## MB3JZP

### Moving eigenvalues with negative real parts of a complex 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 complex skew-Hamiltonian/Hamiltonian pencil aS - bH in
      structured Schur form to the leading principal subpencil, while
      keeping the triangular form. On entry, we have

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

      where A and B are upper triangular.
      S and H are transformed by a unitary matrix Q such that

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

      where Aout and Bout remain in upper triangular form. The notation
      M' denotes the conjugate transpose of the matrix M.
      Optionally, if COMPQ = 'I' or COMPQ = 'U', the unitary matrix Q
      that fulfills (1) is computed.

Specification

          SUBROUTINE MB3JZP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
         $                   LDQ, NEIG, TOL, DWORK, ZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          COMPQ
          INTEGER            INFO, LDA, LDB, LDD, LDF, LDQ, N, NEIG
          DOUBLE PRECISION   TOL
    C     .. Array Arguments ..
          COMPLEX*16         A( LDA, * ), B( LDB, * ), D( LDD, * ),
         $                   F( LDF, * ), Q( LDQ, * ), ZWORK( * )
          DOUBLE PRECISION   DWORK( * )

Arguments

Mode Parameters

      COMPQ   CHARACTER*1
              Specifies whether or not the unitary 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 unitary matrix Q is returned;
              = 'U':  the array Q contains a unitary matrix Q0 on
                      entry, and the matrix Q0*Q is returned, where Q
                      is the product of the unitary 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) COMPLEX*16 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.
              On exit, the leading  N/2-by-N/2 part of this array
              contains the transformed matrix Aout.
              The strictly lower triangular part of this array is not
              referenced.

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

      D       (input/output) COMPLEX*16 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-Hermitian
              matrix D.
              On exit, the leading  N/2-by-N/2 part of this array
              contains the transformed matrix Dout.
              The strictly lower triangular part of this array is not
              referenced.

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

      B       (input/output) COMPLEX*16 array, dimension (LDB, N/2)
              On entry, the leading N/2-by-N/2 part of this array must
              contain the upper triangular matrix B.
              On exit, the leading  N/2-by-N/2 part of this array
              contains the transformed matrix Bout.
              The strictly lower triangular part of this array is not
              referenced.

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

      F       (input/output) COMPLEX*16 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 Hermitian matrix
              F.
              On exit, the leading  N/2-by-N/2 part of this array
              contains the transformed matrix Fout.
              The strictly lower triangular part of this array is not
              referenced.

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

      Q       (input/output) COMPLEX*16 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 unitary 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.

Tolerances

      TOL     DOUBLE PRECISION
              The tolerance used to decide the sign of the eigenvalues.
              If the user sets TOL > 0, then the given value of TOL is
              used. If the user sets TOL <= 0, then an implicitly
              computed, default tolerance, defined by MIN(N,10)*EPS, is
              used instead, where EPS is the machine precision (see
              LAPACK Library routine DLAMCH). A larger value might be
              needed for pencils with multiple eigenvalues.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N/2)

      ZWORK   COMPLEX*16 array, dimension (N/2)

Error Indicator

      INFO    INTEGER
              = 0: succesful exit;
              < 0: if INFO = -i, the i-th argument had an illegal value.

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.
           I. Exchange the eigenvalues between the last diagonal block
              in aA - bB and the last diagonal block in aS - bH.
          II. Move the eigenvalues in the N/2-th place to the (MM+1)-th
              place, where MM denotes the current number of eigenvalues
              with negative real parts in aA - bB.

      The algorithm uses a sequence of unitary transformations as
      described on page 43 in [1]. To achieve those transformations the
      elementary SLICOT Library subroutines MB03DZ and MB03HZ are called
      for the corresponding matrix structures.

References

      [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
          Numerical Computation of Deflating Subspaces of Embedded
          Hamiltonian Pencils.
          Tech. Rep. SFB393/99-15, Technical University Chemnitz,
          Germany, June 1999.

Numerical Aspects

                                                                3
      The algorithm is numerically backward stable and needs O(N )
      complex 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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