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 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 ( 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:
[email protected].
mail [email protected]
send index
The Internet address "[email protected]" 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
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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
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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: [email protected]
EARN/BITNET: netlib%[email protected]net (now livid.uib.no
?)
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For the Pacific, try [email protected] located at the
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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
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about $60; for current information contact
Prime Time Freeware 370 Altair Way, Suite 150 Tel: +1 408-738-4832
[email protected] Sunnyvale, CA 94086 USA Fax: +1 408-738-2050
The following libraries are available from "[email protected]".
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
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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 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.
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 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
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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 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.
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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 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.
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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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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.
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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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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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
Page 20
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
Page 22
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 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.
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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 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.
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COMPLIB.SGIMATH(3F) COMPLIB.SGIMATH(3F)
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 Mby-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
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 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 Mby-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
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COMPLIB.SGIMATH(3F)
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