## MB03FD

### Reducing a real 2-by-2 or 4-by-4 block (anti-)diagonal skew-Hamiltonian/Hamiltonian pencil to generalized Schur form and moving eigenvalues with negative real parts to the top

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

Purpose

      To compute orthogonal matrices Q1 and Q2 for a real 2-by-2 or
      4-by-4 regular pencil

                    ( A11  0  )     (  0  B12 )
        aA - bB = a (         ) - b (         ),                     (1)
                    (  0  A22 )     ( B21  0  )

      such that Q2' A Q1 is upper triangular, Q2' B Q1 is upper quasi-
      triangular, and the eigenvalues with negative real parts (if there
      are any) are allocated on the top. The notation M' denotes the
      transpose of the matrix M. The submatrices A11, A22, and B12 are
      upper triangular. If B21 is 2-by-2, then all the other blocks are
                                     -1        -1
      nonsingular and the product A11   B12 A22   B21 has a pair of
      complex conjugate eigenvalues.

Specification

          SUBROUTINE MB03FD( N, PREC, A, LDA, B, LDB, Q1, LDQ1, Q2, LDQ2,
         $                   DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, N
          DOUBLE PRECISION   PREC
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
         $                   Q1( LDQ1, * ), Q2( LDQ2, * )

Arguments

Input/Output Parameters

      N       (input) INTEGER
              The order of the input pencil, N = 2 or N = 4.

      PREC    (input) DOUBLE PRECISION
              The machine precision, (relative machine precision)*base.
              See the LAPACK Library routine DLAMCH.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A of the pencil aA - bB.
              If N = 2, the diagonal elements only are referenced.
              On exit, if N = 4, the leading N-by-N part of this array
              contains the transformed upper triangular matrix of the
              generalized real Schur form of the pencil aA - bB.
              If N = 2, this array is unchanged on exit.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= N.

      B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
              On entry, the leading N-by-N part of this array must
              contain the matrix B of the pencil aA - bB.
              If N = 2, the anti-diagonal elements only are referenced.
              On exit, if N = 4, the leading N-by-N part of this array
              contains the transformed real Schur matrix of the
              generalized real Schur form of the pencil aA - bB.
              If N = 2, this array is unchanged on exit.

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

      Q1      (output) DOUBLE PRECISION array, dimension (LDQ1, N)
              The leading N-by-N part of this array contains the first
              orthogonal transformation matrix.

      LDQ1    INTEGER
              The leading dimension of the array Q1.  LDQ1 >= N.

      Q2      (output) DOUBLE PRECISION array, dimension (LDQ2, N)
              The leading N-by-N part of this array contains the second
              orthogonal transformation matrix.

      LDQ2    INTEGER
              The leading dimension of the array Q2.  LDQ2 >= N.

Workspace

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

      LDWORK  INTEGER
              The dimension of the array DWORK.
              If N = 4, then LDWORK >= 63. For good performance LDWORK
              should be generally larger.
              If N = 2, then LDWORK >= 0.

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              = 1: the QZ iteration failed in the LAPACK routine DGGES;
              = 2: another error occured during execution of DGGES.

Method

      The algorithm uses orthogonal transformations as described on page
      29 in [2].

References

      [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
          Numerical computation of deflating subspaces of skew-
          Hamiltonian/Hamiltonian pencils.
          SIAM J. Matrix Anal. Appl., 24 (1), pp. 165-190, 2002.

      [2] 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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