CXML
DGEGV (3lapack)
compute for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR)
SYNOPSIS
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB,
* ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
DGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B, the
generalized eigenvalues (alphar +/- alphai*i, 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 eigenvalue
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) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
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) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the second 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 B on exit, see "Further Details",
below.)
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
(output) DOUBLE PRECISION array, dimension (N) 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).
VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See "Purpose",
above.) Real eigenvectors take one column, complex take two
columns, the first for the real part and the second for the
imaginary part. Complex eigenvectors correspond 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 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) DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVL = 'V', the right generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, complex take
two columns, the first for the real part and the second for the
imaginary part. Complex eigenvectors correspond 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 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) DOUBLE PRECISION 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(1,8*N). For good
performance, LWORK must generally be larger. To compute the
optimal value of LWORK, call ILAENV to get blocksizes (for DGEQRF,
DORMQR, and DORGQR.) Then compute: NB -- MAX of the blocksizes
for DGEQRF, DORMQR, and DORGQR; The optimal LWORK is: 2*N + MAX(
6*N, N*(NB+1) ).
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 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 DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed iteration) =N+7:
error return from DTGEVC
=N+8: error return from DGGBAK (computing VL)
=N+9: error return from DGGBAK (computing VR)
=N+10: error return from DLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls DGGBAL 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, DGGBAK 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 DHGEQZ, DGEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
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