dtrevc(3)

NAME

DTREVC - compute some or all of the right and/or left

eigenvectors of a real upper quasi-triangular matrix T

SYNOPSIS

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( * )

PURPOSE

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 factoriza

tion 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. Corre

sponding 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.

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 eigenvec

tors;
= 'B': compute all right and/or left eigenvec

tors, and backtransform them using the input matrices supplied in

VR and/or VL; = 'S': compute selected right and/or left eigen

vectors, 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 refer

enced. 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 eigen

vectors 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 corre

sponding 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 consecu

tive 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 cor

responding 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 consecu

tive 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 col

umn 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 ille

gal 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 com

plex number (x,y) is taken to be |x| + |y|.

LAPACK version 3.0 15 June 2000

DTREVC(3)