dgegs(3P) Sun Performance Library dgegs(3P)NAMEdgegs - routine is deprecated and has been replaced by routine DGGES
SYNOPSIS
SUBROUTINE DGEGS(JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR
INTEGER N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE DGEGS_64(JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LDWORK, INFO)
CHARACTER * 1 JOBVSL, JOBVSR
INTEGER*8 N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
F95 INTERFACE
SUBROUTINE GEGS(JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHAR,
ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR
INTEGER :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
SUBROUTINE GEGS_64(JOBVSL, JOBVSR, [N], A, [LDA], B, [LDB], ALPHAR,
ALPHAI, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], [LDWORK], [INFO])
CHARACTER(LEN=1) :: JOBVSL, JOBVSR
INTEGER(8) :: N, LDA, LDB, LDVSL, LDVSR, LDWORK, INFO
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL(8), DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void dgegs(char jobvsl, char jobvsr, int n, double *a, int lda, double
*b, int ldb, double *alphar, double *alphai, double *beta,
double *vsl, int ldvsl, double *vsr, int ldvsr, int *info);
void dgegs_64(char jobvsl, char jobvsr, long n, double *a, long lda,
double *b, long ldb, double *alphar, double *alphai, double
*beta, double *vsl, long ldvsl, double *vsr, long ldvsr, long
*info);
PURPOSEdgegs routine is deprecated and has been replaced by routine DGGES.
DGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B:
the generalized eigenvalues (alphar +/- alphai*i, beta), the real Schur
form (A, B), and optionally left and/or right Schur vectors (VSL and
VSR).
(If only the generalized eigenvalues are needed, use the driver DGEGV
instead.)
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, 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 interpretation for beta=0, and even for both being
zero. A good beginning reference is the book, "Matrix Computations",
by G. Golub & C. van Loan (Johns Hopkins U. Press)
The (generalized) Schur form of a pair of matrices is the result of
multiplying both matrices on the left by one orthogonal matrix and both
on the right by another orthogonal matrix, these two orthogonal matri‐
ces being chosen so as to bring the pair of matrices into (real) Schur
form.
A pair of matrices A, B is in generalized real Schur form if B is upper
triangular with non-negative diagonal and A is block upper triangular
with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real gener‐
alized eigenvalues, while 2-by-2 blocks of A will be "standardized" by
making the corresponding elements of B have the form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in A and B will have a com‐
plex conjugate pair of generalized eigenvalues.
The left and right Schur vectors are the columns of VSL and VSR,
respectively, where VSL and VSR are the orthogonal matrices which
reduce A and B to Schur form:
Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T B (VSR) )
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.
N (input) The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output)
On entry, the first of the pair of matrices whose generalized
eigenvalues and (optionally) Schur vectors are to be com‐
puted. On exit, the generalized Schur form of A. Note: to
avoid overflow, the Frobenius norm of the matrix A should be
less than the overflow threshold.
LDA (input)
The leading dimension of A. LDA >= max(1,N).
B (input/output)
On entry, the second of the pair of matrices whose general‐
ized eigenvalues and (optionally) Schur vectors are to be
computed. On exit, the generalized Schur form of B. Note:
to avoid overflow, the Frobenius norm of the matrix B should
be less than the overflow threshold.
LDB (input)
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
j=1,...,N and BETA(j),j=1,...,N are the diagonals of the
complex Schur form (A,B) 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 eigen‐
value is real; if positive, then the j-th and (j+1)-st eigen‐
values are a complex conjugate pair, with ALPHAI(j+1) nega‐
tive.
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
alpha/beta. 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).
ALPHAI (output)
See the description for ALPHAR.
BETA (output)
See the description for ALPHAR.
VSL (output)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
(See "Purpose", above.) Not referenced if JOBVSL = 'N'.
LDVSL (output)
The leading dimension of the matrix VSL. LDVSL >=1, and if
JOBVSL = 'V', LDVSL >= N.
VSR (input)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
(See "Purpose", above.) Not referenced if JOBVSR = 'N'.
LDVSR (input)
The leading dimension of the matrix VSR. LDVSR >= 1, and if
JOBVSR = 'V', LDVSR >= N.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.
LDWORK (input)
The dimension of the array WORK. LDWORK >= max(1,4*N). For
good performance, LDWORK must generally be larger. To com‐
pute the optimal value of LDWORK, call ILAENV to get block‐
sizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: NB --
MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR The
optimal LDWORK is 2*N + N*(NB+1).
If LDWORK = -1, then a workspace query is assumed; the rou‐
tine 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 LDWORK is issued by XERBLA.
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: errors that usually indicate
LAPACK problems:
=N+1: error return from DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed iteration)
=N+7: error return from DGGBAK (computing VSL)
=N+8: error return from DGGBAK (computing VSR)
=N+9: error return from SLASCL (various places)
6 Mar 2009 dgegs(3P)
