MB03DD

Exchanging eigenvalues of a real 2-by-2, 3-by-3 or 4-by-4 block upper triangular pencil

Purpose

  To compute orthogonal matrices Q1 and Q2 for a real 2-by-2,
  3-by-3, or 4-by-4 regular block upper triangular pencil

                 ( A11 A12 )     ( B11 B12 )
    aA - bB =  a (         ) - b (         ),                    (1)
                 (  0  A22 )     (  0  B22 )

  such that the pencil a(Q2' A Q1) - b(Q2' B Q1) is still in block
  upper triangular form, but the eigenvalues in Spec(A11, B11),
  Spec(A22, B22) are exchanged, where Spec(X,Y) denotes the spectrum
  of the matrix pencil (X,Y) and the notation M' denotes the
  transpose of the matrix M.

  Optionally, to upper triangularize the real regular pencil in
  block lower triangular form

                ( A11  0  )     ( B11  0  )
    aA - bB = a (         ) - b (         ),                     (2)
                ( A21 A22 )     ( B21 B22 )

  while keeping the eigenvalues in the same diagonal position.

      SUBROUTINE MB03DD( UPLO, N1, N2, PREC, A, LDA, B, LDB, Q1, LDQ1,
     $                   Q2, LDQ2, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, LDQ1, LDQ2, LDWORK, N1, N2
      DOUBLE PRECISION   PREC
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ),
     $                   Q1( LDQ1, * ), Q2( LDQ2, * )

Mode Parameters

  UPLO    CHARACTER*1
          Specifies if the pencil is in lower or upper block
          triangular form on entry, as follows:
          = 'U': Upper block triangular, eigenvalues are exchanged
                 on exit;
          = 'T': Upper block triangular, B triangular, eigenvalues
                 are exchanged on exit;
          = 'L': Lower block triangular, eigenvalues are not
                 exchanged on exit.

  N1      (input/output) INTEGER
          Size of the upper left block, N1 <= 2.
          If UPLO = 'U' or UPLO = 'T' and INFO = 0, or UPLO = 'L'
          and INFO <> 0, N1 and N2 are exchanged on exit; otherwise,
          N1 is unchanged on exit.

  N2      (input/output) INTEGER
          Size of the lower right block, N2 <= 2.
          If UPLO = 'U' or UPLO = 'T' and INFO = 0, or UPLO = 'L'
          and INFO <> 0, N1 and N2 are exchanged on exit; otherwise,
          N2 is unchanged on exit.

  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, N1+N2)
          On entry, the leading (N1+N2)-by-(N1+N2) part of this
          array must contain the matrix A of the pencil aA - bB.
          On exit, if N1 = N2 = 1, this array is unchanged, if
          UPLO = 'U' or UPLO = 'T', but, if UPLO = 'L', it contains
                                       [  0 1 ]
          the matrix J' A J, where J = [ -1 0 ]; otherwise, this
          array contains the transformed quasi-triangular matrix in
          generalized real Schur form.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= N1+N2.

  B       (input/output) DOUBLE PRECISION array, dimension
             (LDB, N1+N2)
          On entry, the leading (N1+N2)-by-(N1+N2) part of this
          array must contain the matrix B of the pencil aA - bB.
          On exit, if N1 = N2 = 1, this array is unchanged, if
          UPLO = 'U' or UPLO = 'T', but, if UPLO = 'L', it contains
          the matrix J' B J; otherwise, this array contains the
          transformed upper triangular matrix in generalized real
          Schur form.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= N1+N2.

  Q1      (output) DOUBLE PRECISION array, dimension (LDQ1, N1+N2)
          The leading (N1+N2)-by-(N1+N2) part of this array contains
          the first orthogonal transformation matrix.

  LDQ1    INTEGER
          The leading dimension of the array Q1.  LDQ1 >= N1+N2.

  Q2      (output) DOUBLE PRECISION array, dimension (LDQ2, N1+N2)
          The leading (N1+N2)-by-(N1+N2) part of this array contains
          the second orthogonal transformation matrix.

  LDQ2    INTEGER
          The leading dimension of the array Q2.  LDQ2 >= N1+N2.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          If N1+N2 = 2 then DWORK is not referenced.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          If N1+N2 = 2, then LDWORK = 0; otherwise,
          LDWORK >= 16*N1 + 10*N2 + 23, if UPLO = 'U';
          LDWORK >=  7*N1 +  7*N2 + 16, if UPLO = 'T';
          LDWORK >= 10*N1 + 16*N2 + 23, if UPLO = 'L'.
          For good performance LDWORK should be generally larger.

  INFO    INTEGER
          = 0: succesful exit;
          = 3: the QZ iteration failed in the LAPACK routine DGGES
               (if UPLO <> 'T') or DHGEQZ (if UPLO = 'T');
          = 4: another error occured during execution of DHGEQZ;
          = 5: reordering of aA - bB in the LAPACK routine DTGSEN
               failed because the transformed matrix pencil aA - bB
               would be too far from generalized Schur form;
               the problem is very ill-conditioned.

  The algorithm uses orthogonal transformations as described in [2]
  (page 30). The QZ algorithm is used for N1 = 2 or N2 = 2, but it
  always acts on an upper block triangular pencil.

  [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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