Contents
cgegv - routine is deprecated and has been replaced by rou-
tine CGGEV
SUBROUTINE CGEGV(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
LDVL, VR, LDVR, WORK, LDWORK, WORK2, INFO)
CHARACTER * 1 JOBVL, JOBVR
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), WORK(*)
INTEGER N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL WORK2(*)
SUBROUTINE CGEGV_64(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
LDVL, VR, LDVR, WORK, LDWORK, WORK2, INFO)
CHARACTER * 1 JOBVL, JOBVR
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), WORK(*)
INTEGER*8 N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL WORK2(*)
F95 INTERFACE
SUBROUTINE GEGV(JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHA, BETA,
VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: JOBVL, JOBVR
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, VL, VR
INTEGER :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL, DIMENSION(:) :: WORK2
SUBROUTINE GEGV_64(JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHA,
BETA, VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: JOBVL, JOBVR
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, VL, VR
INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL, DIMENSION(:) :: WORK2
C INTERFACE
#include <sunperf.h>
void cgegv(char jobvl, char jobvr, int n, complex *a, int
lda, complex *b, int ldb, complex *alpha, complex
*beta, complex *vl, int ldvl, complex *vr, int
ldvr, int *info);
void cgegv_64(char jobvl, char jobvr, long n, complex *a,
long lda, complex *b, long ldb, complex *alpha,
complex *beta, complex *vl, long ldvl, complex
*vr, long ldvr, long *info);
cgegv routine is deprecated and has been replaced by routine
CGGEV.
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.
JOBVL (input)
= 'N': do not compute the left generalized eigen-
vectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input)
= 'N': do not compute the right generalized
eigenvectors;
= 'V': compute the right generalized eigenvec-
tors.
N (input) The order of the matrices A, B, VL, and VR. N >=
0.
A (input/output)
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)
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) 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)
The leading dimension of B. LDB >= max(1,N).
ALPHA (output)
On exit, ALPHA(j)/VL(j), j=1,...,N, will be the
generalized eigenvalues.
Note: the quotients ALPHA(j)/VL(j) may easily
over- or underflow, and VL(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 VL always less than and usually
comparable with norm(B).
VL (output)
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'.
BETA (output)
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)
The leading dimension of the matrix VL. LDVL >= 1,
and if JOBVL = 'V', LDVL >= N.
VR (output)
If JOBVR = '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)
The leading dimension of the matrix VR. LDVR >= 1,
and if JOBVR = 'V', LDVR >= N.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LDWORK.
LDWORK (input)
The dimension of the array WORK. LDWORK >=
max(1,2*N). For good performance, LDWORK must
generally be larger. To compute the optimal value
of LDWORK, call ILAENV to get blocksizes (for
CGEQRF, CUNMQR, and CUNGQR.) Then compute: NB as
the MAX of the blocksizes for CGEQRF, CUNMQR, and
CUNGQR; The optimal LDWORK is MAX( 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.
WORK2 (workspace)
dimension(8*N)
INFO (output)
= 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 VL(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)
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.