## MB03ED

### 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 (factored version)

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

Purpose

      To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2 or
      4-by-4 regular pencil

                     ( A11  0  ) ( B11  0  )     (  0  D12 )
        aAB - bD = a (         ) (         ) - b (         ),        (1)
                     (  0  A22 ) (  0  B22 )     ( D21  0  )

      such that Q3' A Q2 and Q2' B Q1 are upper triangular, Q3' D 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,
      B11, B22 and D12 are upper triangular. If D21 is 2-by-2, then all
      other blocks are nonsingular and the product
         -1        -1    -1        -1
      A22   D21 B11   A11   D12 B22   has a pair of complex conjugate
      eigenvalues.

Specification

          SUBROUTINE MB03ED( N, PREC, A, LDA, B, LDB, D, LDD, Q1, LDQ1, Q2,
         $                   LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
         $                   N
          DOUBLE PRECISION   PREC
    C     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
         $                   DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
         $                   Q3( LDQ3, * )

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) DOUBLE PRECISION array, dimension (LDA, N)
              The leading N-by-N upper triangular part of this array
              must contain the upper triangular matrix A of the pencil
              aAB - bD. The strictly lower triangular part and the
              entries of the (1,2) block are not referenced.

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

      B       (input) DOUBLE PRECISION array, dimension (LDB, N)
              The leading N-by-N upper triangular part of this array
              must contain the upper triangular matrix B of the pencil
              aAB - bD. The strictly lower triangular part and the
              entries of the (1,2) block are not referenced.

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

      D       (input/output) DOUBLE PRECISION array, dimension (LDD, N)
              On entry, the leading N-by-N part of this array must
              contain the matrix D of the pencil aAB - bD.
              On exit, if N = 4, the leading N-by-N part of this array
              contains the transformed matrix D in real Schur form.
              If N = 2, this array is unchanged on exit.

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

      Q3      (output) DOUBLE PRECISION array, dimension (LDQ3, N)
              The leading N-by-N part of this array contains the third
              orthogonal transformation matrix.

      LDQ3    INTEGER
              The leading dimension of the array Q3.  LDQ3 >= 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 >= 79. 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
      20 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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