Contents
sgegs - routine is deprecated and has been replaced by rou-
tine SGGES
SUBROUTINE SGEGS(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
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*),
VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
SUBROUTINE SGEGS_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
REAL 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, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, 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, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, VSL, VSR
C INTERFACE
#include <sunperf.h>
void sgegs(char jobvsl, char jobvsr, int n, float *a, int
lda, float *b, int ldb, float *alphar, float
*alphai, float *beta, float *vsl, int ldvsl, float
*vsr, int ldvsr, int *info);
void sgegs_64(char jobvsl, char jobvsr, long n, float *a,
long lda, float *b, long ldb, float *alphar, float
*alphai, float *beta, float *vsl, long ldvsl,
float *vsr, long ldvsr, long *info);
sgegs routine is deprecated and has been replaced by routine
SGGES.
SGEGS 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 SGEGV 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 interpre-
tation 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 orthogo-
nal matrix, these two orthogonal matrices 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 generalized 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 complex 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) )
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 computed. On exit, the general-
ized 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
generalized eigenvalues and (optionally) Schur
vectors are to be computed. On exit, the general-
ized 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 com-
plex 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 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
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 (input)
If JOBVSL = 'V', VSL will contain the left Schur
vectors. (See "Purpose", above.) Not referenced
if JOBVSL = 'N'.
LDVSL (input)
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 compute the optimal value
of LDWORK, call ILAENV to get blocksizes (for
SGEQRF, SORMQR, and SORGQR.) Then compute: NB --
MAX of the blocksizes for SGEQRF, SORMQR, and
SORGQR The optimal LDWORK is 2*N + N*(NB+1).
If LDWORK = -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 LDWORK is issued by XERBLA.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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 SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed
iteration) =N+7: error return from SGGBAK (comput-
ing VSL)
=N+8: error return from SGGBAK (computing VSR)
=N+9: error return from SLASCL (various places)