ZGEGV(3) LAPACK driver routine (version 3.1) ZGEGV(3)
NAME
ZGEGV - i deprecated and has been replaced by routine ZGGEV
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( * )
PURPOSE
This routine is deprecated and has been replaced by routine ZGGEV.
ZGEGV computes the eigenvalues and, optionally, the left and/or right
eigenvectors of a complex matrix pair (A,B).
Given two square matrices A and B,
the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
eigenvalues lambda and corresponding (non-zero) eigenvectors x such
that
A*x = lambda*B*x.
An alternate form is to find the eigenvalues mu and corresponding
eigenvectors y such that
mu*A*y = B*y.
These two forms are equivalent with mu = 1/lambda and x = y if neither
lambda nor mu is zero. In order to deal with the case that lambda or
mu is zero or small, two values alpha and beta are returned for each
eigenvalue, such that lambda = alpha/beta and
mu = beta/alpha.
The vectors x and y in the above equations are right eigenvectors of
the matrix pair (A,B). Vectors u and v satisfying
u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
are left eigenvectors of (A,B).
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 eigenvectors;
= 'V': compute the left generalized eigenvectors (returned in
VL).
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors (returned in
VR).
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 matrix A. If JOBVL = 'V' or JOBVR = 'V', then on
exit A contains the Schur form of A from the generalized Schur
factorization of the pair (A,B) after balancing. If no eigen-
vectors were computed, then only the diagonal elements of the
Schur form will be correct. See ZGGHRD and ZHGEQZ for details.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on
exit B contains the upper triangular matrix obtained from B in
the generalized Schur factorization of the pair (A,B) after
balancing. If no eigenvectors were computed, then only the
diagonal elements of B will be correct. See ZGGHRD and ZHGEQZ
for details.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
The complex scalars alpha that define the eigenvalues of GNEP.
BETA (output) COMPLEX*16 array, dimension (N)
The complex scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
represent the j-th eigenvalue of the matrix pair (A,B), in one
of the forms lambda = alpha/beta or mu = beta/alpha. Since
either lambda or mu may overflow, they should not, in general,
be computed.
VL (output) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored in the
columns of VL, in the same order as their eigenvalues. Each
eigenvector is scaled so that its largest component has
abs(real part) + abs(imag. part) = 1, except for eigenvectors
corresponding to an eigenvalue with alpha = beta = 0, which are
set to zero. 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 JOBVR = 'V', the right eigenvectors x(j) are stored in the
columns of VR, in the same order as their eigenvalues. Each
eigenvector is scaled so that its largest component has
abs(real part) + abs(imag. part) = 1, except for eigenvectors
corresponding to an eigenvalue with alpha = beta = 0, which are
set to zero. 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 (MAX(1,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 generally be larger. To compute
the optimal value of LWORK, call ILAENV to get blocksizes (for
ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute: NB -- MAX of the
blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR; The optimal LWORK is
MAX( 2*N, N*(NB+1) ).
If LWORK = -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 LWORK is issued by XERBLA.
RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
INFO (output) INTEGER
= 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 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 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.
LAPACK driver routine (version 3.N1o)vember 2006 ZGEGV(3)