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NAME
sgegs - compute for a pair of N-by-N real nonsymmetric
matrices A, B
SYNOPSIS
SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK,
INFO )
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK(
* )
#include <sunperf.h>
void sgegs(char jobvsl, char jobvsr, int n, float *sa, int
lda, float *sb, int ldb, float *alphar, float
*alphai, float *beta, float *vsl, int ldvsl, float
*vsr, int ldvsr, int *info) ;
PURPOSE
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) )
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.
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 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) 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 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) INTEGER
The leading dimension of B. LDB >= max(1,N).
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, 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 eigenvalue is real; if positive, then the j-
th and (j+1)-st eigenvalues are a complex conju-
gate 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).
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur
vectors. (See "Purpose", above.) 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. (See "Purpose", above.) Not referenced
if JOBVSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >=
1, and if JOBVSR = 'V', LDVSR >= N.
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 >=
max(1,4*N). For good performance, LWORK must gen-
erally be larger. To compute the optimal value of
LWORK, call ILAENV to get blocksizes (for SGEQRF,
SORMQR, and SORGQR.) Then compute: NB -- MAX of
the blocksizes for SGEQRF, SORMQR, and SORGQR The
optimal LWORK is 2*N + N*(NB+1).
INFO (output) INTEGER
= 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)
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