DGHFXC

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

                                (  0  I  )
       S = J Z' J' Z, where J = (        ),
                                ( -I  0  )

     to the leading principal subpencil, while keeping the triangular
     form. On entry, we have

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

     where A and B are upper triangular and C is lower triangular.
     Z and H are transformed by a unitary symplectic matrix U and a
     unitary matrix Q such that

                       (  Aout  Dout  )
       Zout = U' Z Q = (              ), and
                       (    0   Cout  )
                                                                    (1)
                            (  Bout  Fout  )
       Hout = J Q' J' H Q = (              ), 
                            (    0  -Bout' )

     where Aout, Bout and Cout remain in 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.
     Optionally, if COMPU = 'I' or COMPU = 'U', the unitary symplectic
     matrix 

           (  U1  U2  )
       U = (          )
           ( -U2  U1  )   

     that fulfills (1) is computed.  
Specification
      SUBROUTINE ZGHFXC( COMPQ, COMPU, N, A, LDA, C, LDC, D, LDD, B,
     $                   LDB, F, LDF, Q, LDQ, U1, LDU1, U2, LDU2, NEIG,
     $                   TOL, INFO )
C
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPU
      INTEGER            INFO, LDA, LDB, LDC, LDD, LDF, LDQ, LDU1, LDU2,
     $                   N, NEIG
      DOUBLE PRECISION   TOL
C
C     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), F( LDF, * ), Q( LDQ, * ),
     $                   U1( LDU1, * ), U2( LDU2, * )
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.

     COMPU   CHARACTER*1
             Specifies whether or not the unitary symplectic
             transformations should be accumulated in the arrays U1 and
             U2, as follows:
             = 'N':  U1 and U2 are not computed;
             = 'I':  the arrays U1 and U2 are initialized internally,
                     and the submatrices U1 and U2 defining the
                     unitary symplectic matrix U are returned;
             = 'U':  the arrays U1 and U2 contain the corresponding
                     submatrices of a unitary symplectic matrix U0
                     on entry, and the updated submatrices U1 and U2
                     of the matrix product U0*U are returned, where U
                     is the product of the unitary symplectic
                     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).

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

     LDC     INTEGER
             The leading dimension of the array C.  LDC >= 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 matrix D.
             On exit, the leading  N/2-by-N/2 part of this array
             contains the transformed matrix Dout.

     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'.

     U1      (input/output) COMPLEX*16 array, dimension (LDU1, N/2)
             On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
             of this array must contain the upper left block of a
             given matrix U0, and on exit, the leading N/2-by-N/2 part
             of this array contains the updated upper left block U1 of
             the product of the input matrix U0 and the transformation
             matrix U used to transform the matrices S and H.
             On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
             of this array contains the upper left block U1 of the
             unitary symplectic transformation matrix U.
             If COMPU = 'N' this array is not referenced.

     LDU1    INTEGER
             The leading dimension of the array U1.
             LDU1 >= 1,           if COMPU = 'N';
             LDU1 >= MAX(1, N/2), if COMPU = 'I' or COMPU = 'U'.

     U2      (input/output) COMPLEX*16 array, dimension (LDU2, N/2)
             On entry, if COMPU = 'U', then the leading N/2-by-N/2 part
             of this array must contain the upper right block of a
             given matrix U0, and on exit, the leading N/2-by-N/2 part
             of this array contains the updated upper right block U2 of
             the product of the input matrix U0 and the transformation
             matrix U used to transform the matrices S and H.
             On exit, if COMPU = 'I', then the leading N/2-by-N/2 part
             of this array contains the upper right block U2 of the
             unitary symplectic transformation matrix U.
             If COMPU = 'N' this array is not referenced.

     LDU2    INTEGER
             The leading dimension of the array U2.
             LDU2 >= 1,           if COMPU = 'N';
             LDU2 >= MAX(1, N/2), if COMPU = 'I' or COMPU = '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 aC'*A - 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 aC'*A - 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 aC'*A - bB.

     The algorithm uses a sequence of unitary transformations as
     described on page 38 in [1]. To achieve those transformations the
     elementary subroutines ZGHFEX and ZGHFEY 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
   
     None.
Example

Program Text

     None.
Program Data
     None.
Program Results
     None.

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