Purpose
=======
LA_GEGV 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
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.
=========
SUBROUTINE LA_GEGV( A, B, , BETA, VL, VR, INFO )
INTEGER, INTENT(OUT), OPTIONAL :: INFO
(), INTENT(INOUT) :: A(:,:), B(:,:)
(), INTENT(OUT), OPTIONAL :: , BETA(:)
(), INTENT(OUT), OPTIONAL :: VL(:,:), VR(:,:)
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
::= ALPHAR, ALPHAI | ALPHA
::= ALPHAR(:), ALPHAI(:) | ALPHA(:)
Arguments
=========
A (input/output) REAL / COMPLEX array, shape (:,:),
SIZE(A,1) == SIZE(A,2) == 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.)
B (input/output) REAL / COMPLEX array, shape (:,:),
SIZE(rBA,1) == SIZE(rBA,2) == 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.)
ALPHAR Only for the real case. Optional (output) REAL array,
ALPHAI shape (:), SIZE(ALPHA') == n. ALPHA ::= ALPHAR | ALPHAI
ALPHA Only for the complex case. Optional (output) COMPLEX
array, shape (:), SIZE(ALPHAI) == n.
BETA Optional (output) REAL / COMPLEX array, shape (:),
SIZE(BETA) == 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 Optional (output) REAL / COMPLEX array, shape (:,:),
SIZE(VL,1) == SIZE(VL,2) == n.
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.
VR Optional (output) REAL / COMPLEX array, shape (:,:),
SIZE(VR,1) == SIZE(VR,2) == n.
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.
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 LA_GGBAL
=n+2: error return from LA_GEQRF
=n+3: error return from LA_ORMQR
=n+4: error return from LA_ORGQR
=n+5: error return from LA_GGHRD
=n+6: error return from LA_HGEQZ (other than failed
iteration)
=n+7: error return from LA_TGEVC
=n+8: error return from LA_GGBAK (computing VL)
=n+9: error return from LA_GGBAK (computing VR)
=n+10: error return from LA_LASCL (various calls)
If INFO is not present and an error occurs, then the program is
terminated with an error message.
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 VL or VR 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 LA_HGEQZ, LA_GEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.