sggesx(3P) Sun Performance Library sggesx(3P)NAMEsggesx - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the real Schur form (S,T), and,
SYNOPSIS
SUBROUTINE SGGESX(JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE,
RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL SELCTG
LOGICAL BWORK(*)
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*),
VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
SUBROUTINE SGGESX_64(JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER*8 IWORK(*)
LOGICAL*8 SELCTG
LOGICAL*8 BWORK(*)
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*),
VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
F95 INTERFACE
SUBROUTINE GGESX(JOBVSL, JOBVSR, SORT, SELCTG, SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR],
RCONDE, RCONDV, [WORK], [LWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL :: SELCTG
LOGICAL, DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, RCONDE, RCONDV, WORK
REAL, DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GGESX_64(JOBVSL, JOBVSR, SORT, SELCTG, SENSE, [N], A, [LDA],
B, [LDB], SDIM, ALPHAR, ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR],
RCONDE, RCONDV, [WORK], [LWORK], [IWORK], [LIWORK], [BWORK],
[INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8) :: SELCTG
LOGICAL(8), DIMENSION(:) :: BWORK
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, RCONDE, RCONDV, WORK
REAL, DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void sggesx(char jobvsl, char jobvsr, char sort,
int(*selctg)(float,float,float), char sense, int n, float *a,
int lda, float *b, int ldb, int *sdim, float *alphar, float
*alphai, float *beta, float *vsl, int ldvsl, float *vsr, int
ldvsr, float *rconde, float *rcondv, int *info);
void sggesx_64(char jobvsl, char jobvsr, char sort,
long(*selctg)(float,float,float), char sense, long n, float
*a, long lda, float *b, long ldb, long *sdim, float *alphar,
float *alphai, float *beta, float *vsl, long ldvsl, float
*vsr, long ldvsr, float *rconde, float *rcondv, long *info);
PURPOSEsggesx computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the real Schur form (S,T), and, option‐
ally, 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; computes a
reciprocal condition number for the average of the selected eigenvalues
(RCONDE); and computes a reciprocal condition number for the right and
left deflating subspaces corresponding to the selected eigenvalues
(RCONDV). The leading columns of VSL and VSR then form an orthonormal
basis for the corresponding left and right eigenspaces (deflating sub‐
spaces).
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
a 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 inter‐
pretation for beta=0 or for 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 trian‐
gular 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 "standard‐
ized" 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 com‐
plex conjugate pair of generalized eigenvalues.
ARGUMENTS
JOBVSL (input)
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input)
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input)
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 a
selected complex eigenvalue may no longer satisfy
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
since ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned), in this
case INFO is set to N+3.
SENSE (input)
Determines which reciprocal condition numbers are computed.
= 'N' : None are computed;
= 'E' : Computed for average of selected eigenvalues only;
= 'V' : Computed for selected deflating subspaces only;
= 'B' : Computed for both. If SENSE = 'E', 'V', or 'B', SORT
must equal 'S'.
N (input) 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 general‐
ized Schur form S.
LDA (input)
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 general‐
ized Schur form T.
LDB (input)
The leading dimension of B. LDB >= max(1,N).
SDIM (output)
If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
genvalues (after sorting) for which SELCTG is true. (Complex
conjugate pairs for which SELCTG is true for either eigenval‐
ue count as 2.)
ALPHAR (output)
REAL array, dimension(N) On exit, (ALPHAR(j) +
ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized ei‐
genvalues. 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 usu‐
ally comparable with norm(A) in magnitude, and BETA always
less than and usually comparable with norm(B).
ALPHAI (output)
REAL array, dimension(N) See the description for ALPHAR.
BETA (output)
REAL array, dimension(N) See the description for ALPHAR.
VSL (output)
REAL array, dimension(LDVSL,N) If JOBVSL = 'V', VSL will con‐
tain the left Schur vectors. Not referenced if JOBVSL = 'N'.
LDVSL (input)
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 con‐
tain the right Schur vectors. Not referenced if JOBVSR =
'N'.
LDVSR (input)
The leading dimension of the matrix VSR. LDVSR >= 1, and if
JOBVSR = 'V', LDVSR >= N.
RCONDE (output)
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
reciprocal condition numbers for the average of the selected
eigenvalues. Not referenced if SENSE = 'N' or 'V'.
RCONDV (output)
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition numbers for the selected deflating sub‐
spaces. Not referenced if SENSE = 'N' or 'E'.
WORK (workspace)
REAL array, dimension(LWORK) On exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= 8*(N+1)+16. If
SENSE = 'E', 'V', or 'B', LWORK >= MAX( 8*(N+1)+16,
2*SDIM*(N-SDIM) ).
IWORK (workspace)
INTEGER array, dimension(LIWORK) Not referenced if SENSE =
'N'.
LIWORK (input)
The dimension of the array WORK. LIWORK >= N+6.
BWORK (workspace)
LOGICAL array, dimension(N) Not referenced if SORT = 'N'.
INFO (output)
= 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 com‐
plex eigenvalues so that leading eigenvalues in the General‐
ized Schur form no longer satisfy SELCTG=.TRUE. This could
also be caused due to scaling. =N+3: reordering failed in
STGSEN.
Further details ===============
An approximate (asymptotic) bound on the average absolute
error of the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum angular
error in the computed deflating subspaces is
EPS * norm((A, B)) / RCONDV( 2 ).
See LAPACK User's Guide, section 4.11 for more information.
6 Mar 2009 sggesx(3P)
