strevc(3F)

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

SYNOPSIS
     SUBROUTINE STREVC( 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( * )

         REAL           T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE
     STREVC 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 SHSEQR), 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.



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/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) REAL 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) REAL 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 SHSEQR).  On exit, if SIDE = 'L' or 'B', VL
             contains:  if HOWMNY = 'A', the matrix Y of left eigenvectors of
             T; 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) REAL 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 SHSEQR).  On exit, if SIDE = 'R' or 'B', VR
             contains:  if HOWMNY = 'A', the matrix X of right eigenvectors of
             T; 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) REAL array, dimension (3*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|.