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