MB03JD

Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil

Purpose

  To move the eigenvalues with strictly negative real parts of an
  N-by-N real skew-Hamiltonian/Hamiltonian pencil aS - bH in
  structured Schur form,

        (  A  D  )      (  B  F  )
    S = (        ), H = (        ),
        (  0  A' )      (  0 -B' )

  with A upper triangular and B upper quasi-triangular, to the
  leading principal subpencil, while keeping the triangular form.
  The notation M' denotes the transpose of the matrix M.
  The matrices S and H are transformed by an orthogonal matrix Q
  such that

                         (  Aout  Dout  )  
    Sout = J Q' J' S Q = (              ),
                         (    0   Aout' )  
                                                                 (1)
                         (  Bout  Fout  )           (  0  I  )
    Hout = J Q' J' H Q = (              ), with J = (        ),
                         (  0    -Bout' )           ( -I  0  )

  where Aout is upper triangular and Bout is upper quasi-triangular.
  Optionally, if COMPQ = 'I' or COMPQ = 'U', the orthogonal matrix Q
  that fulfills (1), is computed.

      SUBROUTINE MB03JD( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
     $                   LDQ, NEIG, IWORK, LIWORK, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ
      INTEGER            INFO, LDA, LDB, LDD, LDF, LDQ, LDWORK, LIWORK,
     $                   N, NEIG
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), D( LDD, * ),
     $                   DWORK( * ),  F( LDF, * ), Q( LDQ, * )

Mode Parameters

  COMPQ   CHARACTER*1
          Specifies whether or not the orthogonal transformations
          should be accumulated in the array Q, as follows:
          = 'N':  Q is not computed;
          = 'I':  the array Q is initialized internally to the unit
                  matrix, and the orthogonal matrix Q is returned;
          = 'U':  the array Q contains an orthogonal matrix Q0 on
                  entry, and the matrix Q0*Q is returned, where Q
                  is the product of the orthogonal transformations
                  that are applied to the pencil aS - bH to reorder
                  the eigenvalues.

  N       (input) INTEGER
          The order of the pencil aS - bH.  N >= 0, even.

  A       (input/output) DOUBLE PRECISION array, dimension
                         (LDA, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the upper triangular matrix A. The elements of the
          strictly lower triangular part of this array are not used.
          On exit, the leading  N/2-by-N/2 part of this array
          contains the transformed matrix Aout.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= MAX(1, N/2).

  D       (input/output) DOUBLE PRECISION array, dimension
                        (LDD, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the upper triangular part of the skew-symmetric
          matrix D. The diagonal need not be set to zero.
          On exit, the leading  N/2-by-N/2 part of this array
          contains the transformed upper triangular part of the
          matrix Dout.
          The strictly lower triangular part of this array is
          not referenced, except for the element D(N/2,N/2-1), but
          its initial value is preserved.

  LDD     INTEGER
          The leading dimension of the array D.  LDD >= MAX(1, N/2).

  B       (input/output) DOUBLE PRECISION array, dimension
                         (LDB, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the upper quasi-triangular matrix B.
          On exit, the leading  N/2-by-N/2 part of this array
          contains the transformed upper quasi-triangular part of
          the matrix Bout.
          The part below the first subdiagonal of this array is
          not referenced.

  LDB     INTEGER
          The leading dimension of the array B.  LDB >= MAX(1, N/2).

  F       (input/output) DOUBLE PRECISION array, dimension
                        (LDF, N/2)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the upper triangular part of the symmetric matrix
          F.
          On exit, the leading  N/2-by-N/2 part of this array
          contains the transformed upper triangular part of the
          matrix Fout.
          The strictly lower triangular part of this array is not
          referenced, except for the element F(N/2,N/2-1), but its
          initial value is preserved.

  LDF     INTEGER
          The leading dimension of the array F.  LDF >= MAX(1, N/2).

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
          On entry, if COMPQ = 'U', then the leading N-by-N part of
          this array must contain a given matrix Q0, and on exit,
          the leading N-by-N part of this array contains the product
          of the input matrix Q0 and the transformation matrix Q
          used to transform the matrices S and H.
          On exit, if COMPQ = 'I', then the leading N-by-N part of
          this array contains the orthogonal transformation matrix
          Q.
          If COMPQ = 'N' this array is not referenced.

  LDQ     INTEGER
          The leading dimension of the array Q.
          LDQ >= 1,         if COMPQ = 'N';
          LDQ >= MAX(1, N), if COMPQ = 'I' or COMPQ = 'U'.

  NEIG    (output) INTEGER
          The number of eigenvalues in aS - bH with strictly
          negative real part.

  IWORK   INTEGER array, dimension (LIWORK)

  LIWORK  INTEGER
          The dimension of the array IWORK.
          LIWORK >= N+1.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

  LDWORK  INTEGER
          The dimension of the array DWORK.
          If COMPQ = 'N',
             LDWORK >= MAX(2*N+32,108);
          if COMPQ = 'I' or COMPQ = 'U',
             LDWORK >= MAX(4*N+32,108).

  INFO    INTEGER
          = 0: succesful exit;
          < 0: if INFO = -i, the i-th argument had an illegal value;
          = 1: error occured during execution of MB03DD;
          = 2: error occured during execution of MB03HD.

  The algorithm reorders the eigenvalues like the following scheme:

  Step 1: Reorder the eigenvalues in the subpencil aA - bB.
       I. Reorder the eigenvalues with negative real parts to the
          top.
      II. Reorder the eigenvalues with positive real parts to the
          bottom.

  Step 2: Reorder the remaining eigenvalues with negative real
          parts in the pencil aS - bH.
       I. Exchange the eigenvalues between the last diagonal block
          in aA - bB and the last diagonal block in aS - bH.
      II. Move the eigenvalues of the R-th block to the (MM+1)-th
          block, where R denotes the number of upper quasi-
          triangular blocks in aA - bB and MM denotes the current
          number of blocks in aA - bB with eigenvalues with negative
          real parts.

  The algorithm uses a sequence of orthogonal transformations as
  described on page 33 in [1]. To achieve those transformations the
  elementary subroutines MB03DD and MB03HD are called for the
  corresponding matrix structures.

  [1] 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.

                                                            3
  The algorithm is numerically backward stable and needs O(N ) real
  floating point operations.

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Program Text

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