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NAME
stgevc - 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 STGEVC( 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( * )
REAL A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ),
WORK( * )
#include <sunperf.h>
void stgevc(char side, char howmny, int *select, int n,
float *sa, int lda, float *sb, int ldb, float *vl,
int ldvl, float *vr, int ldvr, int mm, int *m, int
*info) ;
PURPOSE
STGEVC 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 eigen-
value 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
corresponding eigenvector.
If all eigenvectors are requested, the routine may either
return the matrices X and/or Y of right or left eigenvectors
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 original 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
eigenvectors 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 eigen-
vectors; only one
eigenvector of the pair is computed, namely the one
corresponding to the eigenvalue with positive imaginary
part.
ARGUMENTS
SIDE (input) CHARACTER*1
= '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) REAL 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) REAL 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) REAL 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 SHGEQZ). 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
eigenvalues. If SIDE = 'R', VL is not referenced.
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) COMPLEX 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 SHGEQZ). 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 eigen-
vectors of (A,B) specified by SELECT, stored con-
secutively in the columns of VR, in the same order
as their eigenvalues. If SIDE = 'L', VR is not
referenced.
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) REAL array, dimension (6*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille-
gal value.
> 0: the 2-by-2 block (INFO:INFO+1) does not have
a complex eigenvalue.
FURTHER DETAILS
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diago-
nal
WORK( N+j ) = 1-norm of j-th column of B, above the diag-
onal
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 eigen-
vector
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 portion 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 computed, the
contribution of v(j) times the j-th column of C is added to
the partial sums. Since FORTRAN arrays are stored column-
wise, this has the advantage that at each 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.
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