MB03JP

Moving eigenvalues with negative real parts of a real skew-Hamiltonian/Hamiltonian pencil in structured Schur form to the leading subpencil (applying transformations on panels of columns)

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 MB03JP( 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 >= 3*N-3.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)

  LDWORK  INTEGER
          The dimension of the array DWORK.
          If COMPQ = 'N',
             LDWORK >= MAX(2*N+32,108)+5*N/2;
          if COMPQ = 'I' or COMPQ = 'U',
             LDWORK >= MAX(4*N+32,108)+5*N/2.

  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.

  For large values of N, the routine applies the transformations on
  panels of columns. The user may specify in INFO the desired number
  of columns. If on entry INFO <= 0, then the routine estimates a
  suitable value of this number.

Program Text

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