DTREVC - compute some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T

SUBROUTINE DTREVC(

SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )

CHARACTER HOWMNY, SIDE

INTEGER INFO, LDT, LDVL, LDVR, M, MM, N

LOGICAL SELECT( * )

DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

DTREVC computes some or all of the right and/or left eigenvectors of a real upper quasi-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 orthogonal

matrix. If T was obtained from the real-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.

T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.

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/output) 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 real eigenvector corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select the complex eigenvector corresponding to a complex conjugate pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be set to .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE..

N (input) INTEGER

The order of the matrix T. N >= 0.

T (input) DOUBLE PRECISION array, dimension (LDT,N)

The upper quasi-triangular matrix T in Schur canonical form.

LDT (input) INTEGER

The leading dimension of the array T. LDT >= max(1,N).

VL (input/output) DOUBLE PRECISION 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 orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; VL has the same quasi-lower triangular form as T'. If T(i,i) is a real eigenvalue, then the i-th column VL(i) of VL is its corresponding eigenvector. If T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are complex-conjugate eigenvalues of T, then VL(i)+sqrt(-1)*VL(i+1) is the complex eigenvector corresponding to the eigenvalue with positive real part. 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. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. 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) DOUBLE PRECISION 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 orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; VR has the same quasi-upper triangular form as T. If T(i,i) is a real eigenvalue, then the i-th column VR(i) of VR is its corresponding eigenvector. If T(i:i+1,i:i+1) is a 2-by-2 block whose eigenvalues are complex-conjugate eigenvalues of T, then VR(i)+sqrt(-1)*VR(i+1) is the complex eigenvector corresponding to the eigenvalue with positive real part. 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. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. 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 real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

WORK (workspace) DOUBLE PRECISION array, dimension (3*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|.