MB05MY

Computation of the orthogonal matrix reducing a given matrix to real Schur form T, of the eigenvalues, and of the upper triangular matrix of right eigenvectors of T

Purpose

  To compute, for an N-by-N real nonsymmetric matrix A, the
  orthogonal matrix Q reducing it to real Schur form T, the
  eigenvalues, and the right eigenvectors of T.

  The right eigenvector r(j) of T satisfies
                   T * r(j) = lambda(j) * r(j)
  where lambda(j) is its eigenvalue.

  The matrix of right eigenvectors R is upper triangular, by
  construction.

      SUBROUTINE MB05MY( BALANC, N, A, LDA, WR, WI, R, LDR, Q, LDQ,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         BALANC
      INTEGER           INFO, LDA, LDQ, LDR, LDWORK, N
C     .. Array Arguments ..
      DOUBLE PRECISION  A( LDA, * ), DWORK( * ), Q( LDQ, * ),
     $                  R( LDR, * ), WI( * ), WR( * )

Mode Parameters

  BALANC  CHARACTER*1
          Indicates how the input matrix should be diagonally scaled
          to improve the conditioning of its eigenvalues as follows:
          = 'N':  Do not diagonally scale;
          = 'S':  Diagonally scale the matrix, i.e. replace A by
                  D*A*D**(-1), where D is a diagonal matrix chosen
                  to make the rows and columns of A more equal in
                  norm. Do not permute.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the given matrix A.
          On exit, the leading N-by-N upper quasi-triangular part of
          this array contains the real Schur canonical form of A.

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

  WR      (output) DOUBLE PRECISION array, dimension (N)
  WI      (output) DOUBLE PRECISION array, dimension (N)
          WR and WI contain the real and imaginary parts,
          respectively, of the computed eigenvalues. Complex
          conjugate pairs of eigenvalues appear consecutively
          with the eigenvalue having the positive imaginary part
          first.

  R       (output) DOUBLE PRECISION array, dimension (LDR,N)
          The leading N-by-N upper triangular part of this array
          contains the matrix of right eigenvectors R, in the same
          order as their eigenvalues. The real and imaginary parts
          of a complex eigenvector corresponding to an eigenvalue
          with positive imaginary part are stored in consecutive
          columns. (The corresponding conjugate eigenvector is not
          stored.) The eigenvectors are not backward transformed
          for balancing (when BALANC = 'S').

  LDR     INTEGER
          The leading dimension of array R.  LDR >= max(1,N).

  Q       (output) DOUBLE PRECISION array, dimension (LDQ,N)
          The leading N-by-N part of this array contains the
          orthogonal matrix Q which has reduced A to real Schur
          form.

  LDQ     INTEGER
          The leading dimension of array Q.  LDQ >= MAX(1,N).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
          If BALANC = 'S' and LDWORK > 0, DWORK(2),...,DWORK(N+1)
          return the scaling factors used for balancing.

  LDWORK  INTEGER
          The length of the array DWORK.  LDWORK >= max(1,4*N).
          For good performance, LDWORK must generally be larger.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          > 0:  if INFO = i, the QR algorithm failed to compute all
                the eigenvalues, and no eigenvectors have been
                computed; elements i+1:N of WR and WI contain
                eigenvalues which have converged.

  This routine uses the QR algorithm to obtain the real Schur form
  T of matrix A. Then, the right eigenvectors of T are computed,
  but they are not backtransformed into the eigenvectors of A.
  MB05MY is a modification of the LAPACK driver routine DGEEV.

  [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
      Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
      Ostrouchov, S., and Sorensen, D.
      LAPACK Users' Guide: Second Edition.
      SIAM, Philadelphia, 1995.

                            3
  The algorithm requires 0(N ) operations.

  None

Program Text

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