ZTREVC - compute some or all of the right and/or left
eigenvectors of a complex upper triangular matrix T
SYNOPSIS
SUBROUTINE ZTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL,
LDVL, VR, LDVR, MM, M, WORK, RWORK,
INFO )
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, *
), WORK( * )
PURPOSE
ZTREVC computes some or all of the right and/or left
eigenvectors of a complex upper triangular matrix T. The
right eigenvector x and the left eigenvector y of T corre
sponding 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 eigenvec
tors of T, or the products Q*X and/or Q*Y, where Q is an
input unitary
matrix. If T was obtained from the 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.
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.
If HOWMNY = 'S', SELECT specifies the eigenvectors
to be computed. If HOWMNY = 'A' or 'B', SELECT is
not referenced. To select the eigenvector corre
sponding to the j-th eigenvalue, SELECT(j) must be
set to .TRUE..
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input/output) COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T. T is modified, but
restored on exit.
LDT (input) INTEGER
The leading dimension of the array T. LDT >=
max(1,N).
VL (input/output) COMPLEX*16 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
unitary matrix Q of Schur vectors returned by
ZHSEQR). On exit, if SIDE = 'L' or 'B', VL con
tains: if HOWMNY = 'A', the matrix Y of left
eigenvectors of T; VL is lower triangular. The i-
th column VL(i) of VL is the eigenvector corre
sponding to T(i,i). 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. 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) COMPLEX*16 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
unitary matrix Q of Schur vectors returned by
ZHSEQR). On exit, if SIDE = 'R' or 'B', VR con
tains: if HOWMNY = 'A', the matrix X of right
eigenvectors of T; VR is upper triangular. The i-
th column VR(i) of VR is the eigenvector corre
sponding to T(i,i). 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. If SIDE = 'L', VR is not referenced.
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
eigenvector occupies one column.
WORK (workspace) COMPLEX*16 array, dimension (2*N)
RWORK (workspace) DOUBLE PRECISION array, dimension (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
largest magnitude has magnitude 1; here the magnitude of a
complex number (x,y) is taken to be |x| + |y|.
Man(1) output converted with
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