CTREVC(l) ) CTREVC(l)NAME
CTREVC - compute some or all of the right and/or left eigenvectors of a
complex upper triangular matrix T
SYNOPSIS
SUBROUTINE CTREVC( 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( * )
REAL RWORK( * )
COMPLEX T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
CTREVC 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 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 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 eigenvectors;
= 'B': compute all right and/or left eigenvectors, and back‐
transform 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 com‐
puted. If HOWMNY = 'A' or 'B', SELECT is not referenced. To
select the eigenvector corresponding 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 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 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 CHSEQR). On exit, if SIDE = 'L' or 'B', VL
contains: 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 corresponding to T(i,i). if HOWMNY = 'B', the
matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T speci‐
fied by SELECT, stored consecutively in the columns of VL, in
the same order as their eigenvalues. 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 otherwise.
VR (input/output) COMPLEX 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 CHSEQR). On exit, if SIDE = 'R' or 'B', VR
contains: 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 corresponding to T(i,i). if HOWMNY = 'B', the
matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T speci‐
fied by SELECT, stored consecutively in the columns of VR, in
the same order as their eigenvalues. 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 otherwise.
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 array, dimension (2*N)
RWORK (workspace) REAL 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 version 3.0 15 June 2000 CTREVC(l)
