ZGHUXP

Moving eigenvalues with negative real parts of a complex skew-Hamiltonian/Hamiltonian pencil in
structured Schur form to the leading subpencil (unfactored version)

[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 ZGHUXP( COMPQ, N, A, LDA, D, LDD, B, LDB, F, LDF, Q,
     $                   LDQ, NEIG, TOL, INFO )
C
C     .. Scalar Arguments ..
      CHARACTER          COMPQ
      INTEGER            INFO, LDA, LDB, LDD, LDF, LDQ, N, NEIG
      DOUBLE PRECISION   TOL
C
C     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), B( LDB, * ), D( LDD, * ),
     $                   F( LDF, * ), Q( LDQ, * ) 
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.
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 subroutines ZGHUEX and ZGHUEY 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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