DTGEVC (3lapack)

compute some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B)

SYNOPSIS

  SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR,
                     LDVR, MM, M, WORK, INFO )

      CHARACTER      HOWMNY, SIDE

      INTEGER        INFO, LDA, LDB, LDVL, LDVR, M, MM, N

      LOGICAL        SELECT( * )

      DOUBLE         PRECISION A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR(
                     LDVR, * ), WORK( * )

PURPOSE

  DTGEVC computes some or all of the right and/or left generalized
  eigenvectors of a pair of real upper triangular matrices (A,B).

  The right generalized eigenvector x and the left generalized eigenvector y
  of (A,B) corresponding to a generalized eigenvalue w are defined by:

          (A - wB) * x = 0  and  y**H * (A - wB) = 0

  where y**H denotes the conjugate tranpose of y.

  If an eigenvalue w is determined by zero diagonal elements of both A and B,
  a unit vector is returned as the corresponding eigenvector.

  If all eigenvectors are requested, the routine may either return the
  matrices X and/or Y of right or left eigenvectors of (A,B), or the products
  Z*X and/or Q*Y, where Z and Q are input orthogonal matrices.  If (A,B) was
  obtained from the generalized real-Schur factorization of an original pair
  of matrices
     (A0,B0) = (Q*A*Z**H,Q*B*Z**H),
  then Z*X and Q*Y are the matrices of right or left eigenvectors of A.

  A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks.
  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) 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 the 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..

  N       (input) INTEGER
          The order of the matrices A and B.  N >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
          The upper quasi-triangular matrix A.

  LDA     (input) INTEGER
          The leading dimension of array A.  LDA >= max(1, N).

  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
          The upper triangular matrix B.  If A has a 2-by-2 diagonal block,
          then the corresponding 2-by-2 block of B must be diagonal with
          positive elements.

  LDB     (input) INTEGER
          The leading dimension of array B.  LDB >= 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 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
          (A,B); if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left
          eigenvectors of (A,B) specified by SELECT, stored consecutively in
          the columns of VL, in the same order as their eigenvalues.  If SIDE
          = 'R', VL is not referenced.

          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.

  LDVL    (input) INTEGER
          The leading dimension of array VL.  LDVL >= max(1,N) if SIDE = 'L'
          or 'B'; LDVL >= 1 otherwise.

  VR      (input/output) COMPLEX*16 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 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
          (A,B); if HOWMNY = 'B', the matrix Z*X; if HOWMNY = 'S', the right
          eigenvectors of (A,B) specified by SELECT, stored consecutively in
          the columns of VR, in the same order as their eigenvalues.  If SIDE
          = 'L', VR is not referenced.

          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.

  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 (6*N)

  INFO    (output) 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.

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 LDA (and
  LDB) 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.