MB03CD

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

Purpose

  To compute orthogonal matrices Q1, Q2, Q3 for a real 2-by-2,
  3-by-3, or 4-by-4 regular block upper triangular pencil

                 ( A11 A12 ) ( B11 B12 )     ( D11 D12 )
    aAB - bD = a (         ) (         ) - b (         ),        (1)
                 (  0  A22 ) (  0  B22 )     (  0  D22 )

  such that the pencil a(Q3' A Q2 )(Q2' B Q1 ) - b(Q3' D Q1) is
  still in block upper triangular form, but the eigenvalues in
  Spec(A11 B11, D11), Spec(A22 B22, D22) are exchanged, where
  Spec(X,Y) denotes the spectrum of the matrix pencil (X,Y), and 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  )     ( D11  0  )
  aAB - bD = a (         ) (         ) - b (         ),          (2)
               ( A21 A22 ) ( B21 B22 )     ( D21 D22 )

  while keeping the eigenvalues in the same diagonal position.

      SUBROUTINE MB03CD( UPLO, N1, N2, PREC, A, LDA, B, LDB, D, LDD, Q1,
     $                   LDQ1, Q2, LDQ2, Q3, LDQ3, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, LDD, LDQ1, LDQ2, LDQ3, LDWORK,
     $                   N1, N2
      DOUBLE PRECISION   PREC
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
     $                   DWORK( * ), Q1( LDQ1, * ), Q2( LDQ2, * ),
     $                   Q3( LDQ3, * )

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;
          = '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' 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' 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 or 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 aAB - bD.
          The (2,1) block, if UPLO = 'U', or the (1,2) block, if
          UPLO = 'L', need not be set to zero.
          On exit, if N1 = N2 = 1, this array contains the matrix
                            [  0 1 ]
          J' A J, where J = [ -1 0 ]; otherwise, this array is
          unchanged on exit.

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

  B       (input or 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 aAB - bD.
          The (2,1) block, if UPLO = 'U', or the (1,2) block, if
          UPLO = 'L', need not be set to zero.
          On exit, if N1 = N2 = 1, this array contains the matrix
          J' B J; otherwise, this array is unchanged on exit.

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

  D       (input/output) DOUBLE PRECISION array, dimension
             (LDD, N1+N2)
          On entry, the leading (N1+N2)-by-(N1+N2) part of this
          array must contain the matrix D of the pencil aAB - bD.
          On exit, if N1 = 2 or N2 = 2, the leading
          (N1+N2)-by-(N1+N2) part of this array contains the
          transformed matrix D in real Schur form. If N1 = 1 and
          N2 = 1, this array contains the matrix J' D J.

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

  Q3      (output) DOUBLE PRECISION array, dimension (LDQ3, N1+N2)
          The leading (N1+N2)-by-(N1+N2) part of this array contains
          the third orthogonal transformation matrix.

  LDQ3    INTEGER
          The leading dimension of the array Q3.  LDQ3 >= 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, UPLO = 'U';
          LDWORK >= 10*N1 + 16*N2 + 23, UPLO = 'L'.

  INFO    INTEGER
          = 0: succesful exit;
          = 1: the QZ iteration failed in the LAPACK routine DGGEV;
          = 2: another error occured while executing a routine in
               DGGEV;
          = 3: the QZ iteration failed in the LAPACK routine DGGES;
          = 4: another error occured during execution of DGGES;
          = 5: reordering of aA*B - bD in the LAPACK routine DTGSEN
               failed because the transformed matrix pencil
               aA*B - bD would be too far from generalized Schur
               form; the problem is very ill-conditioned.

  The algorithm uses orthogonal transformations as described in [2]
  (page 21). 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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