## 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)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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.

Specification

          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, * )

Arguments

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.

Input/Output Parameters

      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.

Workspace

      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.

Error Indicator

      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.

Method

      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.

References

      [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.

Numerical Aspects

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

Further Comments

      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.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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