ctrevc (l) - Linux Man Pages

ctrevc: computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T

NAME

CTREVC - computes 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. Matrices of this type are produced by the Schur factorization of a complex general matrix: A = Q*T*Q**H, as computed by CHSEQR.
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 eigenvalues 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 eigenvectors 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 eigenvectors of A.

ARGUMENTS

SIDE (input) CHARACTER*1
= aqRaq: compute right eigenvectors only;
= aqLaq: compute left eigenvectors only;
= aqBaq: compute both right and left eigenvectors.
HOWMNY (input) CHARACTER*1

= aqAaq: compute all right and/or left eigenvectors;
= aqBaq: compute all right and/or left eigenvectors, backtransformed using the matrices supplied in VR and/or VL; = aqSaq: compute selected right and/or left eigenvectors, as indicated by the logical array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = aqSaq, SELECT specifies the eigenvectors to be computed. The eigenvector corresponding to the j-th eigenvalue is computed if SELECT(j) = .TRUE.. Not referenced if HOWMNY = aqAaq or aqBaq.
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 = aqLaq or aqBaq and HOWMNY = aqBaq, VL must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by CHSEQR). On exit, if SIDE = aqLaq or aqBaq, VL contains: if HOWMNY = aqAaq, the matrix Y of left eigenvectors of T; if HOWMNY = aqBaq, the matrix Q*Y; if HOWMNY = aqSaq, 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 = aqRaq.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1, and if SIDE = aqLaq or aqBaq, LDVL >= N.
VR (input/output) COMPLEX array, dimension (LDVR,MM)
On entry, if SIDE = aqRaq or aqBaq and HOWMNY = aqBaq, VR must contain an N-by-N matrix Q (usually the unitary matrix Q of Schur vectors returned by CHSEQR). On exit, if SIDE = aqRaq or aqBaq, VR contains: if HOWMNY = aqAaq, the matrix X of right eigenvectors of T; if HOWMNY = aqBaq, the matrix Q*X; if HOWMNY = aqSaq, the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. Not referenced if SIDE = aqLaq.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1, and if SIDE = aqRaq or aqBaq; 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 = aqAaq or aqBaq, 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) 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|.