dtgevc.f(3)

NAME
dtgevc.f -

SYNOPSIS

  Functions/Subroutines

subroutine dtgevc (SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO)
DTGEVC

Function/Subroutine Documentation

  subroutine dtgevc (characterSIDE, characterHOWMNY, logical, dimension( * )SELECT, integerN, double precision, dimension( lds, * )S, integerLDS, double precision, dimension( ldp, * )P, integerLDP, double precision, dimension( ldvl, * )VL, integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, integerMM, integerM, double precision, dimension( * )WORK, integerINFO)

DTGEVC

Purpose:

     DTGEVC computes some or all of the right and/or left eigenvectors of
     a pair of real matrices (S,P), where S is a quasi-triangular matrix
     and P is upper triangular.  Matrix pairs of this type are produced by
     the generalized Schur factorization of a matrix pair (A,B):

        A = Q*S*Z**T,  B = Q*P*Z**T

     as computed by DGGHRD + DHGEQZ.

     The right eigenvector x and the left eigenvector y of (S,P)
     corresponding to an eigenvalue w are defined by:

        S*x = w*P*x,  (y**H)*S = w*(y**H)*P,

     where y**H denotes the conjugate tranpose of y.
     The eigenvalues are not input to this routine, but are computed
     directly from the diagonal blocks of S and P.

     This routine returns the matrices X and/or Y of right and left
     eigenvectors of (S,P), or the products Z*X and/or Q*Y,
     where Z and Q are input matrices.
     If Q and Z are the orthogonal factors from the generalized Schur
     factorization of a matrix pair (A,B), then Z*X and Q*Y
     are the matrices of right and left eigenvectors of (A,B).

Parameters:
    SIDE

              SIDE is CHARACTER*1
              = 'R': compute right eigenvectors only;
              = 'L': compute left eigenvectors only;
              = 'B': compute both right and left eigenvectors.

    HOWMNY

              HOWMNY is CHARACTER*1
              = 'A': compute all right and/or left eigenvectors;
              = 'B': compute all right and/or left eigenvectors,
                     backtransformed by the matrices in VR and/or VL;
              = 'S': compute selected right and/or left eigenvectors,
                     specified by the logical array SELECT.

    SELECT

              SELECT is LOGICAL array, dimension (N)
              If HOWMNY='S', SELECT specifies the eigenvectors to be
              computed.  If w(j) is a real eigenvalue, the corresponding
              real eigenvector is computed if SELECT(j) is .TRUE..
              If w(j) and w(j+1) are the real and imaginary parts of a
              complex eigenvalue, the corresponding complex eigenvector
              is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
              and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
              set to .FALSE..
              Not referenced if HOWMNY = 'A' or 'B'.

    N

              N is INTEGER
              The order of the matrices S and P.  N >= 0.

    S

              S is DOUBLE PRECISION array, dimension (LDS,N)
              The upper quasi-triangular matrix S from a generalized Schur
              factorization, as computed by DHGEQZ.

    LDS

              LDS is INTEGER
              The leading dimension of array S.  LDS >= max(1,N).

    P

              P is DOUBLE PRECISION array, dimension (LDP,N)
              The upper triangular matrix P from a generalized Schur
              factorization, as computed by DHGEQZ.
              2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
              of S must be in positive diagonal form.

    LDP

              LDP is INTEGER
              The leading dimension of array P.  LDP >= max(1,N).

    VL

              VL is 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 left Schur vectors returned by DHGEQZ).
              On exit, if SIDE = 'L' or 'B', VL contains:
              if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
              if HOWMNY = 'B', the matrix Q*Y;
              if HOWMNY = 'S', the left eigenvectors of (S,P) 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.

              Not referenced if SIDE = 'R'.

    LDVL

              LDVL is INTEGER
              The leading dimension of array VL.  LDVL >= 1, and if
              SIDE = 'L' or 'B', LDVL >= N.

    VR

              VR is DOUBLE PRECISION array, dimension (LDVR,MM)
              On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
              contain an N-by-N matrix Z (usually the orthogonal matrix Z
              of right Schur vectors returned by DHGEQZ).

              On exit, if SIDE = 'R' or 'B', VR contains:
              if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
              if HOWMNY = 'B' or 'b', the matrix Z*X;
              if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
                          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.

              Not referenced if SIDE = 'L'.

    LDVR

              LDVR is INTEGER
              The leading dimension of the array VR.  LDVR >= 1, and if
              SIDE = 'R' or 'B', LDVR >= N.

    MM

              MM is INTEGER
              The number of columns in the arrays VL and/or VR. MM >= M.

    M

              M is 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

              WORK is DOUBLE PRECISION array, dimension (6*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
                    eigenvalue.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

Further Details:

      Allocation of workspace:
      ---------- -- ---------

         WORK( j ) = 1-norm of j-th column of A, above the diagonal
         WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
         WORK( 2*N+1:3*N ) = real part of eigenvector
         WORK( 3*N+1:4*N ) = imaginary part of eigenvector
         WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
         WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector

      Rowwise vs. columnwise solution methods:
      ------- --  ---------- -------- -------

      Finding a generalized eigenvector consists basically of solving the
      singular triangular system

       (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)

      Consider finding the i-th right eigenvector (assume all eigenvalues
      are real). The equation to be solved is:
           n                   i
      0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
          k=j                 k=j

      where  C = (A - w B)  (The components v(i+1:n) are 0.)

      The "rowwise" method is:

      (1)  v(i) := 1
      for j = i-1,. . .,1:
                              i
          (2) compute  s = - sum C(j,k) v(k)   and
                            k=j+1

          (3) v(j) := s / C(j,j)

      Step 2 is sometimes called the "dot product" step, since it is an
      inner product between the j-th row and the portion of the eigenvector
      that has been computed so far.

      The "columnwise" method consists basically in doing the sums
      for all the rows in parallel.  As each v(j) is computed, the
      contribution of v(j) times the j-th column of C is added to the
      partial sums.  Since FORTRAN arrays are stored columnwise, this has
      the advantage that at each step, the elements of C that are accessed
      are adjacent to one another, whereas with the rowwise method, the
      elements accessed at a step are spaced LDS (and LDP) words apart.

      When finding left eigenvectors, the matrix in question is the
      transpose of the one in storage, so the rowwise method then
      actually accesses columns of A and B at each step, and so is the
      preferred method.

Definition at line 295 of file dtgevc.f.

Author

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