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NAME
zgegv - compute for a pair of N-by-N complex nonsymmetric
matrices A and B, the generalized eigenvalues (alpha, beta),
and optionally,
SYNOPSIS
SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA,
BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO
)
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ),
VL( LDVL, * ), VR( LDVR, * ), WORK( * )
#include <sunperf.h>
void zgegv(char jobvl, char jobvr, int n, doublecomplex *za,
int lda, doublecomplex *zb, int ldb, doublecomplex
*zalpha, doublecomplex *zbeta, doublecomplex *vl,
int ldvl, doublecomplex *vr, int ldvr, int *info);
PURPOSE
ZGEGV computes for a pair of N-by-N complex nonsymmetric
matrices A and B, the generalized eigenvalues (alpha, beta),
and optionally, the left and/or right generalized eigenvec-
tors (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) COMPLEX*16 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) COMPLEX*16 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).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues.
Note: the quotients ALPHA(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, ALPHA 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) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors.
(See "Purpose", above.) 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 JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1,
and if JOBVL = 'V', LDVL >= N.
VR (output) COMPLEX*16 array, dimension (LDVR,N)
If JOBVL = 'V', the right generalized eigenvec-
tors. (See "Purpose", above.) 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) COMPLEX*16 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(1,2*N). For good performance, LWORK must gen-
erally be larger. To compute the optimal value of
LWORK, call ILAENV to get blocksizes (for ZGEQRF,
ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of
the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The
optimal LWORK is MAX( 2*N, N*(NB+1) ).
RWORK (workspace/output) DOUBLE PRECISION array, dimen-
sion (8*N)
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 ALPHA(j) and
BETA(j) should be correct for j=INFO+1,...,N. >
N: errors that usually indicate LAPACK problems:
=N+1: error return from ZGGBAL
=N+2: error return from ZGEQRF
=N+3: error return from ZUNMQR
=N+4: error return from ZUNGQR
=N+5: error return from ZGGHRD
=N+6: error return from ZHGEQZ (other than failed
iteration) =N+7: error return from ZTGEVC
=N+8: error return from ZGGBAK (computing VL)
=N+9: error return from ZGGBAK (computing VR)
=N+10: error return from ZLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls ZGGBAL 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, ZGGBAK 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 complex Schur form[*] of the "balanced" versions
of A and B. If no eigenvectors are computed, then only the
diagonal blocks will be correct.
[*] In other words, upper triangular form.
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