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

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.

      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, * )

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.

  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.

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

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

  [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.

  The algorithm is numerically backward stable.

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Program Text

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