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)

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.

      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, * )

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.

  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.

  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
  20 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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