## MB03GD

### Exchanging eigenvalues of a real 2-by-2 or 4-by-4 block upper triangular skew-Hamiltonian/Hamiltonian pencil (factored version)

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

Purpose

      To compute an orthogonal matrix Q and an orthogonal symplectic
      matrix U for a real regular 2-by-2 or 4-by-4 skew-Hamiltonian/
      Hamiltonian pencil a J B' J' B - b D with

            ( B11  B12 )      (  D11  D12  )      (  0  I  )
        B = (          ), D = (            ), J = (        ),
            (  0   B22 )      (   0  -D11' )      ( -I  0  )

      such that J Q' J' D Q and U' B Q keep block triangular form, but
      the eigenvalues are reordered. The notation M' denotes the
      transpose of the matrix M.

Specification

          SUBROUTINE MB03GD( N, B, LDB, D, LDD, MACPAR, Q, LDQ, U, LDU,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDB, LDD, LDQ, LDU, LDWORK, N
    C     .. Array Arguments ..
          DOUBLE PRECISION   B( LDB, * ), D( LDD, * ), DWORK( * ),
         $                   MACPAR( * ), Q( LDQ, * ), U( LDU, * )

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the pencil a J B' J' B - b D. N = 2 or N = 4.

      B       (input) DOUBLE PRECISION array, dimension (LDB, N)
              The leading N-by-N part of this array must contain the
              non-trivial factor of the decomposition of the
              skew-Hamiltonian input matrix J B' J' B. The (2,1) block
              is not referenced.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= N.

      D       (input) DOUBLE PRECISION array, dimension (LDD, N)
              The leading N/2-by-N part of this array must contain the
              first block row of the second matrix of a J B' J' B - b D.
              The matrix D has to be Hamiltonian. The strict lower
              triangle of the (1,2) block is not referenced.

      LDD     INTEGER
              The leading dimension of the array D.  LDD >= N/2.

      MACPAR  (input)  DOUBLE PRECISION array, dimension (2)
              Machine parameters:
              MACPAR(1)  (machine precision)*base, DLAMCH( 'P' );
              MACPAR(2)  safe minimum,             DLAMCH( 'S' ).
              This argument is not used for N = 2.

      Q       (output) DOUBLE PRECISION array, dimension (LDQ, N)
              The leading N-by-N part of this array contains the
              orthogonal transformation matrix Q.

      LDQ     INTEGER
              The leading dimension of the array Q.  LDQ >= N.

      U       (output) DOUBLE PRECISION array, dimension (LDU, N)
              The leading N-by-N part of this array contains the
              orthogonal symplectic transformation matrix U.

      LDU     INTEGER
              The leading dimension of the array U.  LDU >= N.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              If N = 2 then DWORK is not referenced.

      LDWORK  INTEGER
              The length of the array DWORK.
              If N = 2 then LDWORK >= 0; if N = 4 then LDWORK >= 12.

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              = 1: B11 or B22 is a (numerically) singular matrix.

Method

      The algorithm uses orthogonal transformations as described on page
      22 in [1], but with an improved implementation.

References

      [1] Benner, P., Byers, R., Losse, P., Mehrmann, V. and Xu, H.
          Numerical Solution of Real Skew-Hamiltonian/Hamiltonian
          Eigenproblems.
          Tech. Rep., Technical University Chemnitz, Germany,
          Nov. 2007.

Numerical Aspects

      The algorithm is numerically backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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