CSPFA. CSVDC - CSVDC is a subroutine to reduce a complex NxP matrix X by unitary transformations U and V to diagonal form. The diagonal elements S(I) are the singular values of X. The columns of U are the corresponding left singular vectors, and the columns of V the right singular vectors. CTRCO - CTRCO estimates the condition of a complex triangular matrix. CTRDI - CTRDI computes the determinant and inverse of a complex triangular matrix. CTRSL - CTRSL solves systems of the form T * X = B or CTRANS(T) * X = B where T is a triangular matrix of order N. Here CTRANS(T) denotes the conjugate transpose of the matrix T. DCHDC - DCHDC computes the Cholesky decomposition of a positive definite matrix. A pivoting option allows the user to estimate the condition of a positive definite matrix or determine the rank of a positive semidefinite matrix. PPPPaaaaggggeeee 11115555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DCHDD - DCHDD downdates an augmented Cholesky decomposition or the triangular factor of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, DCHDD determines an orthogonal matrix U and a scalar ZETA such that (R Z ) (RR ZZ) U * ( ) = ( ) , (0 ZETA) ( X Y) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) removed. In this case, if RHO is the norm of the residual vector, then the norm of the residual vector of the downdated problem is DSQRT(RHO**2 - ZETA**2). DCHDD will simultaneously downdate several triplets (Z,Y,RHO) along with R. For a less terse description of what DCHDD does and how it may be applied, see the LINPACK guide. DCHEX - DCHEX updates the Cholesky factorization A = TRANS(R)*R of a positive definite matrix A of order P under diagonal permutations of the form TRANS(E)*A*E where E is a permutation matrix. Specifically, given an upper triangular matrix R and a permutation matrix E (which is specified by K, L, and JOB), DCHEX determines an orthogonal matrix U such that U*R*E = RR, where RR is upper triangular. At the users option, the transformation U will be multiplied into the array Z. If A = TRANS(X)*X, so that R is the triangular part of the QR factorization of X, then RR is the triangular part of the QR factorization of X*E, i.e. X with its columns permuted. For a less terse description of what DCHEX does and how it may be applied, see the LINPACK guide. DCHUD - DCHUD updates an augmented Cholesky decomposition of the triangular part of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, DCHUD determines a untiary matrix U and a scalar ZETA such that (R Z) (RR ZZ ) U * ( ) = ( ) , (X Y) ( 0 ZETA) PPPPaaaaggggeeee 11116666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) appended. In this case, if RHO is the norm of the residual vector, then the norm of the residual vector of the updated problem is DSQRT(RHO**2 + ZETA**2). DCHUD will simultaneously update several triplets (Z,Y,RHO). For a less terse description of what DCHUD does and how it may be applied, see the LINPACK guide. DGBCO - DGBCO factors a double precision band matrix by Gaussian elimination and estimates the condition of the matrix. DGBDI - DGBDI computes the determinant of a band matrix using the factors computed by DGBCO or DGBFA. If the inverse is needed, use DGBSL N times. DGBFA - DGBFA factors a double precision band matrix by elimination. DGBSL - DGBSL solves the double precision band system A * X = B or TRANS(A) * X = B using the factors computed by DGBCO or DGBFA. DGECO - DGECO factors a double precision matrix by Gaussian elimination and estimates the condition of the matrix. DGEDI - DGEDI computes the determinant and inverse of a matrix using the factors computed by DGECO or DGEFA. DGEFA - DGEFA factors a double precision matrix by Gaussian elimination. DGESL - DGESL solves the double precision system A * X = B or TRANS(A) * X = B using the factors computed by DGECO or DGEFA. DGTSL - DGTSL given a general tridiagonal matrix and a right hand side will find the solution. DPBCO - DPBCO factors a double precision symmetric positive definite matrix stored in band form and estimates the condition of the matrix. DPBDI - DPBDI computes the determinant of a double precision symmetric positive definite band matrix using the factors computed by DPBCO or DPBFA. If the inverse is needed, use DPBSL N times. DPBFA - DPBFA factors a double precision symmetric positive definite matrix stored in band form. DPBSL - DPBSL solves the double precision symmetric positive definite band system A*X = B using the factors computed by DPBCO or DPBFA. DPOCO - DPOCO factors a double precision symmetric positive definite matrix and estimates the condition of the matrix. PPPPaaaaggggeeee 11117777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DPODI - DPODI computes the determinant and inverse of a certain double precision symmetric positive definite matrix (see below) using the factors computed by DPOCO, DPOFA or DQRDC. DPOFA - DPOFA factors a double precision symmetric positive definite matrix. DPOSL - DPOSL solves the double precision symmetric positive definite system A * X = B using the factors computed by DPOCO or DPOFA. DPPCO - DPPCO factors a double precision symmetric positive definite matrix stored in packed form and estimates the condition of the matrix. DPPDI - DPPDI computes the determinant and inverse of a double precision symmetric positive definite matrix using the factors computed by DPPCO or DPPFA . DPPFA - DPPFA factors a double precision symmetric positive definite matrix stored in packed form. DPPSL - DPPSL solves the double precision symmetric positive definite system A * X = B using the factors computed by DPPCO or DPPFA. DPTSL - DPTSL, given a positive definite symmetric tridiagonal matrix and a right hand side, will find the solution. DQRDC - DQRDC uses Householder transformations to compute the QR factorization of an N by P matrix X. Column pivoting based on the 2- norms of the reduced columns may be performed at the user's option. DQRSL - DQRSL applies the output of DQRDC to compute coordinate transformations, projections, and least squares solutions. For K .LE. MIN(N,P), let XK be the matrix XK = (X(JPVT(1)),X(JPVT(2)), ... ,X(JPVT(K))) formed from columnns JPVT(1), ... ,JPVT(K) of the original N X P matrix X that was input to DQRDC (if no pivoting was done, XK consists of the first K columns of X in their original order). DQRDC produces a factored orthogonal matrix Q and an upper triangular matrix R such that XK = Q * (R) (0) This information is contained in coded form in the arrays X and QRAUX. DSICO - DSICO factors a double precision symmetric matrix by elimination with symmetric pivoting and estimates the condition of the matrix. DSIDI - DSIDI computes the determinant, inertia and inverse of a double precision symmetric matrix using the factors from DSIFA. PPPPaaaaggggeeee 11118888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DSIFA - DSIFA factors a double precision symmetric matrix by elimination with symmetric pivoting. DSISL - DSISL solves the double precision symmetric system A * X = B using the factors computed by DSIFA. DSPCO - DSPCO factors a double precision symmetric matrix stored in packed form by elimination with symmetric pivoting and estimates the condition of the matrix. DSPDI - DSPDI computes the determinant, inertia and inverse of a double precision symmetric matrix using the factors from DSPFA, where the matrix is stored in packed form. DSPFA - DSPFA factors a double precision symmetric matrix stored in packed form by elimination with symmetric pivoting. DSPSL - DSISL solves the double precision symmetric system A * X = B using the factors computed by DSPFA. DSVDC - DSVDC is a subroutine to reduce a double precision NxP matrix X by orthogonal transformations U and V to diagonal form. The diagonal elements S(I) are the singular values of X. The columns of U are the corresponding left singular vectors, and the columns of V the right singular vectors. DTRCO - DTRCO estimates the condition of a double precision triangular matrix. DTRDI - DTRDI computes the determinant and inverse of a double precision triangular matrix. DTRSL - DTRSL solves systems of the form T * X = B or TRANS(T) * X = B where T is a triangular matrix of order N. Here TRANS(T) denotes the transpose of the matrix T. SCHDC - SCHDC computes the Cholesky decomposition of a positive definite matrix. A pivoting option allows the user to estimate the condition of a positive definite matrix or determine the rank of a positive semidefinite matrix. SCHDD - SCHDD downdates an augmented Cholesky decomposition or the triangular factor of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, SCHDD determines an orthogonal matrix U and a scalar ZETA such that (R Z ) (RR ZZ) PPPPaaaaggggeeee 11119999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) U * ( ) = ( ) , (0 ZETA) ( X Y) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) removed. In this case, if RHO is the norm of the residual vector, then the norm of the residual vector of the downdated problem is SQRT(RHO**2 - ZETA**2). SCHDD will simultaneously downdate several triplets (Z,Y,RHO) along with R. For a less terse description of what SCHDD does and how it may be applied, see the LINPACK guide. SCHEX - SCHEX updates the Cholesky factorization A = TRANS(R)*R of a positive definite matrix A of order P under diagonal permutations of the form TRANS(E)*A*E where E is a permutation matrix. Specifically, given an upper triangular matrix R and a permutation matrix E (which is specified by K, L, and JOB), SCHEX determines an orthogonal matrix U such that U*R*E = RR, where RR is upper triangular. At the users option, the transformation U will be multiplied into the array Z. If A = TRANS(X)*X, so that R is the triangular part of the QR factorization of X, then RR is the triangular part of the QR factorization of X*E, i.e., X with its columns permuted. For a less terse description of what SCHEX does and how it may be applied, see the LINPACK guide. SCHUD - SCHUD updates an augmented Cholesky decomposition of the triangular part of an augmented QR decomposition. Specifically, given an upper triangular matrix R of order P, a row vector X, a column vector Z, and a scalar Y, SCHUD determines a unitary matrix U and a scalar ZETA such that (R Z) (RR ZZ ) U * ( ) = ( ) , (X Y) ( 0 ZETA) where RR is upper triangular. If R and Z have been obtained from the factorization of a least squares problem, then RR and ZZ are the factors corresponding to the problem with the observation (X,Y) appended. In this case, if RHO is the norm of the residual vector, then the norm of the residual vector of the updated problem is SQRT(RHO**2 + ZETA**2). SCHUD will simultaneously update several triplets (Z,Y,RHO). For a less terse description of what SCHUD does and how it may be applied, see the PPPPaaaaggggeeee 22220000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) LINPACK guide. SGBCO - SBGCO factors a real band matrix by Gaussian elimination and estimates the condition of the matrix. SGBDI - SGBDI computes the determinant of a band matrix using the factors computed by SBGCO or SGBFA. If the inverse is needed, use SGBSL N times. SGBFA - SGBFA factors a real band matrix by elimination. SGBSL - SGBSL solves the real band system A * X = B or TRANS(A) * X = B using the factors computed by SBGCO or SGBFA. SGECO - SGECO factors a real matrix by Gaussian elimination and estimates the condition of the matrix. SGEDI - SGEDI computes the determinant and inverse of a matrix using the factors computed by SGECO or SGEFA. SGEFA - SGEFA factors a real matrix by Gaussian elimination. SGESL - SGESL solves the real system A * X = B or TRANS(A) * X = B using the factors computed by SGECO or SGEFA. SGTSL - SGTSL given a general tridiagonal matrix and a right hand side will find the solution. SPBCO - SPBCO factors a real symmetric positive definite matrix stored in band form and estimates the condition of the matrix. SPBDI - SPBDI computes the determinant of a real symmetric positive definite band matrix using the factors computed by SPBCO or SPBFA. If the inverse is needed, use SPBSL N times. SPBFA - SPBFA factors a real symmetric positive definite matrix stored in band form. SPBSL - SPBSL solves the real symmetric positive definite band system A*X = B using the factors computed by SPBCO or SPBFA. SPOCO - SPOCO factors a real symmetric positive definite matrix and estimates the condition of the matrix. SPODI - SPODI computes the determinant and inverse of a certain real symmetric positive definite matrix (see below) using the factors computed by SPOCO, SPOFA or SQRDC. SPOFA - SPOFA factors a real symmetric positive definite matrix. SPOSL - SPOSL solves the real symmetric positive definite system A * X = B using the factors computed by SPOCO or SPOFA. PPPPaaaaggggeeee 22221111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SPPCO - SPPCO factors a real symmetric positive definite matrix stored in packed form and estimates the condition of the matrix. SPPDI - SPPDI computes the determinant and inverse of a real symmetric positive definite matrix using the factors computed by SPPCO or SPPFA . SPPFA - SPPFA factors a real symmetric positive definite matrix stored in packed form. SPPSL - SPPSL solves the real symmetric positive definite system A * X = B using the factors computed by SPPCO or SPPFA. SPTSL - SPTSL given a positive definite tridiagonal matrix and a right hand side will find the solution. SQRDC - SQRDC uses Householder transformations to compute the QR factorization of an N by P matrix X. Column pivoting based on the 2- norms of the reduced columns may be performed at the user's option. SQRSL - SQRSL applies the output of SQRDC to compute coordinate transformations, projections, and least squares solutions. For K .LE. MIN(N,P), let XK be the matrix XK = (X(JPVT(1)),X(JPVT(2)), ... ,X(JPVT(K))) formed from columnns JPVT(1), ... ,JPVT(K) of the original N x P matrix X that was input to SQRDC (if no pivoting was done, XK consists of the first K columns of X in their original order). SQRDC produces a factored orthogonal matrix Q and an upper triangular matrix R such that XK = Q * (R) (0) This information is contained in coded form in the arrays X and QRAUX. SSICO - SSICO factors a real symmetric matrix by elimination with symmetric pivoting and estimates the condition of the matrix. SSIDI - SSIDI computes the determinant, inertia and inverse of a real symmetric matrix using the factors from SSIFA. SSIFA - SSIFA factors a real symmetric matrix by elimination with symmetric pivoting. SSISL - SSISL solves the real symmetric system A * X = B using the factors computed by SSIFA. SSPCO - SSPCO factors a real symmetric matrix stored in packed form by elimination with symmetric pivoting and estimates the condition of the matrix. SSPDI - SSPDI computes the determinant, inertia and inverse of a real PPPPaaaaggggeeee 22222222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) symmetric matrix using the factors from SSPFA, where the matrix is stored in packed form. SSPFA - SSPFA factors a real symmetric matrix stored in packed form by elimination with symmetric pivoting. SSPSL - SSISL solves the real symmetric system A * X = B using the factors computed by SSPFA. SSVDC - SSVDC is a subroutine to reduce a real NxP matrix X by orthogonal transformations U and V to diagonal form. The diagonal elements S(I) are the singular values of X. The columns of U are the corresponding left singular vectors, and the columns of V the right singular vectors. STRCO - STRCO estimates the condition of a real triangular matrix. STRDI - STRDI computes the determinant and inverse of a real triangular matrix. STRSL - STRSL solves systems of the form T * X = B or TRANS(T) * X = B where T is a triangular matrix of order N. Here TRANS(T) denotes the transpose of the matrix T. LLLLAAAAPPPPAAAACCCCKKKK LLLLIIIIBBBBRRRRAAAARRRRYYYY SBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices. CGBCON estimates the reciprocal of the condition number of a complex general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by CGBTRF. CGBEQU computes row and column scalings intended to equilibrate an M by N band matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. CGBRFS 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. CGBSV computes the solution to a complex system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 22223333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CGBSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. CGBTF2 computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. CGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. CGBTRS solves a system of linear equations A * X = B, A**T * X = B, or A**H * X = B with a general band matrix A using the LU factorization computed by CGBTRF. CGEBAK forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by CGEBAL. CGEBAL balances a general complex matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. CGEBD2 reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation: Q' * A * P = B. CGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation: Q**H * A * P = B. CGECON estimates the reciprocal of the condition number of a general complex matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by CGETRF. CGEEQU computes row and column scalings intended to equilibrate an M by N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. CGEES computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). CGEESX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). CGEEV computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. PPPPaaaaggggeeee 22224444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) CGEHD2 reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q' * A * Q = H . CGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q' * A * Q = H . CGELQ2 computes an LQ factorization of a complex m by n matrix A: A = L * Q. CGELQF computes an LQ factorization of a complex M-by-N matrix A: A = L * Q. CGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. CGELSS computes the minimum norm solution to a complex linear least squares problem: Minimize 2-norm(| b - A*x |). CGELSX computes the minimum-norm solution to a complex linear least squares problem: minimize || A * X - B || CGEQL2 computes a QL factorization of a complex m by n matrix A: A = Q * L. CGEQLF computes a QL factorization of a complex M-by-N matrix A: A = Q * L. CGEQPF computes a QR factorization with column pivoting of a complex M- by-N matrix A: A*P = Q*R. CGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. CGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. CGERFS improves the computed solution to a system of linear equations and PPPPaaaaggggeeee 22225555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) provides error bounds and backward error estimates for the solution. CGERQ2 computes an RQ factorization of a complex m by n matrix A: A = R * Q. CGERQF computes an RQ factorization of a complex M-by-N matrix A: A = R * Q. CGESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. CGESVD computes the singular value decomposition (SVD) of a complex M- by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * conjugate-transpose(V) CGESVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. CGETF2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row interchanges. CGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. CGETRI computes the inverse of a matrix using the LU factorization computed by CGETRF. CGETRS solves a system of linear equations A * X = B, A**T * X = B, or A**H * X = B with a general N-by-N matrix A using the LU factorization computed by CGETRF. CGGBAK forms the right or left eigenvectors of the generalized eigenvalue problem by backward transformation on the computed eigenvectors of the balanced matrix output by CGGBAL. CGGBAL balances a pair of general complex matrices (A,B) for the generalized eigenvalue problem A*X = lambda*B*X. This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity CGGGLM solves a generalized linear regression model (GLM) problem: minimize y'*y subject to d = A*x + B*y CGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary similarity transformations, where A is a PPPPaaaaggggeeee 22226666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) (generally non-symmetric) square matrix and B is upper triangular. More precisely, CGGHRD simultaneously decomposes A into Q H Z* and B into Q T Z* , where H is upper Hessenberg, T is upper triangular, Q and Z are unitary, and * means conjugate transpose. CGGLSE solves the linear equality constrained least squares (LSE) problem: minimize || A*x - c ||_2 subject to B*x = d CGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = Q*R, B = Q*T*Z, CGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B: A = R*Q, B = Z*T*Q, CGGSVD computes the generalized singular value decomposition (GSVD) of the M-by-N complex matrix A and P-by-N complex matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1) where U, V and Q are unitary matrices, R is an upper triangular matrix, and Z' means the conjugate transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively: CGGSVP computes unitary matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the conjugate transpose of Z. CGTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by CGTTRF. CGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. CGTSV solves the equation where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting. CGTSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 22227777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CGTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges. CGTTRS solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF. CHBEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. CHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. CHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. CHECON estimates the reciprocal of the condition number of a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by CHETRF. CHEEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. CHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. CHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. CHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. CHEGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian and B is also CHERFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, and provides error bounds and backward error estimates for the solution. CHESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N- by-NRHS matrices. CHESVX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N PPPPaaaaggggeeee 22228888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) Hermitian matrix and X and B are N-by-NRHS matrices. CHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q' * A * Q = T. CHETF2 computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. CHETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is CHETRI computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by CHETRF. CHETRS solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by CHETRF. CHGEQZ implements a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N). CHPCON estimates the reciprocal of the condition number of a complex Hermitian packed matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by CHPTRF. CHPEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage. CHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. CHPGST reduces a complex Hermitian-definite generalized eigenproblem to standard form, using packed storage. CHPGV computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. CHPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite and packed, and provides error bounds and backward error estimates for the solution. PPPPaaaaggggeeee 22229999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CHPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices. CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or A = L*D*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices. CHPTRD reduces a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. CHPTRF computes the factorization of a complex Hermitian packed matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**H or A = L*D*L**H CHPTRI computes the inverse of a complex Hermitian indefinite matrix A in packed storage using the factorization A = U*D*U**H or A = L*D*L**H computed by CHPTRF. CHPTRS solves a system of linear equations A*X = B with a complex Hermitian matrix A stored in packed format using the factorization A = U*D*U**H or A = L*D*L**H computed by CHPTRF. CHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H. CHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. CLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. CLACGV conjugates a complex vector of length N. CLACON estimates the 1-norm of a square, complex matrix A. Reverse communication is used for evaluating matrix-vector products. CLACPY copies all or part of a two-dimensional matrix A to another matrix B. CLACRT applies a plane rotation, where the cos and sin (C and S) are complex and the vectors CX and CY are complex. CLADIV := X / Y, where X and Y are complex. The computation of X / Y will not overflow on an intermediary step unless the results overflows. PPPPaaaaggggeeee 33330000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H. CLAESY computes the eigendecomposition of a 2x2 symmetric matrix ( ( A, B );( B, C ) ) provided the norm of the matrix of eigenvectors is larger than some threshold value. CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such that if ( UPPER ) then ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV ) CLAGTM performs a matrix-vector product of the form CLAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: CLAHQR is an auxiliary routine called by CHSEQR to update the eigenvalues and Schur decomposition already computed by CHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. CLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by a unitary similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. CLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that CLANGB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n band matrix A, with kl sub-diagonals and ku super-diagonals. CLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex matrix A. CLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex tridiagonal matrix A. PPPPaaaaggggeeee 33331111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CLANHB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n hermitian band matrix A, with k super-diagonals. CLANHE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A. CLANHP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A, supplied in packed form. CLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. CLANHT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix A. CLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals. CLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A, supplied in packed form. CLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A. CLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals. CLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form. CLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. Given two column vectors X and Y, let The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y. CLAPMT rearranges the columns of the M by N matrix X as specified by the PPPPaaaaggggeeee 33332222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: CLAQGB equilibrates a general M by N band matrix A with KL subdiagonals and KU superdiagonals using the row and scaling factors in the vectors R and C. CLAQGE equilibrates a general M by N matrix A using the row and scaling factors in the vectors R and C. CLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S. CLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S. CLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S. CLAR2V applies a vector of complex plane rotations with real cosines from both sides to a sequence of 2-by-2 complex Hermitian matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := CLARF applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form CLARFB applies a complex block reflector H or its transpose H' to a complex M-by-N matrix C, from either the left or the right. CLARFG generates a complex elementary reflector H of order n, such that ( x ) ( 0 ) CLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. CLARFX applies a complex elementary reflector H to a complex m by n matrix C, from either the left or the right. H is represented in the form CLARGV generates a vector of complex plane rotations with real cosines, determined by elements of the complex vectors x and y. For i = 1,2,...,n CLARNV returns a vector of n random complex numbers from a uniform or normal distribution. CLARTG generates a plane rotation so that [ -SN CS ] [ G ] [ 0 ] CLARTV applies a vector of complex plane rotations with real cosines to elements of the complex vectors x and y. For i = 1,2,...,n PPPPaaaaggggeeee 33333333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ( x(i) ) := ( c(i) s(i) ) ( x(i) ) CLASCL multiplies the M by N complex matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded. CLASET initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. CLASR performs the transformation consisting of a sequence of plane rotations determined by the parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l' and z = n when SIDE = 'R' or 'r' ): CLASSQ returns the values scl and ssq such that where x( i ) = abs( X( 1 + ( i - 1 )*INCX ) ). The value of sumsq is assumed to be at least unity and the value of ssq will then satisfy 1.0 .le. ssq .le. ( sumsq + 2*n ). CLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A. CLASYF computes a partial factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: CLATBS solves one of the triangular systems with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A' denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. CLATPS solves one of the triangular systems with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. CLATRD reduces NB rows and columns of a complex Hermitian matrix A to PPPPaaaaggggeeee 33334444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) Hermitian tridiagonal form by a unitary similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. CLATRS solves one of the triangular systems with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. CLATZM applies a Householder matrix generated by CTZRQF to a matrix. CLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. CLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. CLAZRO initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. CPBCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite band matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPBTRF. CPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. CPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution. CPBSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. CPBTF2 computes the Cholesky factorization of a complex Hermitian PPPPaaaaggggeeee 33335555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) positive definite band matrix A. CPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. CPBTRS solves a system of linear equations A*X = B with a Hermitian positive definite band matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPBTRF. CPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPOTRF. CPOEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. CPORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and provides error bounds and backward error estimates for the solution. CPOSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. CPOTF2 computes the Cholesky factorization of a complex Hermitian positive definite matrix A. CPOTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. CPOTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPOTRF. CPOTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPOTRF. CPPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite packed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPPTRF. PPPPaaaaggggeeee 33336666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CPPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. CPPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and packed, and provides error bounds and backward error estimates for the solution. CPPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. CPPTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in packed format. CPPTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPPTRF. CPPTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A in packed storage using the Cholesky factorization A = U**H*U or A = L*L**H computed by CPPTRF. CPTCON computes the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite tridiagonal matrix using the factorization A = L*D*L**T or A = U**T*D*U computed by CPTTRF. CPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF and then calling CBDSQR to compute the singular values of the bidiagonal factor. CPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution. CPTSV computes the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. CPTSVX uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N PPPPaaaaggggeeee 33337777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. CPTTRF computes the factorization of a complex Hermitian positive definite tridiagonal matrix A. CPTTRS solves a system of linear equations A * X = B with a Hermitian positive definite tridiagonal matrix A using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF. CROT applies a plane rotation, where the cos (C) is real and the sin (S) is complex, and the vectors CX and CY are complex. CSPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSPTRF. CSPMV performs the matrix-vector operation where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form. CSPR performs the symmetric rank 1 operation where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix, supplied in packed form. CSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution. CSPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. CSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. CSPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T CSPTRI computes the inverse of a complex symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by CSPTRF. CSPTRS solves a system of linear equations A*X = B with a complex symmetric matrix A stored in packed format using the factorization A = U*D*U**T or A = L*D*L**T computed by CSPTRF. PPPPaaaaggggeeee 33338888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CSRSCL multiplies an n-element complex vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow. CSTEIN computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. CSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band complex Hermitian matrix can also be found if CSYTRD or CSPTRD or CSBTRD has been used to reduce this matrix to tridiagonal form. CSYCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. CSYMV performs the matrix-vector operation where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix. CSYR performs the symmetric rank 1 operation where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix. CSYRFS 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. CSYSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N- by-NRHS matrices. CSYSVX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. CSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' CSYTRF computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is CSYTRI computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. CSYTRS solves a system of linear equations A*X = B with a complex PPPPaaaaggggeeee 33339999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by CSYTRF. CTBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1-norm or the infinity-norm. CTBRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix. CTBTRS solves a triangular system of the form where A is a triangular band matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. CTGEVC computes selected left and/or right generalized eigenvectors of a pair of complex upper triangular matrices (A,B). The j-th generalized left and right eigenvectors are y and x, resp., such that: CTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. CTPCON estimates the reciprocal of the condition number of a packed triangular matrix A, in either the 1-norm or the infinity-norm. CTPRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular packed coefficient matrix. CTPTRI computes the inverse of a complex upper or lower triangular matrix A stored in packed format. CTPTRS solves a triangular system of the form where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. CTRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm. CTREVC computes all or some right and/or left eigenvectors of a complex upper triangular matrix T. CTREXC reorders the Schur factorization of a complex matrix A = Q*T*Q**H, so that the diagonal element of T with row index IFST is moved to row ILST. CTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. CTRSEN reorders the Schur factorization of a complex matrix A = Q*T*Q**H, PPPPaaaaggggeeee 44440000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) so that a selected cluster of eigenvalues appears in the leading positions on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. CTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary). CTRSYL solves the complex Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or CTRTI2 computes the inverse of a complex upper or lower triangular matrix. CTRTRI computes the inverse of a complex upper or lower triangular matrix A. CTRTRS solves a triangular system of the form where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. CTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations. CUNG2L 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 CUNG2R generates an m by n complex matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m CUNGBR generates one of the matrices Q or P**H determined by CGEBRD when reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. CUNGHR generates a complex unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by CGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). CUNGL2 generates an m-by-n complex matrix Q with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of order n CUNGLQ generates an M-by-N complex matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N PPPPaaaaggggeeee 44441111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CUNGQL 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 CUNGQR generates an M-by-N complex matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M CUNGR2 generates an m by n complex matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n CUNGRQ generates an M-by-N complex matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N CUNGTR generates a complex unitary matrix Q which is defined as the product of n-1 elementary reflectors of order N, as returned by CHETRD: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), CUNM2L overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors CUNM2R overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors If VECT = 'Q', CUNMBR overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H CUNMHR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H CUNML2 overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors CUNMLQ overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H CUNMQL overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H CUNMQR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H PPPPaaaaggggeeee 44442222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CUNMR2 overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors CUNMRQ overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H CUNMTR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H CUPGTR generates a complex unitary matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by CHPTRD using packed storage: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), CUPMTR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H DBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices. DGBCON estimates the reciprocal of the condition number of a real general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by DGBTRF. DGBEQU computes row and column scalings intended to equilibrate an M by N band matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. DGBRFS 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. DGBSV computes the solution to a real system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. DGBSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. DGBTF2 computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges. DGBTRF computes an LU factorization of a real m-by-n band matrix A using PPPPaaaaggggeeee 44443333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) partial pivoting with row interchanges. DGBTRS solves a system of linear equations A * X = B or A' * X = B with a general band matrix A using the LU factorization computed by DGBTRF. DGEBAK forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by DGEBAL. DGEBAL balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. DGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q' * A * P = B. DGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. DGECON estimates the reciprocal of the condition number of a general real matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by DGETRF. DGEEQU computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. DGEES computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). DGEESX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). DGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. DGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. For a pair of N-by-N real nonsymmetric matrices A, B: compute the generalized eigenvalues (alphar +/- alphai*i, beta) compute the real Schur form (A,B) PPPPaaaaggggeeee 44444444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) For a pair of N-by-N real nonsymmetric matrices A, B: compute the generalized eigenvalues (alphar +/- alphai*i, beta) compute the left and/or right generalized eigenvectors (VL and VR) DGEHD2 reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . DGELQ2 computes an LQ factorization of a real m by n matrix A: A = L * Q. DGELQF computes an LQ factorization of a real M-by-N matrix A: A = L * Q. DGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. DGELSS computes the minimum norm solution to a real linear least squares problem: Minimize 2-norm(| b - A*x |). DGELSX computes the minimum-norm solution to a real linear least squares problem: minimize || A * X - B || DGEQL2 computes a QL factorization of a real m by n matrix A: A = Q * L. DGEQLF computes a QL factorization of a real M-by-N matrix A: A = Q * L. DGEQPF computes a QR factorization with column pivoting of a real M-by-N matrix A: A*P = Q*R. DGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. DGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. DGERQ2 computes an RQ factorization of a real m by n matrix A: A = R * Q. DGERQF computes an RQ factorization of a real M-by-N matrix A: A = R * Q. DGESV computes the solution to a real system of linear equations PPPPaaaaggggeeee 44445555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. DGESVD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) DGESVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. DGETF2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row interchanges. DGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. DGETRI computes the inverse of a matrix using the LU factorization computed by DGETRF. DGETRS solves a system of linear equations A * X = B or A' * X = B with a general N-by-N matrix A using the LU factorization computed by DGETRF. DGGBAK forms the right or left eigenvectors of the generalized eigenvalue problem by backward transformation on the computed eigenvectors of the balanced matrix output by DGGBAL. DGGBAL balances a pair of general real matrices (A,B) for the generalized eigenvalue problem A*X = lambda*B*X. This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity DGGGLM solves a generalized linear regression model (GLM) problem: minimize y'*y subject to d = A*x + B*y DGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal similarity transformations, where A is a (generally non-symmetric) square matrix and B is upper triangular. More precisely, DGGHRD simultaneously decomposes A into Q H Z' and B into Q T Z' , where H is upper Hessenberg, T is upper triangular, Q and Z are orthogonal, and ' means transpose. DGGLSE solves the linear equality constrained least squares (LSE) problem: minimize || A*x - c ||_2 subject to B*x = d PPPPaaaaggggeeee 44446666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = Q*R, B = Q*T*Z, DGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B: A = R*Q, B = Z*T*Q, DGGSVD computes the generalized singular value decomposition (GSVD) of the M-by-N matrix A and P-by-N matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1) where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper tridiagonal matrix, D1 and D2 are "diagonal" matrices, and of the following structures, respectively: DGGSVP computes orthogonal matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of (M+P)-by-N matrix (A',B')'. Z' denotes the transpose of Z. DGTCON estimates the reciprocal of the condition number of a real tridiagonal matrix A using the LU factorization as computed by DGTTRF. DGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. DGTSV solves the equation where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting. DGTSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B or A**T * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices. DGTTRF computes an LU factorization of a real tridiagonal matrix A using elimination with partial pivoting and row interchanges. DGTTRS solves one of the systems of equations A*X = B or A'*X = B, with a tridiagonal matrix A using the LU factorization computed by DGTTRF. DHGEQZ implements a single-/double-shift version of the QZ method for finding the generalized eigenvalues B is upper triangular, and A is block upper triangular, where the diagonal blocks are either 1x1 or 2x2, the 2x2 blocks having complex generalized eigenvalues (see the description of the argument JOB.) PPPPaaaaggggeeee 44447777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form using one orthogonal tranformation (usually called Q) on the left and another (usually called Z) on the right. The 2x2 upper-triangular diagonal blocks of B corresponding to 2x2 blocks of A will be reduced to positive diagonal matrices. (I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.) DHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H. DHSEQR computes the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. DLABAD takes as input the values computed by SLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by DLAMCH. This subroutine is needed because DLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray. DLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. DLACON estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products. DLACPY copies all or part of a two-dimensional matrix A to another matrix B. DLADIV performs complex division in real arithmetic in D. Knuth, The art of Computer Programming, Vol.2, p.195 DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value. DLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: DLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H. PPPPaaaaggggeeee 44448888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in an upper quasi-triangular matrix T by an orthogonal similarity transformation. DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow. DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such that if ( UPPER ) then DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column. DLAGTM performs a matrix-vector product of the form DLAGTS may be used to solve one of the systems of equations where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambda*I) as DLAHQR is an auxiliary routine called by DHSEQR to update the eigenvalues and Schur decomposition already computed by DHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. DLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that DLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A' - w D) X = s B with possible scaling ("s") and perturbation of A. (A' PPPPaaaaggggeeee 44449999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) means A-transpose.) A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices -- real if w is real, complex if w is complex. NA may be 1 or 2. DLAMCH determines double precision machine parameters. DLANGB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n band matrix A, with kl sub-diagonals and ku super-diagonals. DLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A. DLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real tridiagonal matrix A. DLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. DLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals. DLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A, supplied in packed form. DLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A. DLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A. DLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals. DLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form. DLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. PPPPaaaaggggeeee 55550000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form: [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] Given two column vectors X and Y, let The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y. DLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: DLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow. DLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow. DLAQGB equilibrates a general M by N band matrix A with KL subdiagonals and KU superdiagonals using the row and scaling factors in the vectors R and C. DLAQGE equilibrates a general M by N matrix A using the row and scaling factors in the vectors R and C. DLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S. DLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S. DLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S. DLAQTR solves the real quasi-triangular system or the complex quasi-triangular systems DLAR2V applies a vector of real plane rotations from both sides to a sequence of 2-by-2 real symmetric matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) ) ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) ) DLARF applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form PPPPaaaaggggeeee 55551111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) H = I - tau * v * v' DLARFB applies a real block reflector H or its transpose H' to a real m by n matrix C, from either the left or the right. DLARFG generates a real elementary reflector H of order n, such that ( x ) ( 0 ) DLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. DLARFX applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form DLARGV generates a vector of real plane rotations, determined by elements of the real vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( a(i) ) DLARNV returns a vector of n random real numbers from a uniform or normal distribution. DLARTG generate a plane rotation so that [ -SN CS ] [ G ] [ 0 ] DLARTV applies a vector of real plane rotations to elements of the real vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) DLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). DLAS2 computes the singular values of the 2-by-2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value. DLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded. DLASET initializes an m-by-n matrix A to BETA on the diagonal and ALPHA on the offdiagonals. DLASR performs the transformation consisting of a sequence of plane rotations determined by the parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l' and z = n when SIDE = 'R' or 'r' ): DLASSQ returns the values scl and smsq such that PPPPaaaaggggeeee 55552222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value DLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ] [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ]. DLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A. DLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or -1. op(T) = T or T', where T' denotes the transpose of T. DLASYF computes a partial factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: DLATBS solves one of the triangular systems are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. DLATPS solves one of the triangular systems transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. DLATRS solves one of the triangular systems triangular matrix, A' denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a PPPPaaaaggggeeee 55553333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) non-trivial solution to A*x = 0 is returned. DLATZM applies a Householder matrix generated by DTZRQF to a matrix. DLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. DLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. DLAZRO initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. DOPGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by DSPTRD using packed storage: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), DOPMTR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORG2L generates an m by n real matrix Q with orthonormal columns, which is defined as the last n columns of a product of k elementary reflectors of order m DORG2R generates an m by n real matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m DORGBR generates one of the matrices Q or P**T determined by DGEBRD when reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) or G(i) respectively. DORGHR generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). DORGL2 generates an m by n real matrix Q with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of order n DORGLQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N DORGQL generates an M-by-N real matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M PPPPaaaaggggeeee 55554444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DORGQR generates an M-by-N real matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M DORGR2 generates an m by n real matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n DORGRQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N DORGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order N, as returned by DSYTRD: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), DORM2L overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors DORM2R overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q**T * C C * Q**T DORMHR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORML2 overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors DORMLQ overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORMQL overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORMQR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORMR2 overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors PPPPaaaaggggeeee 55555555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DORMRQ overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DORMTR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T DPBCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite band matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF. DPBEQU computes row and column scalings intended to equilibrate a symmetric positive definite band matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. DPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and banded, and provides error bounds and backward error estimates for the solution. DPBSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. DPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. DPBTF2 computes the Cholesky factorization of a real symmetric positive definite band matrix A. DPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A. DPBTRS solves a system of linear equations A*X = B with a symmetric positive definite band matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF. DPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. DPOEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. PPPPaaaaggggeeee 55556666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. DPOSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices. DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices. DPOTF2 computes the Cholesky factorization of a real symmetric positive definite matrix A. DPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A. DPOTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. DPOTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. DPPCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite packed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPPTRF. DPPEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. DPPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and packed, and provides error bounds and backward error estimates for the solution. DPPSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 55557777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DPPTRF computes the Cholesky factorization of a real symmetric positive definite matrix A stored in packed format. DPPTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPPTRF. DPPTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A in packed storage using the Cholesky factorization A = U**T*U or A = L*L**T computed by DPPTRF. DPTCON computes the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite tridiagonal matrix using the factorization A = L*D*L**T or A = U**T*D*U computed by DPTTRF. DPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using DPTTRF, and then calling DBDSQR to compute the singular values of the bidiagonal factor. DPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution. DPTSV computes the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. DPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. DPTTRF computes the factorization of a real symmetric positive definite tridiagonal matrix A. DPTTRS solves a system of linear equations A * X = B with a symmetric positive definite tridiagonal matrix A using the factorization A = L*D*L**T or A = U**T*D*U computed by DPTTRF. (The two forms are equivalent if A is real.) DRSCL multiplies an n-element real vector x by the real scalar 1/a. This is done without overflow or underflow as long as DSBEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. PPPPaaaaggggeeee 55558888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DSBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. DSPCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSPTRF. DSPEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. DSPEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. DSPGST reduces a real symmetric-definite generalized eigenproblem to standard form, using packed storage. DSPGV computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. DSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution. DSPSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. DSPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. DSPTRF computes the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T DSPTRI computes the inverse of a real symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by DSPTRF. DSPTRS solves a system of linear equations A*X = B with a real symmetric matrix A stored in packed format using the factorization A = U*D*U**T or PPPPaaaaggggeeee 55559999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A = L*D*L**T computed by DSPTRF. DSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. DSTEIN computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. DSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to tridiagonal form. DSTERF computes all eigenvalues of a symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of the QL or QR algorithm. DSTEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. DSYCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. DSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. DSYEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. DSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. DSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. DSYGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric and B is also DSYRFS 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. PPPPaaaaggggeeee 66660000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) DSYSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N- by-NRHS matrices. DSYSVX uses the diagonal pivoting factorization to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * A * Q = T. DSYTF2 computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' DSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. DSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is DSYTRI computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. DSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by DSYTRF. DTBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1-norm or the infinity-norm. DTBRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix. DTBTRS solves a triangular system of the form where A is a triangular band matrix of order N, and B is an N-by NRHS matrix. A check is made to verify that A is nonsingular. DTGEVC computes selected left and/or right generalized eigenvectors of a pair of real upper triangular matrices (A,B). The j-th generalized left and right eigenvectors are y and x, resp., such that: DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper ``triangular (or trapezoidal)'' matrices A and B. DTPCON estimates the reciprocal of the condition number of a packed triangular matrix A, in either the 1-norm or the infinity-norm. DTPRFS provides error bounds and backward error estimates for the PPPPaaaaggggeeee 66661111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) solution to a system of linear equations with a triangular packed coefficient matrix. DTPTRI computes the inverse of a real upper or lower triangular matrix A stored in packed format. DTPTRS solves a triangular system of the form where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. DTRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm. DTREVC computes all or some right and/or left eigenvectors of a real upper quasi-triangular matrix T. DTREXC reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that the diagonal block of T with row index IFST is moved to row ILST. DTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. 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 diagonal blocks of the upper quasi-triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. DTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal). DTRSYL solves the real Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or DTRTI2 computes the inverse of a real upper or lower triangular matrix. DTRTRI computes the inverse of a real upper or lower triangular matrix A. DTRTRS solves a triangular system of the form where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. DTZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations. DZSUM1 takes the sum of the absolute values of a complex vector and PPPPaaaaggggeeee 66662222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) returns a double precision result. ICMAX1 finds the index of the element whose real part has maximum absolute value. ILAENV is called from the LAPACK routines to choose problem-dependent parameters for the local environment. See ISPEC for a description of the parameters. LSAME returns .TRUE. if CA is the same letter as CB regardless of case. LSAMEN tests if the first N letters of CA are the same as the first N letters of CB, regardless of case. LSAMEN returns .TRUE. if CA and CB are equivalent except for case and .FALSE. otherwise. LSAMEN also returns .FALSE. if LEN( CA ) or LEN( CB ) is less than N. SBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices. SCSUM1 takes the sum of the absolute values of a complex vector and returns a single precision result. SGBCON estimates the reciprocal of the condition number of a real general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by SGBTRF. SGBEQU computes row and column scalings intended to equilibrate an M by N band matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. SGBRFS 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. SGBSV computes the solution to a real system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. SGBSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. SGBTF2 computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges. SGBTRF computes an LU factorization of a real m-by-n band matrix A using partial pivoting with row interchanges. PPPPaaaaggggeeee 66663333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SGBTRS solves a system of linear equations A * X = B or A' * X = B with a general band matrix A using the LU factorization computed by SGBTRF. SGEBAK forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by SGEBAL. SGEBAL balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. SGEBD2 reduces a real general m by n matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q' * A * P = B. SGEBRD reduces a general real M-by-N matrix A to upper or lower bidiagonal form B by an orthogonal transformation: Q**T * A * P = B. SGECON estimates the reciprocal of the condition number of a general real matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by SGETRF. SGEEQU computes row and column scalings intended to equilibrate an M-by-N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. SGEES computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). SGEESX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). SGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. SGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. For a pair of N-by-N real nonsymmetric matrices A, B: compute the generalized eigenvalues (alphar +/- alphai*i, beta) compute the real Schur form (A,B) For a pair of N-by-N real nonsymmetric matrices A, B: PPPPaaaaggggeeee 66664444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) compute the generalized eigenvalues (alphar +/- alphai*i, beta) compute the left and/or right generalized eigenvectors (VL and VR) SGEHD2 reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . SGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H . SGELQ2 computes an LQ factorization of a real m by n matrix A: A = L * Q. SGELQF computes an LQ factorization of a real M-by-N matrix A: A = L * Q. SGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. SGELSS computes the minimum norm solution to a real linear least squares problem: Minimize 2-norm(| b - A*x |). SGELSX computes the minimum-norm solution to a real linear least squares problem: minimize || A * X - B || SGEQL2 computes a QL factorization of a real m by n matrix A: A = Q * L. SGEQLF computes a QL factorization of a real M-by-N matrix A: A = Q * L. SGEQPF computes a QR factorization with column pivoting of a real M-by-N matrix A: A*P = Q*R. SGEQR2 computes a QR factorization of a real m by n matrix A: A = Q * R. SGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R. SGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. SGERQ2 computes an RQ factorization of a real m by n matrix A: A = R * Q. SGERQF computes an RQ factorization of a real M-by-N matrix A: A = R * Q. SGESV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 66665555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SGESVD computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * transpose(V) SGESVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. SGETF2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row interchanges. SGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. SGETRI computes the inverse of a matrix using the LU factorization computed by SGETRF. SGETRS solves a system of linear equations A * X = B or A' * X = B with a general N-by-N matrix A using the LU factorization computed by SGETRF. SGGBAK forms the right or left eigenvectors of the generalized eigenvalue problem by backward transformation on the computed eigenvectors of the balanced matrix output by SGGBAL. SGGBAL balances a pair of general real matrices (A,B) for the generalized eigenvalue problem A*X = lambda*B*X. This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity SGGGLM solves a generalized linear regression model (GLM) problem: minimize y'*y subject to d = A*x + B*y SGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal similarity transformations, where A is a (generally non-symmetric) square matrix and B is upper triangular. More precisely, SGGHRD simultaneously decomposes A into Q H Z' and B into Q T Z' , where H is upper Hessenberg, T is upper triangular, Q and Z are orthogonal, and ' means transpose. SGGLSE solves the linear equality constrained least squares (LSE) problem: minimize || A*x - c ||_2 subject to B*x = d SGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: PPPPaaaaggggeeee 66666666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A = Q*R, B = Q*T*Z, SGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B: A = R*Q, B = Z*T*Q, SGGSVD computes the generalized singular value decomposition (GSVD) of the M-by-N matrix A and P-by-N matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1) where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper tridiagonal matrix, D1 and D2 are "diagonal" matrices, and of the following structures, respectively: SGGSVP computes orthogonal matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of (M+P)-by-N matrix (A',B')'. Z' denotes the transpose of Z. SGTCON estimates the reciprocal of the condition number of a real tridiagonal matrix A using the LU factorization as computed by SGTTRF. SGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. SGTSV solves the equation where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting. SGTSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B or A**T * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices. SGTTRF computes an LU factorization of a real tridiagonal matrix A using elimination with partial pivoting and row interchanges. SGTTRS solves one of the systems of equations A*X = B or A'*X = B, with a tridiagonal matrix A using the LU factorization computed by SGTTRF. SHGEQZ implements a single-/double-shift version of the QZ method for finding the generalized eigenvalues B is upper triangular, and A is block upper triangular, where the diagonal blocks are either 1x1 or 2x2, the 2x2 blocks having complex generalized eigenvalues (see the description of the argument JOB.) If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form using one orthogonal tranformation (usually called Q) on the left and PPPPaaaaggggeeee 66667777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) another (usually called Z) on the right. The 2x2 upper-triangular diagonal blocks of B corresponding to 2x2 blocks of A will be reduced to positive diagonal matrices. (I.e., if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1) will be positive.) SHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a real upper Hessenberg matrix H. SHSEQR computes the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. SLABAD takes as input the values computed by SLAMCH for underflow and overflow, and returns the square root of each of these values if the log of LARGE is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by SLAMCH. This subroutine is needed because SLAMCH does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray. SLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. SLACON estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products. SLACPY copies all or part of a two-dimensional matrix A to another matrix B. SLADIV performs complex division in real arithmetic in D. Knuth, The art of Computer Programming, Vol.2, p.195 SLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, and RT2 is the eigenvalue of smaller absolute value. SLAEBZ contains the iteration loops which compute and use the function N(w), which is the count of eigenvalues of a symmetric tridiagonal matrix T less than or equal to its argument w. It performs a choice of two types of loops: SLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H. SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix PPPPaaaaggggeeee 66668888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) [ A B ] [ B C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in an upper quasi-triangular matrix T by an orthogonal similarity transformation. SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow. SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such that if ( UPPER ) then SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n tridiagonal matrix and lambda is a scalar, as where P is a permutation matrix, L is a unit lower tridiagonal matrix with at most one non-zero sub-diagonal elements per column and U is an upper triangular matrix with at most two non-zero super-diagonal elements per column. SLAGTM performs a matrix-vector product of the form SLAGTS may be used to solve one of the systems of equations where T is an n by n tridiagonal matrix, for x, following the factorization of (T - lambda*I) as SLAHQR is an auxiliary routine called by SHSEQR to update the eigenvalues and Schur decomposition already computed by SHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. SLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. SLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that SLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A' - w D) X = s B with possible scaling ("s") and perturbation of A. (A' means A-transpose.) PPPPaaaaggggeeee 66669999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices -- real if w is real, complex if w is complex. NA may be 1 or 2. SLAMCH determines single precision machine parameters. SLANGB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n band matrix A, with kl sub-diagonals and ku super-diagonals. SLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A. SLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real tridiagonal matrix A. SLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. SLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals. SLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A, supplied in packed form. SLANST returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric tridiagonal matrix A. SLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A. SLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals. SLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form. SLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric PPPPaaaaggggeeee 77770000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) matrix in standard form: [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] Given two column vectors X and Y, let The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y. SLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: SLAPY2 returns sqrt(x**2+y**2), taking care not to cause unnecessary overflow. SLAPY3 returns sqrt(x**2+y**2+z**2), taking care not to cause unnecessary overflow. SLAQGB equilibrates a general M by N band matrix A with KL subdiagonals and KU superdiagonals using the row and scaling factors in the vectors R and C. SLAQGE equilibrates a general M by N matrix A using the row and scaling factors in the vectors R and C. SLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S. SLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S. SLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S. SLAQTR solves the real quasi-triangular system or the complex quasi-triangular systems SLAR2V applies a vector of real plane rotations from both sides to a sequence of 2-by-2 real symmetric matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) ) ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) ) SLARF applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form PPPPaaaaggggeeee 77771111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) H = I - tau * v * v' SLARFB applies a real block reflector H or its transpose H' to a real m by n matrix C, from either the left or the right. SLARFG generates a real elementary reflector H of order n, such that ( x ) ( 0 ) SLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. SLARFX applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form SLARGV generates a vector of real plane rotations, determined by elements of the real vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( a(i) ) SLARNV returns a vector of n random real numbers from a uniform or normal distribution. SLARTG generate a plane rotation so that [ -SN CS ] [ G ] [ 0 ] SLARTV applies a vector of real plane rotations to elements of the real vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) SLARUV returns a vector of n random real numbers from a uniform (0,1) distribution (n <= 128). SLAS2 computes the singular values of the 2-by-2 matrix [ F G ] [ 0 H ]. On return, SSMIN is the smaller singular value and SSMAX is the larger singular value. SLASCL multiplies the M by N real matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded. SLASET initializes an m-by-n matrix A to BETA on the diagonal and ALPHA on the offdiagonals. SLASR performs the transformation consisting of a sequence of plane rotations determined by the parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l' and z = n when SIDE = 'R' or 'r' ): SLASSQ returns the values scl and smsq such that PPPPaaaaggggeeee 77772222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is assumed to be non-negative and scl returns the value SLASV2 computes the singular value decomposition of a 2-by-2 triangular matrix [ F G ] [ 0 H ]. On return, abs(SSMAX) is the larger singular value, abs(SSMIN) is the smaller singular value, and (CSL,SNL) and (CSR,SNR) are the left and right singular vectors for abs(SSMAX), giving the decomposition [ CSL SNL ] [ F G ] [ CSR -SNR ] = [ SSMAX 0 ] [-SNL CSL ] [ 0 H ] [ SNR CSR ] [ 0 SSMIN ]. SLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A. SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or -1. op(T) = T or T', where T' denotes the transpose of T. SLASYF computes a partial factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: SLATBS solves one of the triangular systems are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. SLATPS solves one of the triangular systems transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. SLATRS solves one of the triangular systems triangular matrix, A' denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a PPPPaaaaggggeeee 77773333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) non-trivial solution to A*x = 0 is returned. SLATZM applies a Householder matrix generated by STZRQF to a matrix. SLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. SLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. SLAZRO initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. SOPGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by SSPTRD using packed storage: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), SOPMTR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORG2L generates an m by n real matrix Q with orthonormal columns, which is defined as the last n columns of a product of k elementary reflectors of order m SORG2R generates an m by n real matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m SORGBR generates one of the matrices Q or P**T determined by SGEBRD when reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and P**T are defined as products of elementary reflectors H(i) or G(i) respectively. SORGHR generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by SGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). SORGL2 generates an m by n real matrix Q with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of order n SORGLQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N SORGQL generates an M-by-N real matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M PPPPaaaaggggeeee 77774444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SORGQR generates an M-by-N real matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M SORGR2 generates an m by n real matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n SORGRQ generates an M-by-N real matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N SORGTR generates a real orthogonal matrix Q which is defined as the product of n-1 elementary reflectors of order N, as returned by SSYTRD: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), SORM2L overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors SORM2R overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors If VECT = 'Q', SORMBR overwrites the general real M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'T': Q**T * C C * Q**T SORMHR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORML2 overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors SORMLQ overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORMQL overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORMQR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORMR2 overwrites the general real m by n matrix C with where Q is a real orthogonal matrix defined as the product of k elementary reflectors PPPPaaaaggggeeee 77775555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SORMRQ overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SORMTR overwrites the general real M-by-N matrix C with TRANS = 'T': Q**T * C C * Q**T SPBCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite band matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF. SPBEQU computes row and column scalings intended to equilibrate a symmetric positive definite band matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. SPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and banded, and provides error bounds and backward error estimates for the solution. SPBSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. SPBSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band matrix and X and B are N-by-NRHS matrices. SPBTF2 computes the Cholesky factorization of a real symmetric positive definite band matrix A. SPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A. SPBTRS solves a system of linear equations A*X = B with a symmetric positive definite band matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPBTRF. SPOCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF. SPOEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. PPPPaaaaggggeeee 77776666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SPORFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite, and provides error bounds and backward error estimates for the solution. SPOSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices. SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix and X and B are N-by-NRHS matrices. SPOTF2 computes the Cholesky factorization of a real symmetric positive definite matrix A. SPOTRF computes the Cholesky factorization of a real symmetric positive definite matrix A. SPOTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF. SPOTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPOTRF. SPPCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite packed matrix using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPPTRF. SPPEQU computes row and column scalings intended to equilibrate a symmetric positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. SPPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and packed, and provides error bounds and backward error estimates for the solution. SPPSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 77777777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SPPTRF computes the Cholesky factorization of a real symmetric positive definite matrix A stored in packed format. SPPTRI computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPPTRF. SPPTRS solves a system of linear equations A*X = B with a symmetric positive definite matrix A in packed storage using the Cholesky factorization A = U**T*U or A = L*L**T computed by SPPTRF. SPTCON computes the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite tridiagonal matrix using the factorization A = L*D*L**T or A = U**T*D*U computed by SPTTRF. SPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using SPTTRF, and then calling SBDSQR to compute the singular values of the bidiagonal factor. SPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution. SPTSV computes the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. SPTSVX uses the factorization A = L*D*L**T to compute the solution to a real system of linear equations A*X = B, where A is an N-by-N symmetric positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. SPTTRF computes the factorization of a real symmetric positive definite tridiagonal matrix A. SPTTRS solves a system of linear equations A * X = B with a symmetric positive definite tridiagonal matrix A using the factorization A = L*D*L**T or A = U**T*D*U computed by SPTTRF. (The two forms are equivalent if A is real.) SRSCL multiplies an n-element real vector x by the real scalar 1/a. This is done without overflow or underflow as long as SSBEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. SSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. PPPPaaaaggggeeee 77778888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SSBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. SSPCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSPTRF. SSPEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. SSPEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. SSPGST reduces a real symmetric-definite generalized eigenproblem to standard form, using packed storage. SSPGV computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. SSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution. SSPSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. SSPTRD reduces a real symmetric matrix A stored in packed form to symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. SSPTRF computes the factorization of a real symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T SSPTRI computes the inverse of a real symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by SSPTRF. SSPTRS solves a system of linear equations A*X = B with a real symmetric matrix A stored in packed format using the factorization A = U*D*U**T or PPPPaaaaggggeeee 77779999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A = L*D*L**T computed by SSPTRF. SSTEBZ computes the eigenvalues of a symmetric tridiagonal matrix T. The user may ask for all eigenvalues, all eigenvalues in the half-open interval (VL, VU], or the IL-th through IU-th eigenvalues. SSTEIN computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. SSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band symmetric matrix can also be found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to tridiagonal form. SSTERF computes all eigenvalues of a symmetric tridiagonal matrix using the Pal-Walker-Kahan variant of the QL or QR algorithm. SSTEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. SSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. SSYCON estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF. SSYEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. SSYEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. SSYGS2 reduces a real symmetric-definite generalized eigenproblem to standard form. SSYGST reduces a real symmetric-definite generalized eigenproblem to standard form. SSYGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric and B is also SSYRFS 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. PPPPaaaaggggeeee 88880000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) SSYSV computes the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N- by-NRHS matrices. SSYSVX uses the diagonal pivoting factorization to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. SSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: Q' * A * Q = T. SSYTF2 computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' SSYTRD reduces a real symmetric matrix A to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q**T * A * Q = T. SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is SSYTRI computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF. SSYTRS solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by SSYTRF. STBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1-norm or the infinity-norm. STBRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix. STBTRS solves a triangular system of the form where A is a triangular band matrix of order N, and B is an N-by NRHS matrix. A check is made to verify that A is nonsingular. STGEVC computes selected left and/or right generalized eigenvectors of a pair of real upper triangular matrices (A,B). The j-th generalized left and right eigenvectors are y and x, resp., such that: STGSJA computes the generalized singular value decomposition (GSVD) of two real upper ``triangular (or trapezoidal)'' matrices A and B. STPCON estimates the reciprocal of the condition number of a packed triangular matrix A, in either the 1-norm or the infinity-norm. STPRFS provides error bounds and backward error estimates for the PPPPaaaaggggeeee 88881111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) solution to a system of linear equations with a triangular packed coefficient matrix. STPTRI computes the inverse of a real upper or lower triangular matrix A stored in packed format. STPTRS solves a triangular system of the form where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. STRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm. STREVC computes all or some right and/or left eigenvectors of a real upper quasi-triangular matrix T. STREXC reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that the diagonal block of T with row index IFST is moved to row ILST. STRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. STRSEN 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 diagonal blocks of the upper quasi-triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. STRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal). STRSYL solves the real Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or STRTI2 computes the inverse of a real upper or lower triangular matrix. STRTRI computes the inverse of a real upper or lower triangular matrix A. STRTRS solves a triangular system of the form where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. STZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations. XERBLA is an error handler for the LAPACK routines. It is called by an PPPPaaaaggggeeee 88882222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) LAPACK routine if an input parameter has an invalid value. A message is printed and execution stops. DBDSQR computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the transpose of P), where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and Q and P are orthogonal matrices. ZDRSCL multiplies an n-element complex vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow. ZGBCON estimates the reciprocal of the condition number of a complex general band matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by ZGBTRF. ZGBEQU computes row and column scalings intended to equilibrate an M by N band matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest element in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. ZGBRFS 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. ZGBSV computes the solution to a complex system of linear equations A * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. ZGBSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by-NRHS matrices. ZGBTF2 computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. ZGBTRF computes an LU factorization of a complex m-by-n band matrix A using partial pivoting with row interchanges. ZGBTRS solves a system of linear equations A * X = B, A**T * X = B, or A**H * X = B with a general band matrix A using the LU factorization computed by ZGBTRF. ZGEBAK forms the right or left eigenvectors of a complex general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by ZGEBAL. ZGEBAL balances a general complex matrix A. This involves, first, PPPPaaaaggggeeee 88883333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ILO to IHI to make the rows and columns as close in norm as possible. Both steps are optional. ZGEBD2 reduces a complex general m by n matrix A to upper or lower real bidiagonal form B by a unitary transformation: Q' * A * P = B. ZGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation: Q**H * A * P = B. ZGECON estimates the reciprocal of the condition number of a general complex matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by ZGETRF. ZGEEQU computes row and column scalings intended to equilibrate an M by N matrix A and reduce its condition number. R returns the row scale factors and C the column scale factors, chosen to try to make the largest entry in each row and column of the matrix B with elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1. ZGEES computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H). ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q' * A * Q = H . ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similarity transformation: Q' * A * Q = H . ZGELQ2 computes an LQ factorization of a complex m by n matrix A: A = L * Q. PPPPaaaaggggeeee 88884444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZGELQF computes an LQ factorization of a complex M-by-N matrix A: A = L * Q. ZGELS solves overdetermined or underdetermined complex linear systems involving an M-by-N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. ZGELSS computes the minimum norm solution to a complex linear least squares problem: Minimize 2-norm(| b - A*x |). ZGELSX computes the minimum-norm solution to a complex linear least squares problem: minimize || A * X - B || ZGEQL2 computes a QL factorization of a complex m by n matrix A: A = Q * L. ZGEQLF computes a QL factorization of a complex M-by-N matrix A: A = Q * L. ZGEQPF computes a QR factorization with column pivoting of a complex M- by-N matrix A: A*P = Q*R. ZGEQR2 computes a QR factorization of a complex m by n matrix A: A = Q * R. ZGEQRF computes a QR factorization of a complex M-by-N matrix A: A = Q * R. ZGERFS improves the computed solution to a system of linear equations and provides error bounds and backward error estimates for the solution. ZGERQ2 computes an RQ factorization of a complex m by n matrix A: A = R * Q. ZGERQF computes an RQ factorization of a complex M-by-N matrix A: A = R * Q. ZGESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. ZGESVD computes the singular value decomposition (SVD) of a complex M- by-N matrix A, optionally computing the left and/or right singular vectors. The SVD is written A = U * SIGMA * conjugate-transpose(V) ZGESVX uses the LU factorization to compute the solution to a complex system of linear equations PPPPaaaaggggeeee 88885555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS matrices. ZGETF2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row interchanges. ZGETRF computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges. ZGETRI computes the inverse of a matrix using the LU factorization computed by ZGETRF. ZGETRS solves a system of linear equations A * X = B, A**T * X = B, or A**H * X = B with a general N-by-N matrix A using the LU factorization computed by ZGETRF. ZGGBAK forms the right or left eigenvectors of the generalized eigenvalue problem by backward transformation on the computed eigenvectors of the balanced matrix output by ZGGBAL. ZGGBAL balances a pair of general complex matrices (A,B) for the generalized eigenvalue problem A*X = lambda*B*X. This involves, first, permuting A and B by similarity transformations to isolate eigenvalues in the first 1 to ILO-1 and last IHI+1 to N elements on the diagonal; and second, applying a diagonal similarity ZGGGLM solves a generalized linear regression model (GLM) problem: minimize y'*y subject to d = A*x + B*y ZGGHRD reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary similarity transformations, where A is a (generally non-symmetric) square matrix and B is upper triangular. More precisely, ZGGHRD simultaneously decomposes A into Q H Z* and B into Q T Z* , where H is upper Hessenberg, T is upper triangular, Q and Z are unitary, and * means conjugate transpose. ZGGLSE solves the linear equality constrained least squares (LSE) problem: minimize || A*x - c ||_2 subject to B*x = d ZGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = Q*R, B = Q*T*Z, ZGGRQF computes a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B: A = R*Q, B = Z*T*Q, PPPPaaaaggggeeee 88886666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZGGSVD computes the generalized singular value decomposition (GSVD) of the M-by-N complex matrix A and P-by-N complex matrix B: U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) (1) where U, V and Q are unitary matrices, R is an upper triangular matrix, and Z' means the conjugate transpose of Z. Let K+L = the numerical effective rank of the matrix (A',B')', then D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively: ZGGSVP computes unitary matrices U, V and Q such that A23 is upper trapezoidal. K+L = the effective rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the conjugate transpose of Z. ZGTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by ZGTTRF. ZGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. ZGTSV solves the equation where A is an N-by-N tridiagonal matrix, by Gaussian elimination with partial pivoting. ZGTSVX uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a tridiagonal matrix of order N and X and B are N-by-NRHS matrices. ZGTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges. ZGTTRS solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by ZGTTRF. ZHBEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. ZHBEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ZHBTRD reduces a complex Hermitian band matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. ZHECON estimates the reciprocal of the condition number of a complex PPPPaaaaggggeeee 88887777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF. ZHEEV computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. ZHEEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ZHEGS2 reduces a complex Hermitian-definite generalized eigenproblem to standard form. ZHEGST reduces a complex Hermitian-definite generalized eigenproblem to standard form. ZHEGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian and B is also ZHERFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite, and provides error bounds and backward error estimates for the solution. ZHESV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N- by-NRHS matrices. ZHESVX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix and X and B are N-by-NRHS matrices. ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q' * A * Q = T. ZHETF2 computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' ZHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. ZHETRF computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is ZHETRI computes the inverse of a complex Hermitian indefinite matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF. PPPPaaaaggggeeee 88888888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZHETRS solves a system of linear equations A*X = B with a complex Hermitian matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHETRF. ZHGEQZ implements a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N). ZHPCON estimates the reciprocal of the condition number of a complex Hermitian packed matrix A using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. ZHPEV computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix in packed storage. ZHPEVX computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ZHPGST reduces a complex Hermitian-definite generalized eigenproblem to standard form, using packed storage. ZHPGV computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite. ZHPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian indefinite and packed, and provides error bounds and backward error estimates for the solution. ZHPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices. ZHPSVX uses the diagonal pivoting factorization A = U*D*U**H or A = L*D*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian matrix stored in packed format and X and B are N-by-NRHS matrices. ZHPTRD reduces a complex Hermitian matrix A stored in packed form to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. ZHPTRF computes the factorization of a complex Hermitian packed matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**H or A = L*D*L**H ZHPTRI computes the inverse of a complex Hermitian indefinite matrix A in PPPPaaaaggggeeee 88889999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) packed storage using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. ZHPTRS solves a system of linear equations A*X = B with a complex Hermitian matrix A stored in packed format using the factorization A = U*D*U**H or A = L*D*L**H computed by ZHPTRF. ZHSEIN uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H. ZHSEQR computes the eigenvalues of a complex upper Hessenberg matrix H, and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**H, where T is an upper triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q' * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. ZLACGV conjugates a complex vector of length N. ZLACON estimates the 1-norm of a square, complex matrix A. Reverse communication is used for evaluating matrix-vector products. ZLACPY copies all or part of a two-dimensional matrix A to another matrix B. ZLACRT applies a plane rotation, where the cos and sin (C and S) are complex and the vectors CX and CY are complex. ZLADIV := X / Y, where X and Y are complex. The computation of X / Y will not overflow on an intermediary step unless the results overflows. ZLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue W of a complex upper Hessenberg matrix H. ZLAESY computes the eigendecomposition of a 2x2 symmetric matrix ( ( A, B );( B, C ) ) provided the norm of the matrix of eigenvectors is larger than some threshold value. ZLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]. On return, RT1 is the eigenvalue of larger absolute value, RT2 is the eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right eigenvector for RT1, giving the decomposition ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such that if ( UPPER ) then ( -CONJG(SNU) CSU ) ( -CONJG(SNV) CSV ) ZLAGTM performs a matrix-vector product of the form PPPPaaaaggggeeee 99990000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZLAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: ZLAHQR is an auxiliary routine called by ZHSEQR to update the eigenvalues and Schur decomposition already computed by ZHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by a unitary similarity transformation Q' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. ZLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that ZLANGB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n band matrix A, with kl sub-diagonals and ku super-diagonals. ZLANGE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex matrix A. ZLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex tridiagonal matrix A. ZLANHB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n hermitian band matrix A, with k super-diagonals. ZLANHE returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A. ZLANHP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex hermitian matrix A, supplied in packed form. ZLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. ZLANHT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix A. PPPPaaaaggggeeee 99991111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals. ZLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A, supplied in packed form. ZLANSY returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex symmetric matrix A. ZLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals. ZLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form. ZLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. Given two column vectors X and Y, let The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y. ZLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: ZLAQGB equilibrates a general M by N band matrix A with KL subdiagonals and KU superdiagonals using the row and scaling factors in the vectors R and C. ZLAQGE equilibrates a general M by N matrix A using the row and scaling factors in the vectors R and C. ZLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S. ZLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S. ZLAQSY equilibrates a symmetric matrix A using the scaling factors in the vector S. ZLAR2V applies a vector of complex plane rotations with real cosines from PPPPaaaaggggeeee 99992222 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) both sides to a sequence of 2-by-2 complex Hermitian matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ZLARF applies a complex elementary reflector H to a complex M-by-N matrix C, from either the left or the right. H is represented in the form ZLARFB applies a complex block reflector H or its transpose H' to a complex M-by-N matrix C, from either the left or the right. ZLARFG generates a complex elementary reflector H of order n, such that ( x ) ( 0 ) ZLARFT forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors. ZLARFX applies a complex elementary reflector H to a complex m by n matrix C, from either the left or the right. H is represented in the form ZLARGV generates a vector of complex plane rotations with real cosines, determined by elements of the complex vectors x and y. For i = 1,2,...,n ZLARNV returns a vector of n random complex numbers from a uniform or normal distribution. ZLARTG generates a plane rotation so that [ -SN CS ] [ G ] [ 0 ] ZLARTV applies a vector of complex plane rotations with real cosines to elements of the complex vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ZLASCL multiplies the M by N complex matrix A by the real scalar CTO/CFROM. This is done without over/underflow as long as the final result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded. ZLASET initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. ZLASR performs the transformation consisting of a sequence of plane rotations determined by the parameters PIVOT and DIRECT as follows ( z = m when SIDE = 'L' or 'l' and z = n when SIDE = 'R' or 'r' ): ZLASSQ returns the values scl and ssq such that where x( i ) = abs( X( 1 + ( i - 1 )*INCX ) ). The value of sumsq is assumed to be at least unity and the value of ssq will then satisfy PPPPaaaaggggeeee 99993333 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) 1.0 .le. ssq .le. ( sumsq + 2*n ). ZLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A. ZLASYF computes a partial factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The partial factorization has the form: ZLATBS solves one of the triangular systems with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A' denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. ZLATPS solves one of the triangular systems with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A. ZLATRS solves one of the triangular systems with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. ZLATZM applies a Householder matrix generated by ZTZRQF to a matrix. ZLAUU2 computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. PPPPaaaaggggeeee 99994444 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZLAUUM computes the product U * U' or L' * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. ZLAZRO initializes a 2-D array A to BETA on the diagonal and ALPHA on the offdiagonals. ZPBCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite band matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPBTRF. ZPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. ZPBRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and banded, and provides error bounds and backward error estimates for the solution. ZPBSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices. ZPBTF2 computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. ZPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. ZPBTRS solves a system of linear equations A*X = B with a Hermitian positive definite band matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPBTRF. ZPOCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPOTRF. ZPOEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. PPPPaaaaggggeeee 99995555 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZPORFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite, and provides error bounds and backward error estimates for the solution. ZPOSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix and X and B are N-by-NRHS matrices. ZPOTF2 computes the Cholesky factorization of a complex Hermitian positive definite matrix A. ZPOTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix A. ZPOTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPOTRF. ZPOTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPOTRF. ZPPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite packed matrix using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPPTRF. ZPPEQU computes row and column scalings intended to equilibrate a Hermitian positive definite matrix A in packed storage and reduce its condition number (with respect to the two-norm). S contains the scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings. ZPPRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and packed, and provides error bounds and backward error estimates for the solution. ZPPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. ZPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N Hermitian positive definite matrix stored in packed format and X and B are N-by-NRHS matrices. PPPPaaaaggggeeee 99996666 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZPPTRF computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in packed format. ZPPTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPPTRF. ZPPTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A in packed storage using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPPTRF. ZPTCON computes the reciprocal of the condition number (in the 1-norm) of a complex Hermitian positive definite tridiagonal matrix using the factorization A = L*D*L**T or A = U**T*D*U computed by ZPTTRF. ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using DPTTRF and then calling ZBDSQR to compute the singular values of the bidiagonal factor. ZPTRFS improves the computed solution to a system of linear equations when the coefficient matrix is Hermitian positive definite and tridiagonal, and provides error bounds and backward error estimates for the solution. ZPTSV computes the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. ZPTSVX uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. ZPTTRF computes the factorization of a complex Hermitian positive definite tridiagonal matrix A. ZPTTRS solves a system of linear equations A * X = B with a Hermitian positive definite tridiagonal matrix A using the factorization A = U**H*D*U or A = L*D*L**H computed by ZPTTRF. ZROT applies a plane rotation, where the cos (C) is real and the sin (S) is complex, and the vectors CX and CY are complex. ZSPCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric packed matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. ZSPMV performs the matrix-vector operation where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix, supplied in packed form. PPPPaaaaggggeeee 99997777 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZSPR performs the symmetric rank 1 operation where alpha is a complex scalar, x is an n element vector and A is an n by n symmetric matrix, supplied in packed form. ZSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, and provides error bounds and backward error estimates for the solution. ZSPSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices. ZSPTRF computes the factorization of a complex symmetric matrix A stored in packed format using the Bunch-Kaufman diagonal pivoting method: A = U*D*U**T or A = L*D*L**T ZSPTRI computes the inverse of a complex symmetric indefinite matrix A in packed storage using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. ZSPTRS solves a system of linear equations A*X = B with a complex symmetric matrix A stored in packed format using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSPTRF. ZSTEIN computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration. ZSTEQR computes all eigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR method. The eigenvectors of a full or band complex Hermitian matrix can also be found if ZSYTRD or ZSPTRD or ZSBTRD has been used to reduce this matrix to tridiagonal form. ZSYCON estimates the reciprocal of the condition number (in the 1-norm) of a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. ZSYMV performs the matrix-vector operation where alpha and beta are scalars, x and y are n element vectors and A is an n by n symmetric matrix. ZSYR performs the symmetric rank 1 operation where alpha is a complex scalar, x is an n element vector and A is an n PPPPaaaaggggeeee 99998888 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) by n symmetric matrix. ZSYRFS 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. ZSYSV computes the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N- by-NRHS matrices. ZSYSVX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. ZSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method: A = U*D*U' or A = L*D*L' ZSYTRF computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is ZSYTRI computes the inverse of a complex symmetric indefinite matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. ZSYTRS solves a system of linear equations A*X = B with a complex symmetric matrix A using the factorization A = U*D*U**T or A = L*D*L**T computed by ZSYTRF. ZTBCON estimates the reciprocal of the condition number of a triangular band matrix A, in either the 1-norm or the infinity-norm. ZTBRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular band coefficient matrix. ZTBTRS solves a triangular system of the form where A is a triangular band matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. ZTGEVC computes selected left and/or right generalized eigenvectors of a pair of complex upper triangular matrices (A,B). The j-th generalized left and right eigenvectors are y and x, resp., such that: ZTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. ZTPCON estimates the reciprocal of the condition number of a packed triangular matrix A, in either the 1-norm or the infinity-norm. PPPPaaaaggggeeee 99999999 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZTPRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular packed coefficient matrix. ZTPTRI computes the inverse of a complex upper or lower triangular matrix A stored in packed format. ZTPTRS solves a triangular system of the form where A is a triangular matrix of order N stored in packed format, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. ZTRCON estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm. ZTREVC computes all or some right and/or left eigenvectors of a complex upper triangular matrix T. ZTREXC reorders the Schur factorization of a complex matrix A = Q*T*Q**H, so that the diagonal element of T with row index IFST is moved to row ILST. ZTRRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular coefficient matrix. ZTRSEN 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 on the diagonal of the upper triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace. ZTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q unitary). ZTRSYL solves the complex Sylvester matrix equation: op(A)*X + X*op(B) = scale*C or ZTRTI2 computes the inverse of a complex upper or lower triangular matrix. ZTRTRI computes the inverse of a complex upper or lower triangular matrix A. ZTRTRS solves a triangular system of the form where A is a triangular matrix of order N, and B is an N-by-NRHS matrix. A check is made to verify that A is nonsingular. ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A to PPPPaaaaggggeeee 111100000000 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) upper triangular form by means of unitary transformations. ZUNG2L 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 ZUNG2R generates an m by n complex matrix Q with orthonormal columns, which is defined as the first n columns of a product of k elementary reflectors of order m ZUNGBR generates one of the matrices Q or P**H determined by ZGEBRD when reducing a complex matrix A to bidiagonal form: A = Q * B * P**H. ZUNGHR generates a complex unitary matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by ZGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1). ZUNGL2 generates an m-by-n complex matrix Q with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of order n ZUNGLQ generates an M-by-N complex matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N ZUNGQL 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 ZUNGQR generates an M-by-N complex matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M ZUNGR2 generates an m by n complex matrix Q with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n ZUNGRQ generates an M-by-N complex matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N ZUNGTR generates a complex unitary matrix Q which is defined as the product of n-1 elementary reflectors of order N, as returned by ZHETRD: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), ZUNM2L overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors PPPPaaaaggggeeee 111100001111 CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) CCCCOOOOMMMMPPPPLLLLIIIIBBBB....SSSSGGGGIIIIMMMMAAAATTTTHHHH((((3333FFFF)))) ZUNM2R overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors If VECT = 'Q', ZUNMBR overwrites the general complex M-by-N matrix C with SIDE = 'L' SIDE = 'R' TRANS = 'N': Q * C C * Q TRANS = 'C': Q**H * C C * Q**H ZUNMHR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUNML2 overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors ZUNMLQ overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUNMQL overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUNMQR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUNMR2 overwrites the general complex m-by-n matrix C with where Q is a complex unitary matrix defined as the product of k elementary reflectors ZUNMRQ overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUNMTR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H ZUPGTR generates a complex unitary matrix Q which is defined as the product of n-1 elementary reflectors of order n, as returned by ZHPTRD using packed storage: if UPLO = 'U', Q = H(n-1) . . . H(2) H(1), ZUPMTR overwrites the general complex M-by-N matrix C with TRANS = 'C': Q**H * C C * Q**H PPPPaaaaggggeeee 111100002222