CTREVC - compute some or all of the right and/or left eigenvectors of a complex upper triangular matrix T

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( * )

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.

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 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 contain 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 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 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 contain 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 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 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

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|.