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dtrevc


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

 SYNOPSIS
      SUBROUTINE DTREVC( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL,
                         VR, LDVR, MM, M, WORK, INFO )

          CHARACTER      HOWMNY, JOB

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

          LOGICAL        SELECT( * )

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

 PURPOSE
      DTREVC computes all or some 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**H*T = w*y**H.

      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**T, then Q*X and/or 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 diag-
      onal blocks; each 2-by-2 diagonal block has its diagonal
      elements equal and its off-diagonal elements of opposite
      sign.

 ARGUMENTS
      JOB     (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;
              = 'O':  compute all right and/or left eigenvectors,
              multiplied on the left by an input (generally
              orthogonal) matrix; = 'S':  compute some 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. To select the real eigenvector
              corresponding to a real eigenvalue w(j), SELECT(j)
              must be set to .TRUE.. To select the complex eigen-
              vector 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..  If HOWMNY = 'A'
              or 'O', SELECT is not referenced.

      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 canoni-
              cal form.

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

 (LDVL,MM)
      VL      (input/output) DOUBLE PRECISION array, dimension
              On entry, if JOB = 'L' or 'B' and HOWMNY = 'O', VL
              must contain an N-by-N matrix Q (usually the orthog-
              onal matrix Q of Schur vectors returned by DHSEQR).
              On exit, if JOB = 'L' or 'B', VL contains: if HOWMNY
              = 'A', the matrix Y of left eigenvectors of T; if
              HOWMNY = 'O', 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 JOB =
              'R', VL is not referenced.

      LDVL    (input) INTEGER
              The leading dimension of the array VL.  LDVL >=
              max(1,N).

 (LDVR,MM)
      VR      (input/output) DOUBLE PRECISION array, dimension
              On entry, if JOB = 'R' or 'B' and HOWMNY = 'O', VR
              must contain an N-by-N matrix Q (usually the orthog-
              onal matrix Q of Schur vectors returned by DHSEQR).
              On exit, if JOB = 'R' or 'B', VR contains: if HOWMNY
              = 'A', the matrix X of right eigenvectors of T; if
              HOWMNY = 'O', 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 JOB =
              'L', VR is not referenced.

      LDVR    (input) INTEGER
              The leading dimension of the array VR.  LDVR >=
              max(1,N).

      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
              required to store the eigenvectors; each selected
              real eigenvector occupies one column and each
              selected complex eigenvector occupies two columns.
              If HOWMNY = 'A' or 'O', M is set to N.

      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

 FURTHER DETAILS
      The algorithm used in this program is basically backward
      (forward) substitution, with scaling to make the code robust
      against possible overflow.

      Each eigenvector is normalized so that the element of larg-
      est magnitude has magnitude 1; here the magnitude of a com-
      plex number (x,y) is taken to be |x| + |y|.