sgegv(3P) Sun Performance Library sgegv(3P)NAMEsgegv - routine is deprecated and has been replaced by routine SGGEV
SYNOPSIS
SUBROUTINE SGEGV(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
BETA, VL, LDVL, VR, LDVR, WORK, LDWORK, INFO)
CHARACTER * 1 JOBVL, JOBVR
INTEGER N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), WORK(*)
SUBROUTINE SGEGV_64(JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
BETA, VL, LDVL, VR, LDVR, WORK, LDWORK, INFO)
CHARACTER * 1 JOBVL, JOBVR
INTEGER*8 N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VL(LDVL,*),
VR(LDVR,*), WORK(*)
F95 INTERFACE
SUBROUTINE GEGV(JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHAR,
ALPHAI, BETA, VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [INFO])
CHARACTER(LEN=1) :: JOBVL, JOBVR
INTEGER :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, VL, VR
SUBROUTINE GEGV_64(JOBVL, JOBVR, [N], A, [LDA], B, [LDB], ALPHAR,
ALPHAI, BETA, VL, [LDVL], VR, [LDVR], [WORK], [LDWORK], [INFO])
CHARACTER(LEN=1) :: JOBVL, JOBVR
INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, LDWORK, INFO
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, WORK
REAL, DIMENSION(:,:) :: A, B, VL, VR
C INTERFACE
#include <sunperf.h>
void sgegv(char jobvl, char jobvr, int n, float *a, int lda, float *b,
int ldb, float *alphar, float *alphai, float *beta, float
*vl, int ldvl, float *vr, int ldvr, int *info);
void sgegv_64(char jobvl, char jobvr, long n, float *a, long lda, float
*b, long ldb, float *alphar, float *alphai, float *beta,
float *vl, long ldvl, float *vr, long ldvr, long *info);
PURPOSEsgegv routine is deprecated and has been replaced by routine SGGEV.
SGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and option‐
ally, 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).
ALPHAR (output)
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 positive, 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).
ALPHAI (output)
See the description of ALPHAR.
BETA (output)
See the description of ALPHAR.
VL (output)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, com‐
plex take two columns, the first for the real part and the
second for the imaginary part. Complex eigenvectors corre‐
spond 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 ei‐
genvalues 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.) Real eigenvectors take one column, com‐
plex take two columns, the first for the real part and the
second for the imaginary part. Complex eigenvectors corre‐
spond 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 ei‐
genvalues 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(1,8*N). For
good performance, LDWORK must generally be larger. To com‐
pute the optimal value of LDWORK, call ILAENV to get block‐
sizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: NB --
MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR; The
optimal LDWORK is: 2*N + MAX( 6*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.
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 ALPHAR(j), ALPHAI(j), and BETA(j) should
be correct for j=INFO+1,...,N. > N: errors that usually
indicate 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 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 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 Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
6 Mar 2009 sgegv(3P)
