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NAME
sgegv - compute for a pair of n-by-n real nonsymmetric
matrices A and B, the generalized eigenvalues (alphar +/-
alphai*i, beta), and optionally, the left and/or right gen-
eralized eigenvectors (VL and VR)
SYNOPSIS
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR,
ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK,
INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
#include <sunperf.h>
void sgegv(char jobvl, char jobvr, int n, float *sa, int
lda, float *sb, int ldb, float *alphar, float
*alphai, float *beta, float *vl, int ldvl, float
*vr, int ldvr, int *info);
PURPOSE
SGEGV computes for a pair of n-by-n real nonsymmetric
matrices A and B, the generalized eigenvalues (alphar +/-
alphai*i, beta), and optionally, the left and/or right gen-
eralized eigenvectors (VL and VR).
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)
A right generalized eigenvector corresponding to a general-
ized eigenvalue w for a pair of matrices (A,B) is a vector
r such that (A - w B) r = 0 . A left generalized eigen-
vector is a vector l such that l**H * (A - w B) = 0, where
l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B --
see "Further Details", below.
ARGUMENTS
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigen-
vectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized
eigenvectors;
= 'V': compute the right generalized eigenvec-
tors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >=
0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices whose
generalized eigenvalues and (optionally) general-
ized eigenvectors are to be computed. On exit,
the contents will have been destroyed. (For a
description of the contents of A on exit, see
"Further Details", below.)
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) general-
ized eigenvectors are to be computed. On exit,
the contents will have been destroyed. (For a
description of the contents of B on exit, see
"Further Details", below.)
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. If ALPHAI(j) is
zero, then the j-th eigenvalue is real; if posi-
tive, 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).
VL (output) REAL array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors.
(See "Purpose", above.) Real eigenvectors take
one column, complex take two columns, the first
for the real part and the second for the imaginary
part. Complex eigenvectors correspond to an
eigenvalue with positive imaginary part. Each
eigenvector will be scaled so the largest com-
ponent will have abs(real part) + abs(imag. part)
= 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as
the corresponding eigenvector. Not referenced if
JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1,
and if JOBVL = 'V', LDVL >= N.
VR (output) REAL array, dimension (LDVR,N)
If JOBVL = 'V', the right generalized eigenvec-
tors. (See "Purpose", above.) Real eigenvectors
take one column, complex take two columns, the
first for the real part and the second for the
imaginary part. Complex eigenvectors correspond
to an eigenvalue with positive imaginary part.
Each eigenvector will be scaled so the largest
component will have abs(real part) + abs(imag.
part) = 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as
the corresponding eigenvector. Not referenced if
JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1,
and if JOBVR = 'V', LDVR >= 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(1,8*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 + MAX( 6*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. No eigenvec-
tors have been calculated, but ALPHAR(j),
ALPHAI(j), and BETA(j) should be correct for
j=INFO+1,...,N. > N: errors that usually indi-
cate 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 STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls SGGBAL to both permute and scale rows and
columns of A and B. The permutations PL and PR are chosen
so that PL*A*PR and PL*B*R will be upper triangular except
for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i
and j as close together as possible. The diagonal scaling
matrices DL and DR are chosen so that the pair
DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one
(except for the elements that start out zero.)
After the eigenvalues and eigenvectors of the balanced
matrices have been computed, SGGBAK transforms the eigenvec-
tors back to what they would have been (in perfect arith-
metic) if they had not been balanced.
Contents of A and B on Exit
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or
JOBVR='V' or both), then on exit the arrays A and B will
contain the real Schur form[*] of the "balanced" versions of
A and B. If no eigenvectors are computed, then only the
diagonal blocks will be correct.
[*] See SHGEQZ, SGEGS, or read the book "Matrix Computa-
tions",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
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