NAME

CTGEVC - compute some or all of the right and/or left generalized eigenvectors of a pair of complex upper triangular matrices (A,B)

SYNOPSIS

SUBROUTINE CTGEVC(
SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, INFO )

CHARACTER HOWMNY, SIDE

INTEGER INFO, LDA, LDB, LDVL, LDVR, M, MM, N

LOGICAL SELECT( ∗ )

REAL RWORK( ∗ )

COMPLEX A( LDA, ∗ ), B( LDB, ∗ ), VL( LDVL, ∗ ), VR( LDVR, ∗ ), WORK( ∗ )

PURPOSE

CTGEVC computes some or all of the right and/or left generalized eigenvectors of a pair of complex upper triangular matrices (A,B). 
 
The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by:
 
        (A - wB) ∗ x = 0  and  y∗∗H ∗ (A - wB) = 0
 
where y∗∗H denotes the conjugate tranpose of y.
 
If an eigenvalue w is determined by zero diagonal elements of both A and B, a unit vector is returned as the corresponding eigenvector.
 
If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of (A,B), or the products Z∗X and/or Q∗Y, where Z and Q are input unitary matrices.  If (A,B) was obtained from the generalized Schur factorization of an original pair of matrices
   (A0,B0) = (Q∗A∗Z∗∗H,Q∗B∗Z∗∗H),
then Z∗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 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) 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 eigenvector corresponding to the j-th eigenvalue, SELECT(j) must be set to .TRUE..

N       (input) INTEGER
The order of the matrices A and B.  N >= 0.

A       (input) COMPLEX array, dimension (LDA,N)
The upper triangular matrix A.

LDA     (input) INTEGER
The leading dimension of array A.  LDA >= max(1,N).

B       (input) COMPLEX array, dimension (LDB,N)
The upper triangular matrix B.  B must have real diagonal elements.

LDB     (input) INTEGER
The leading dimension of array B.  LDB >= max(1,N).

VL      (input/output) COMPLEX 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 left Schur vectors returned by CHGEQZ). On exit, if SIDE = ’L’ or ’B’, VL contains: if HOWMNY = ’A’, the matrix Y of left eigenvectors of (A,B); if HOWMNY = ’B’, the matrix Q∗Y; if HOWMNY = ’S’, the left eigenvectors of (A,B) specified 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 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 contain an N-by-N matrix Q (usually the unitary matrix Z of right Schur vectors returned by CHGEQZ). On exit, if SIDE = ’R’ or ’B’, VR contains: if HOWMNY = ’A’, the matrix X of right eigenvectors of (A,B); if HOWMNY = ’B’, the matrix Z∗X; if HOWMNY = ’S’, the right eigenvectors of (A,B) specified 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 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 (2∗N)

INFO    (output) INTEGER
= 0:  successful exit.
< 0:  if INFO = -i, the i-th argument had an illegal value.

  —  LAPACK version 2.0  —  08 October 1994