COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
complib, complib.sgimath, sgimath  Scientific and Mathematical Library
The Silicon Graphics Scientific Mathematical Library, complib.sgimath, is
a comprehensive collection of highperformance 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 ( man blas ), fft
routines ( man fft ), convolution routines ( man conv ) and LAPACK ( man
lapack ) 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
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
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someone will be paying for long distance 1200bps phone calls, so keep
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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 readonly 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
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?)
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University of Wollongong, NSW, Australia.
The contents of netlib (other than toms) is available on CDROM from
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about $60; for current information contact
Prime Time Freeware 370 Altair Way, Suite 150 Tel: +1 4087384832
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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 highperformance 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
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
addition, there are two routines which use the singular value
decomposition to solve certain least squares problems.
BLAS LIBRARY  Basic Linear Algebra Subprograms
BLAS Level 1
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
BLAS Level 2 dgemv, sgemv, zgemv, cgemv  BLAS Level Two MatrixVector
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 MatrixVector 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 MatrixVector Product
dtrsv, strsv, ztrsv, ctrsv  BLAS Level Two Solution of triangular
system of equations.
dgbmv, sgbmv, zgbmv, cgbmv  BLAS Level Two MatrixVector 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
MatrixVector Product and Solution of System of Equations.
BLAS Level 3 dtrmm, strmm, ztrmm, ctrmm  BLAS level three Matrix
Product
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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
EISPACK LIBRARY [Toc] [Back]
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 nonsymmetric 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
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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
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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 offdiagonal elements are all nonnegative,
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 offdiagonal elements are all nonnegative, 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.
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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.
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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 quasitriangular 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
quasitriangular form and the other in upper triangular form. It reduces
the quasitriangular matrix further, so that any remaining 2by2 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
quasitriangular form (in which each 2by2 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.
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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.
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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
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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
onedimensional 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.
LINPACK LIBRARY [Toc] [Back]
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
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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)
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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)
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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)
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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.
LAPACK LIBRARY [Toc] [Back]
SBDSQR computes the singular value decomposition (SVD) of a real NbyN
(upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the
transpose of P), where S is a diagonal matrix with nonnegative 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 1norm or the infinitynorm, 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 NbyNRHS matrices.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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 NbyNRHS matrices.
CGBTF2 computes an LU factorization of a complex mbyn band matrix A
using partial pivoting with row interchanges.
CGBTRF computes an LU factorization of a complex mbyn 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 ILO1 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 MbyN 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 1norm or the infinitynorm, 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 NbyN 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 NbyN 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 NbyN complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
CGEEVX computes for an NbyN complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
For a pair of NbyN complex nonsymmetric matrices A, B:
compute the generalized eigenvalues (alpha, beta)
For a pair of NbyN 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 MbyN matrix A: A = L
* Q.
CGELS solves overdetermined or underdetermined complex linear systems
involving an MbyN matrix A, or its conjugatetranspose, 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 2norm( b  A*x ).
CGELSX computes the minimumnorm 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 MbyN matrix A: A = Q *
L.
CGEQPF computes a QR factorization with column pivoting of a complex MbyN
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 MbyN matrix A: A = Q *
R.
CGERFS improves the computed solution to a system of linear equations and
Page 25
COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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 MbyN matrix A: A = R
* Q.
CGESV computes the solution to a complex system of linear equations
A * X = B, where A is an NbyN matrix and X and B are NbyNRHS
matrices.
CGESVD computes the singular value decomposition (SVD) of a complex MbyN
matrix A, optionally computing the left and/or right singular
vectors. The SVD is written
A = U * SIGMA * conjugatetranspose(V)
CGESVX uses the LU factorization to compute the solution to a complex
system of linear equations
A * X = B, where A is an NbyN matrix and X and B are NbyNRHS
matrices.
CGETF2 computes an LU factorization of a general mbyn matrix A using
partial pivoting with row interchanges.
CGETRF computes an LU factorization of a general MbyN 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 NbyN
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 ILO1 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
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COMPLIB.SGIMATH(3F)
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