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)

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.

      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( * )

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.

  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.

  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.

  DWORK   DOUBLE PRECISION array, dimension (N/2)

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

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

  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.

  [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.

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