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NAME
dtrevc - compute some or all of the right and/or left eigen-
vectors of a real upper quasi-triangular matrix T
SYNOPSIS
SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL,
LDVL, VR, LDVR, MM, M, WORK, INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
WORK( * )
#include <sunperf.h>
void dtrevc(char side, char howmny, int *select, int n, dou-
ble *t, int ldt, double *vl, int ldvl, double *vr,
int ldvr, int mm, int *m, int *info) ;
PURPOSE
DTREVC computes some or all of the right and/or left eigen-
vectors of a real upper quasi-triangular matrix T.
The right eigenvector x and the left eigenvector y of T
corresponding to an eigenvalue w are defined by:
T*x = w*x, y'*T = w*y'
where y' denotes the conjugate transpose of the vector y.
If all eigenvectors are requested, the routine may either
return the matrices X and/or Y of right or left eigenvectors
of T, or the products Q*X and/or Q*Y, where Q is an input
orthogonal
matrix. If T was obtained from the real-Schur factorization
of an original matrix A = Q*T*Q', then Q*X and Q*Y are the
matrices of right or left eigenvectors of A.
T must be in Schur canonical form (as returned by DHSEQR),
that is, block upper triangular with 1-by-1 and 2-by-2 diag-
onal blocks; each 2-by-2 diagonal block has its diagonal
elements equal and its off-diagonal elements of opposite
sign. Corresponding to each 2-by-2 diagonal block is a com-
plex conjugate pair of eigenvalues and eigenvectors; 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 eigenvec-
tors;
= 'B': compute all right and/or left eigenvec-
tors, 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/output) 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 a 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.; then on exit
SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE..
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input) DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur
canonical form.
LDT (input) INTEGER
The leading dimension of the array T. LDT >=
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 Schur vectors returned by
DHSEQR). On exit, if SIDE = 'L' or 'B', VL con-
tains: if HOWMNY = 'A', the matrix Y of left
eigenvectors of T; if HOWMNY = 'B', the matrix
Q*Y; if HOWMNY = 'S', the left eigenvectors of T
specified by SELECT, stored consecutively in the
columns of VL, in the same order as their eigen-
values. 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. If SIDE = 'R', VL is
not referenced.
LDVL (input) INTEGER
The leading dimension of the 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 Q of Schur vectors returned by
DHSEQR). On exit, if SIDE = 'R' or 'B', VR con-
tains: if HOWMNY = 'A', the matrix X of right
eigenvectors of T; if HOWMNY = 'B', the matrix
Q*X; if HOWMNY = 'S', the right eigenvectors of T
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their eigen-
values. 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. If SIDE = 'L', VR is
not referenced.
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
(3*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value
FURTHER DETAILS
The algorithm used in this program is basically backward
(forward) substitution, with scaling to make the the code
robust against possible overflow.
Each eigenvector is normalized so that the element of larg-
est magnitude has magnitude 1; here the magnitude of a com-
plex number (x,y) is taken to be |x| + |y|.
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