dtgevc.f - Man Page

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

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:

December 2016

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 297 of file dtgevc.f.