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NAME
cgegv - compute for a pair of N-by-N complex nonsymmetric
matrices A and B, the generalized eigenvalues (alpha, beta),
and optionally,
SYNOPSIS
SUBROUTINE CGEGV( 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
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
LDVL, * ), VR( LDVR, * ), WORK( * )
#include <sunperf.h>
void cgegv(char jobvl, char jobvr, int n, complex *ca, int
lda, complex *cb, int ldb, complex *calpha, com-
plex *cbeta, complex *vl, int ldvl, complex *vr,
int ldvr, int *info);
PURPOSE
CGEGV 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 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 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 array, dimension (N)
BETA (output) COMPLEX 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 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 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 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 CGEQRF,
CUNMQR, and CUNGQR.) Then compute: NB -- MAX of
the blocksizes for CGEQRF, CUNMQR, and CUNGQR; The
optimal LWORK is MAX( 2*N, N*(NB+1) ).
RWORK (workspace/output) REAL array, dimension (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 CGGBAL
=N+2: error return from CGEQRF
=N+3: error return from CUNMQR
=N+4: error return from CUNGQR
=N+5: error return from CGGHRD
=N+6: error return from CHGEQZ (other than failed
iteration) =N+7: error return from CTGEVC
=N+8: error return from CGGBAK (computing VL)
=N+9: error return from CGGBAK (computing VR)
=N+10: error return from CLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls CGGBAL 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, CGGBAK 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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