ZTREVC(1) LAPACK routine (version 3.2) ZTREVC(1)NAME
ZTREVC - computes 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. Matrices of this type are produced
by the Schur factorization of a complex general matrix: A = Q*T*Q**H,
as computed by ZHSEQR.
The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:
T*x = w*x, (y**H)*T = w*(y**H)
where y**H denotes the conjugate transpose of the vector y. The eigen‐
values are not input to this routine, but are read directly from the
diagonal of T.
This routine returns the matrices X and/or Y of right and left eigen‐
vectors of T, or the products Q*X and/or Q*Y, where Q is an input
matrix. If Q is the unitary factor that reduces a matrix A to Schur
form T, then Q*X and Q*Y are the matrices of right and left eigenvec‐
tors 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 eigenvectors;
= 'B': compute all right and/or left eigenvectors, backtrans‐
formed using the matrices supplied in VR and/or VL; = 'S':
compute selected right and/or left eigenvectors, as indicated
by the logical array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be com‐
puted. The eigenvector corresponding to the j-th eigenvalue is
computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = 'A'
or 'B'.
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 con‐
tain 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
contains: 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 eigenvalues. Not
referenced if SIDE = 'R'.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1, and if SIDE
= 'L' or 'B', LDVL >= N.
VR (input/output) COMPLEX*16 array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must con‐
tain 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
contains: 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 consecu‐
tively in the columns of VR, in the same order as their eigen‐
values. Not referenced if SIDE = 'L'.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1, and if SIDE
= 'R' or 'B'; LDVR >= N.
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 illegal value
FURTHER DETAILS
The algorithm used in this program is basically backward (forward) sub‐
stitution, 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|.
LAPACK routine (version 3.2) November 2008 ZTREVC(1)