SGGES(3S)SGGES(3S)NAMESGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
SYNOPSIS
SUBROUTINE SGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM,
ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
LWORK, BWORK, INFO )
CHARACTER JOBVSL, JOBVSR, SORT
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
LOGICAL BWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )
LOGICAL SELCTG
EXTERNAL SELCTG
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSESGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the
generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR). This gives the generalized Schur factorization
(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster of
eigenvalues appears in the leading diagonal blocks of the upper quasi-
triangular matrix S and the upper triangular matrix T.The leading columns
of VSL and VSR then form an orthonormal basis for the corresponding left
and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver SGGEV
instead, which is faster.)
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a
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ratio alpha/beta = w, such that A - w*B is singular. It is usually
represented as the pair (alpha,beta), as there is a reasonable
interpretation for beta=0 or both being zero.
A pair of matrices (S,T) is in generalized real Schur form if T is upper
triangular with non-negative diagonal and S is block upper triangular
with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real
generalized eigenvalues, while 2-by-2 blocks of S will be "standardized"
by making the corresponding elements of T have the form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
ARGUMENTS
JOBVSL (input) CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input) CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the diagonal
of the generalized Schur form. = 'N': Eigenvalues are not
ordered;
= 'S': Eigenvalues are ordered (see SELCTG);
SELCTG (input) LOGICAL FUNCTION of three REAL arguments
SELCTG must be declared EXTERNAL in the calling subroutine. If
SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is
used to select eigenvalues to sort to the top left of the Schur
form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one
of a complex conjugate pair of eigenvalues is selected, then both
complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in
this case.
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices. On exit, A has been
overwritten by its generalized Schur form S.
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LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices. On exit, B has
been overwritten by its generalized Schur form T.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM (output) INTEGER
If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of
eigenvalues (after sorting) for which SELCTG is true. (Complex
conjugate pairs for which SELCTG is true for either eigenvalue
count as 2.)
ALPHAR (output) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA (output) REAL
array, dimension (N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) +
ALPHAI(j)*i, and BETA(j),j=1,...,N are the diagonals of the
complex Schur form (S,T) that would result if the 2-by-2 diagonal
blocks of the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations. If
ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive,
then the j-th and (j+1)-st eigenvalues are a complex conjugate
pair, with ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
easily over- or underflow, and BETA(j) may even be zero. Thus,
the user should avoid naively computing the ratio. However,
ALPHAR and ALPHAI will be always less than and usually comparable
with norm(A) in magnitude, and BETA always less than and usually
comparable with norm(B).
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors. Not
referenced if JOBVSL = 'N'.
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and if JOBVSL
= 'V', LDVSL >= N.
VSR (output) REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors. Not
referenced if JOBVSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and if
JOBVSR = 'V', LDVSR >= N.
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WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 8*N+16.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns this
value as the first entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
BWORK (workspace) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1,...,N: The QZ iteration failed. (A,B) are not in Schur
form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for
j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in
SHGEQZ.
=N+2: after reordering, roundoff changed values of some complex
eigenvalues so that leading eigenvalues in the Generalized Schur
form no longer satisfy SELCTG=.TRUE. This could also be caused
due to scaling. =N+3: reordering failed in STGSEN.
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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