MB03HD

Exchanging eigenvalues of a real 2-by-2 or 4-by-4 skew-Hamiltonian/Hamiltonian pencil in structured Schur form

Purpose

  To determine an orthogonal matrix Q, for a real regular 2-by-2 or
  4-by-4 skew-Hamiltonian/Hamiltonian pencil

                  ( A11 A12  )     ( B11  B12  )
      aA - bB = a (          ) - b (           )
                  (  0  A11' )     (  0  -B11' )

  in structured Schur form, such that  J Q' J' (aA - bB) Q  is still
  in structured Schur form but the eigenvalues are exchanged. The
  notation M' denotes the transpose of the matrix M.

      SUBROUTINE MB03HD( N, A, LDA, B, LDB, MACPAR, Q, LDQ, DWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDQ, N
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
     $                   MACPAR( * ), Q( LDQ, * )

Input/Output Parameters

  N       (input) INTEGER
          The order of the pencil aA - bB.  N = 2 or N = 4.

  A       (input) DOUBLE PRECISION array, dimension (LDA, N)
          If N = 4, the leading N/2-by-N upper trapezoidal part of
          this array must contain the first block row of the skew-
          Hamiltonian matrix A of the pencil aA - bB in structured
          Schur form. Only the entries (1,1), (1,2), (1,4), and
          (2,2) are referenced.
          If N = 2, this array is not referenced.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= N/2.

  B       (input) DOUBLE PRECISION array, dimension (LDB, N)
          The leading N/2-by-N part of this array must contain the
          first block row of the Hamiltonian matrix B of the
          pencil aA - bB in structured Schur form. The entry (2,3)
          is not referenced.

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

  DWORK   DOUBLE PRECISION array, dimension (24)
          If N = 2, then DWORK is not referenced.

  INFO    INTEGER
          = 0: succesful exit;
          = 1: the leading N/2-by-N/2 block of the matrix B is
               numerically singular, but slightly perturbed values
               have been used. This is a warning.

  The algorithm uses orthogonal transformations as described on page
  31 in [2]. The structure is exploited.

  [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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