## MB4DPZ

### Balancing a complex skew-Hamiltonian/Hamiltonian pencil, exploiting the structure

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To balance the 2*N-by-2*N complex skew-Hamiltonian/Hamiltonian
      pencil aS - bH, with

            (  A  D  )         (  C  V  )
        S = (        ) and H = (        ),  A, C N-by-N,             (1)
            (  E  A' )         (  W -C' )

      where D and E are skew-Hermitian, V and W are Hermitian matrices,
      and ' denotes conjugate transpose. This involves, first, permuting
      aS - bH by a symplectic equivalence transformation to isolate
      eigenvalues in the first 1:ILO-1 elements on the diagonal of A
      and C; and second, applying a diagonal equivalence transformation
      to make the pairs of rows and columns ILO:N and N+ILO:2*N as close
      in 1-norm as possible. Both steps are optional. Balancing may
      reduce the 1-norms of the matrices S and H.

Specification

          SUBROUTINE MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
         $                   LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER          JOB
          INTEGER            ILO, INFO, IWARN, LDA, LDC, LDDE, LDVW, N
          DOUBLE PRECISION   THRESH
    C     .. Array Arguments ..
          DOUBLE PRECISION   DWORK(*), LSCALE(*), RSCALE(*)
          COMPLEX*16         A(LDA,*), C(LDC,*), DE(LDDE,*), VW(LDVW,*)

Arguments

Mode Parameters

      JOB     CHARACTER*1
              Specifies the operations to be performed on S and H:
              = 'N':  none:  simply set ILO = 1, LSCALE(I) = 1.0 and
                      RSCALE(I) = 1.0 for i = 1,...,N.
              = 'P':  permute only;
              = 'S':  scale only;
              = 'B':  both permute and scale.

Input/Output Parameters

      N       (input) INTEGER
              The order of matrices A, D, E, C, V, and W.  N >= 0.

      THRESH  (input) DOUBLE PRECISION
              If JOB = 'S' or JOB = 'B', and THRESH >= 0, threshold
              value for magnitude of the elements to be considered in
              the scaling process: elements with magnitude less than or
              equal to THRESH*MXNORM are ignored for scaling, where
              MXNORM is the maximum of the 1-norms of the original
              submatrices S(s,s) and H(s,s), with s = [ILO:N,N+ILO:2*N].
              If THRESH < 0, the subroutine finds the scaling factors
              for which some conditions, detailed below, are fulfilled.
              A sequence of increasing strictly positive threshold
              values is used.
              If THRESH = -1, the condition is that
                 max( norm(H(s,s),1)/norm(S(s,s),1),
                      norm(S(s,s),1)/norm(H(s,s),1) )                (1)
              has the smallest value, for the threshold values used,
              where S(s,s) and H(s,s) are the scaled submatrices.
              If THRESH = -2, the norm ratio reduction (1) is tried, but
              the subroutine may return IWARN = 1 and reset the scaling
              factors to 1, if this seems suitable. See the description
              of the argument IWARN and FURTHER COMMENTS.
              If THRESH = -3, the condition is that
                 norm(H(s,s),1)*norm(S(s,s),1)                       (2)
              has the smallest value for the scaled submatrices.
              If THRESH = -4, the norm reduction in (2) is tried, but
              the subroutine may return IWARN = 1 and reset the scaling
              factors to 1, as for THRESH = -2 above.
              If THRESH = -VALUE, with VALUE >= 10, the condition
              numbers of the left and right scaling transformations will
              be bounded by VALUE, i.e., the ratios between the largest
              and smallest entries in [LSCALE(ILO:N); RSCALE(ILO:N)]
              will be at most VALUE. VALUE should be a power of 10.
              If JOB = 'N' or JOB = 'P', the value of THRESH is
              irrelevant.

      A       (input/output) COMPLEX*16 array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A.
              On exit, the leading N-by-N part of this array contains
              the matrix A of the balanced skew-Hamiltonian matrix S.
              In particular, the strictly lower triangular part of the
              first ILO-1 columns of A is zero.

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

      DE      (input/output) COMPLEX*16 array, dimension (LDDE, N+1)
              On entry, the leading N-by-N lower triangular part of
              this array must contain the lower triangular part of the
              skew-Hermitian matrix E, and the N-by-N upper triangular
              part of the submatrix in the columns 2 to N+1 of this
              array must contain the upper triangular part of the
              skew-Hermitian matrix D. The real parts of the entries on
              the diagonal and the first superdiagonal of this array
              should be zero.
              On exit, the leading N-by-N lower triangular part of this
              array contains the lower triangular part of the balanced
              matrix E, and the N-by-N upper triangular part of the
              submatrix in the columns 2 to N+1 of this array contains
              the upper triangular part of the balanced matrix D.
              In particular, the lower triangular part of the first
              ILO-1 columns of DE is zero.

      LDDE    INTEGER
              The leading dimension of the array DE.  LDDE >= MAX(1, N).

      C       (input/output) COMPLEX*16 array, dimension (LDC, N)
              On entry, the leading N-by-N part of this array must
              contain the matrix C.
              On exit, the leading N-by-N part of this array contains
              the matrix C of the balanced Hamiltonian matrix H.
              In particular, the strictly lower triangular part of the
              first ILO-1 columns of C is zero.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= MAX(1, N).

      VW      (input/output) COMPLEX*16 array, dimension (LDVW, N+1)
              On entry, the leading N-by-N lower triangular part of
              this array must contain the lower triangular part of the
              Hermitian matrix W, and the N-by-N upper triangular
              part of the submatrix in the columns 2 to N+1 of this
              array must contain the upper triangular part of the
              Hermitian matrix V. The imaginary parts of the entries on
              the diagonal and the first superdiagonal of this array
              should be zero.
              On exit, the leading N-by-N lower triangular part of this
              array contains the lower triangular part of the balanced
              matrix W, and the N-by-N upper triangular part of the
              submatrix in the columns 2 to N+1 of this array contains
              the upper triangular part of the balanced matrix V. In
              particular, the lower triangular part of the first ILO-1
              columns of VW is zero.

      LDVW    INTEGER
              The leading dimension of the array VW.  LDVW >= MAX(1, N).

      ILO     (output) INTEGER
              ILO-1 is the number of deflated eigenvalues in the
              balanced skew-Hamiltonian/Hamiltonian matrix pencil.
              ILO is set to 1 if JOB = 'N' or JOB = 'S'.

      LSCALE  (output) DOUBLE PRECISION array, dimension (N)
              Details of the permutations of S and H and scaling applied
              to A, D, C, and V from the left. For j = 1,...,ILO-1 let
              P(j) = LSCALE(j). If P(j) <= N, then rows and columns P(j)
              and P(j)+N are interchanged with rows and columns j and
              j+N, respectively. If P(j) > N, then row and column P(j)-N
              are interchanged with row and column j+N by a generalized
              symplectic permutation. For j = ILO,...,N the j-th element
              of LSCALE contains the factor of the scaling applied to
              row j of the matrices A, D, C, and V.

      RSCALE  (output) DOUBLE PRECISION array, dimension (N)
              Details of the permutations of S and H and scaling applied
              to A, E, C, and W from the right. For j = 1,...,ILO-1 let
              P(j) = RSCALE(j). If P(j) <= N, then rows and columns P(j)
              and P(j)+N are interchanged with rows and columns j and
              j+N, respectively. If P(j) > N, then row and column P(j)-N
              are interchanged with row and column j+N by a generalized
              symplectic permutation. For j = ILO,...,N the j-th element
              of RSCALE contains the factor of the scaling applied to
              column j of the matrices A, E, C, and W.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK) where
              LDWORK = 0,   if  JOB = 'N' or JOB = 'P', or N = 0;
              LDWORK = 6*N, if (JOB = 'S' or JOB = 'B') and THRESH >= 0;
              LDWORK = 8*N, if (JOB = 'S' or JOB = 'B') and THRESH <  0.
              On exit, if JOB = 'S' or JOB = 'B', DWORK(1) and DWORK(2)
              contain the initial 1-norms of S(s,s) and H(s,s), and
              DWORK(3) and DWORK(4) contain their final 1-norms,
              respectively. Moreover, DWORK(5) contains the THRESH value
              used (irrelevant if IWARN = 1 or ILO = N).

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = 1:  scaling has been requested, for THRESH = -2 or
                    THRESH = -4, but it most probably would not improve
                    the accuracy of the computed solution for a related
                    eigenproblem (since maximum norm increased
                    significantly compared to the original pencil
                    matrices and (very) high and/or small scaling
                    factors occurred). The returned scaling factors have
                    been reset to 1, but information about permutations,
                    if requested, has been preserved.

Error Indicator

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

Method

      Balancing consists of applying a (symplectic) equivalence
      transformation to isolate eigenvalues and/or to make the 1-norms
      of each pair of rows and columns indexed by s of S and H nearly
      equal. If THRESH < 0, a search is performed to find those scaling
      factors giving the smallest norm ratio or product defined above
      (see the description of the parameter THRESH).

      Assuming JOB = 'S', let Dl and Dr be diagonal matrices containing
      the vectors LSCALE and RSCALE, respectively. The returned matrices
      are obtained using the equivalence transformation

        ( Dl  0 ) ( A  D  ) ( Dr  0 )   ( Dl  0 ) ( C  V  ) ( Dr  0 )
        (       ) (       ) (       ),  (       ) (       ) (       ).
        ( 0  Dr ) ( E  A' ) ( 0  Dl )   ( 0  Dr ) ( W -C' ) ( 0  Dl )

      For THRESH = 0, the routine returns essentially the same results
      as the LAPACK subroutine ZGGBAL [1]. Setting THRESH < 0, usually
      gives better results than ZGGBAL for badly scaled matrix pencils.

References

      [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
          Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
          Ostrouchov, S., and Sorensen, D.
          LAPACK Users' Guide: Second Edition.
          SIAM, Philadelphia, 1995.

      [2] Benner, P.
          Symplectic balancing of Hamiltonian matrices.
          SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.

Numerical Aspects

      The transformations used preserve the skew-Hamiltonian/Hamiltonian
      structure and do not introduce significant rounding errors.
      No rounding errors appear if JOB = 'P'. If T is the global
      transformation matrix applied to the right, then J'*T*J is the
      global transformation matrix applied to the left, where
      J = [ 0 I; -I 0 ], with blocks of order N.

Further Comments

      If THRESH = -2, the increase of the maximum norm of the scaled
      submatrices, compared to the maximum norm of the initial
      submatrices, is bounded by MXGAIN = 100.
      If THRESH = -2, or THRESH = -4, the maximum condition number of
      the scaling transformations is bounded by MXCOND = 1/SQRT(EPS),
      where EPS is the machine precision (see LAPACK Library routine
      DLAMCH).

Example

Program Text

    *     MB4DPZ EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 10 )
          INTEGER          LDA, LDC, LDDE, LDVW
          PARAMETER        ( LDA  = NMAX, LDC = NMAX, LDDE = NMAX,
         $                   LDVW = NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      JOB
          INTEGER          I, ILO, INFO, IWARN, J, N
          DOUBLE PRECISION THRESH
    *     .. Local Arrays ..
          COMPLEX*16       A(LDA, NMAX ), C( LDC, NMAX ), DE(LDDE, NMAX),
         $                 VW(LDVW, NMAX)
          DOUBLE PRECISION DWORK(8*NMAX), LSCALE(NMAX), RSCALE(NMAX)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         MB4DPZ
    *     .. Executable Statements ..
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * )  N, JOB, THRESH
          IF( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99985 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J),  J = 1,N ),   I = 1,N )
             READ ( NIN, FMT = * ) ( ( DE(I,J), J = 1,N+1 ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( C(I,J),  J = 1,N ),   I = 1,N )
             READ ( NIN, FMT = * ) ( ( VW(I,J), J = 1,N+1 ), I = 1,N )
             CALL MB4DPZ( JOB, N, THRESH, A, LDA, DE, LDDE, C, LDC, VW,
         $                LDVW, ILO, LSCALE, RSCALE, DWORK, IWARN, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                DO 10  I = 1, N
                   WRITE ( NOUT, FMT = 99993 ) ( A(I,J), J = 1,N )
    10          CONTINUE
                WRITE ( NOUT, FMT = 99996 )
                DO 20  I = 1, N
                   WRITE ( NOUT, FMT = 99993 ) ( DE(I,J), J = 1,N+1 )
    20          CONTINUE
                WRITE ( NOUT, FMT = 99995 )
                DO 30  I = 1, N
                   WRITE ( NOUT, FMT = 99993 ) ( C(I,J), J = 1,N )
    30          CONTINUE
                WRITE ( NOUT, FMT = 99994 )
                DO 40  I = 1, N
                   WRITE ( NOUT, FMT = 99993 ) ( VW(I,J), J = 1,N+1 )
    40          CONTINUE
                WRITE ( NOUT, FMT = 99992 )  ILO
                WRITE ( NOUT, FMT = 99991 )
                WRITE ( NOUT, FMT = 99984 ) ( LSCALE(I), I = 1,N )
                WRITE ( NOUT, FMT = 99990 )
                WRITE ( NOUT, FMT = 99984 ) ( RSCALE(I), I = 1,N )
                IF ( LSAME( JOB, 'S' ) .OR. LSAME( JOB, 'B' ) ) THEN
                   IF ( .NOT.( THRESH.EQ.-2 .OR. THRESH.EQ.-4 ) ) THEN
                      WRITE ( NOUT, FMT = 99989 )
                      WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 1,2 )
                      WRITE ( NOUT, FMT = 99988 )
                      WRITE ( NOUT, FMT = 99984 ) ( DWORK(I), I = 3,4 )
                      WRITE ( NOUT, FMT = 99987 )
                      WRITE ( NOUT, FMT = 99984 ) ( DWORK(5) )
                   ELSE
                      WRITE ( NOUT, FMT = 99986 ) IWARN
                   END IF
                END IF
             END IF
          END IF
    *
    99999 FORMAT (' MB4DPZ EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB4DPZ = ',I2)
    99997 FORMAT (' The balanced matrix A is ')
    99996 FORMAT (/' The balanced matrix DE is ')
    99995 FORMAT (' The balanced matrix C is ')
    99994 FORMAT (/' The balanced matrix VW is ')
    99993 FORMAT (20( 1X, G11.4, SP, F9.3, S, 'i ') )
    99992 FORMAT (/' ILO = ',I4)
    99991 FORMAT (/' The permutations and left scaling factors are ')
    99990 FORMAT (/' The permutations and right scaling factors are ')
    99989 FORMAT (/' The initial 1-norms of the (sub)matrices are ')
    99988 FORMAT (/' The final 1-norms of the (sub)matrices are ')
    99987 FORMAT (/' The threshold value finally used is ')
    99986 FORMAT (/' IWARN on exit from MB4DPZ = ',I2)
    99985 FORMAT (/' N is out of range.',/' N = ',I5)
    99984 FORMAT (20(1X,G11.4))
          END

Program Data

    MB4DPZ EXAMPLE PROGRAM DATA
           2       B      -3
       (1,0.5)         0
             0   (1,0.5)
             0         0         0
             0         0         0
       (1,0.5)         0
             0   (-2,-1)
             1   -1.e-12         0
      (-1,0.5)        -1         0

Program Results

     MB4DPZ EXAMPLE PROGRAM RESULTS

     The balanced matrix A is
       1.000       -0.500i    0.000        0.000i
       0.000        0.000i    1.000       +0.500i

     The balanced matrix DE is
       0.000        0.000i    0.000       +0.000i    0.000       +0.000i
       0.000       +0.000i    0.000       +0.000i    0.000       +0.000i
     The balanced matrix C is
       2.000       -1.000i    1.000       -0.500i
       0.000        0.000i    1.000       +0.500i

     The balanced matrix VW is
       0.000        0.000i    1.000       +0.000i    0.000        0.000i
       0.000        0.000i    1.000       +0.000i  -0.1000E-11   +0.000i

     ILO =    2

     The permutations and left scaling factors are
       4.000       1.000

     The permutations and right scaling factors are
       4.000       1.000

     The initial 1-norms of the (sub)matrices are
       1.118       2.118

     The final 1-norms of the (sub)matrices are
       1.118       2.118

     The threshold value finally used is
      -3.000

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