## 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

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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.

Specification

          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( * )

Arguments

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.

Input/Output Parameters

      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).

Workspace

      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.

Error Indicator

      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.

Method

      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.

References

      [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.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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