cgegv(3P) Sun Performance Library cgegv(3P)NAMEcgegv - routine is deprecated and has been replaced by routine CGGEV
SYNOPSIS
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);
PURPOSEcgegv routine is deprecated and has been replaced by routine CGGEV.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 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 interpretation 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 generalized eigen‐
value w for a pair of matrices (A,B) is a vector r such that (A -
w B) r = 0 . A left generalized eigenvector 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)
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input)
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
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) generalized 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 general‐
ized eigenvalues and (optionally) generalized 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 under‐
flow, 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 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 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 com‐
pute the optimal value of LDWORK, call ILAENV to get block‐
sizes (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 rou‐
tine 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 illegal value.
=1,...,N: The QZ iteration failed. No eigenvectors 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)
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 eigenvectors back to what they
would have been (in perfect arithmetic) 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.
6 Mar 2009 cgegv(3P)
