MB03JZ

Moving eigenvalues with negative real parts of a complex 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 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 MB03JZ( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
     $                   LDQ, NEIG, TOL, 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, * ) 

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.

  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.

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Program Text

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