DTGEVC - compute some or all of the right and/or left gen
eralized eigenvectors of a pair of real upper triangular
matrices (A,B)
SYNOPSIS
SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, A, LDA, B,
LDB, VL, LDVL, VR, LDVR, MM, M, WORK,
INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDA, LDB, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), VL(
LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
DTGEVC computes some or all of the right and/or left gen
eralized eigenvectors of a pair of real upper triangular
matrices (A,B). The right generalized eigenvector x and
the left generalized eigenvector y of (A,B) corresponding
to a generalized eigenvalue w are defined by:
(A - wB) * x = 0 and y**H * (A - wB) = 0
where y**H denotes the conjugate tranpose of y.
If an eigenvalue w is determined by zero diagonal elements
of both A and B, a unit vector is returned as the corre
sponding eigenvector.
If all eigenvectors are requested, the routine may either
return the matrices X and/or Y of right or left eigenvec
tors of (A,B), or the products Z*X and/or Q*Y, where Z and
Q are input orthogonal matrices. If (A,B) was obtained
from the generalized real-Schur factorization of an origi
nal pair of matrices
(A0,B0) = (Q*A*Z**H,Q*B*Z**H),
then Z*X and Q*Y are the matrices of right or left eigen
vectors of A.
A must be block upper triangular, with 1-by-1 and 2-by-2
diagonal blocks. Corresponding to each 2-by-2 diagonal
block is a complex conjugate pair of eigenvalues and
eigenvectors; only one
eigenvector of the pair is computed, namely the one corre
sponding to the eigenvalue with positive imaginary part.
ARGUMENTS
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors.
HOWMNY (input) CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
and backtransform them using the input matrices
supplied in VR and/or VL; = 'S': compute selected
right and/or left eigenvectors, specified by the
logical array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors
to be computed. If HOWMNY='A' or 'B', SELECT is
not referenced. To select the real eigenvector
corresponding to the real eigenvalue w(j),
SELECT(j) must be set to .TRUE. To select the
complex eigenvector corresponding to a complex
conjugate pair w(j) and w(j+1), either SELECT(j)
or SELECT(j+1) must be set to .TRUE..
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The upper quasi-triangular matrix A.
LDA (input) INTEGER
The leading dimension of array A. LDA >= max(1,
N).
B (input) DOUBLE PRECISION array, dimension (LDB,N)
The upper triangular matrix B. If A has a 2-by-2
diagonal block, then the corresponding 2-by-2
block of B must be diagonal with positive ele
ments.
LDB (input) INTEGER
The leading dimension of array B. LDB >=
max(1,N).
VL (input/output) DOUBLE PRECISION array,
dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B',
VL must contain an N-by-N matrix Q (usually the
orthogonal matrix Q of left Schur vectors returned
by DHGEQZ). On exit, if SIDE = 'L' or 'B', VL
contains: if HOWMNY = 'A', the matrix Y of left
eigenvectors of (A,B); if HOWMNY = 'B', the matrix
Q*Y; if HOWMNY = 'S', the left eigenvectors of
(A,B) specified by SELECT, stored consecutively in
the columns of VL, in the same order as their
A complex eigenvector corresponding to a complex
eigenvalue is stored in two consecutive columns,
the first holding the real part, and the second
the imaginary part.
LDVL (input) INTEGER
The leading dimension of array VL. LDVL >=
max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 other
wise.
VR (input/output) DOUBLE PRECISION array,
dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B',
VR must contain an N-by-N matrix Q (usually the
orthogonal matrix Z of right Schur vectors
returned by DHGEQZ). On exit, if SIDE = 'R' or
'B', VR contains: if HOWMNY = 'A', the matrix X of
right eigenvectors of (A,B); if HOWMNY = 'B', the
matrix Z*X; if HOWMNY = 'S', the right eigenvec
tors of (A,B) specified by SELECT, stored consecu
tively in the columns of VR, in the same order as
their eigenvalues. If SIDE = 'L', VR is not ref
erenced.
A complex eigenvector corresponding to a complex
eigenvalue is stored in two consecutive columns,
the first holding the real part and the second the
imaginary part.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >=
max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 other
wise.
MM (input) INTEGER
The number of columns in the arrays VL and/or VR.
MM >= M.
M (output) INTEGER
The number of columns in the arrays VL and/or VR
actually used to store the eigenvectors. If
HOWMNY = 'A' or 'B', M is set to N. Each selected
real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
WORK (workspace) DOUBLE PRECISION array, dimension
(6*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal value.
a complex eigenvalue.
FURTHER DETAILS
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diag
onal
WORK( N+j ) = 1-norm of j-th column of B, above the
diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed
eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed
eigenvector
Rowwise vs. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of
solving the singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0
(for left)
Consider finding the i-th right eigenvector (assume all
eigenvalues are real). The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. .
.,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0.)
The "rowwise" method is:
(1) v(i) := 1
for j = i-1,. . .,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the "dot product" step, since
it is an inner product between the j-th row and the por
tion of the eigenvector that has been computed so far.
The "columnwise" method consists basically in doing the
sums for all the rows in parallel. As each v(j) is com
puted, the contribution of v(j) times the j-th column of C
is added to the partial sums. Since FORTRAN arrays are
step, the elements of C that are accessed are adjacent to
one another, whereas with the rowwise method, the elements
accessed at a step are spaced LDA (and LDB) words apart.
When finding left eigenvectors, the matrix in question is
the transpose of the one in storage, so the rowwise method
then actually accesses columns of A and B at each step,
and so is the preferred method.
Man(1) output converted with
man2html