## MB04DS

### Balancing a real skew-Hamiltonian matrix

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

Purpose

      To balance a real skew-Hamiltonian matrix

                    [  A   G  ]
               S =  [       T ] ,
                    [  Q   A  ]

      where A is an N-by-N matrix and G, Q are N-by-N skew-symmetric
      matrices. This involves, first, permuting S by a symplectic
      similarity transformation to isolate eigenvalues in the first
      1:ILO-1 elements on the diagonal of A; and second, applying a
      diagonal similarity transformation to rows and columns
      ILO:N, N+ILO:2*N to make the rows and columns as close in 1-norm
      as possible. Both steps are optional.

Specification

          SUBROUTINE MB04DS( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         JOB
          INTEGER           ILO, INFO, LDA, LDQG, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), QG(LDQG,*), SCALE(*)

Arguments

Mode Parameters

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

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A. N >= 0.

      A       (input/output) DOUBLE PRECISION 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. 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).

      QG      (input/output) DOUBLE PRECISION array, dimension
                             (LDQG,N+1)
              On entry, the leading N-by-N+1 part of this array must
              contain in columns 1:N the strictly lower triangular part
              of the matrix Q and in columns 2:N+1 the strictly upper
              triangular part of the matrix G. The parts containing the
              diagonal and the first supdiagonal of this array are not
              referenced.
              On exit, the leading N-by-N+1 part of this array contains
              the strictly lower and strictly upper triangular parts of
              the matrices Q and G, respectively, of the balanced
              skew-Hamiltonian. In particular, the strictly lower
              triangular part of the first ILO-1 columns of QG is zero.

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

      ILO     (output) INTEGER
              ILO-1 is the number of deflated eigenvalues in the
              balanced skew-Hamiltonian matrix.

      SCALE   (output) DOUBLE PRECISION array of dimension (N)
              Details of the permutations and scaling factors applied to
              S.  For j = 1,...,ILO-1 let P(j) = SCALE(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 SCALE
              contains the factor of the scaling applied to row and
              column j.

Error Indicator

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

References

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

Further Comments

      None

Example

Program Text

    *     MB04DS EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 100 )
          INTEGER          LDA, LDQG
          PARAMETER        ( LDA = NMAX, LDQG = NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      JOB
          INTEGER          I, ILO, INFO, J, N
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA, NMAX), DUMMY(1), QG(LDQG, NMAX+1),
         $                 SCALE(NMAX)
    *     .. External Functions ..
          DOUBLE PRECISION DLANTR, DLAPY2
          EXTERNAL         DLANTR, DLAPY2
    *     .. External Subroutines ..
          EXTERNAL         MB04DS
    *     .. 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
          IF( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             READ ( NIN, FMT = * ) ( ( QG(I,J), J = 1,N+1 ), I = 1,N )
             CALL MB04DS( JOB, N, A, LDA, QG, LDQG, ILO, SCALE, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                DO 30  I = 1, N
                   WRITE (NOUT, FMT = 99995) ( A(I,J), J = 1,N )
    30          CONTINUE
                WRITE ( NOUT, FMT = 99996 )
                DO 40  I = 1, N
                   WRITE (NOUT, FMT = 99995) ( QG(I,J), J = 1,N+1 )
    40          CONTINUE
                WRITE (NOUT, FMT = 99993)  ILO
                IF ( ILO.GT.1 ) THEN
                    WRITE (NOUT, FMT = 99992) DLAPY2( DLANTR( 'Frobenius',
         $                 'Lower', 'No Unit', N-1, ILO-1, A(2,1), LDA,
         $                 DUMMY ), DLANTR( 'Frobenius', 'Lower', 'No Unit',
         $                 N-1, ILO-1, QG(2,1), LDQG, DUMMY ) )
                END IF
             END IF
          END IF
    *
    99999 FORMAT (' MB04DS EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from MB04DS = ',I2)
    99997 FORMAT (' The balanced matrix A is ')
    99996 FORMAT (/' The balanced matrix QG is ')
    99995 FORMAT (20(1X,F9.4))
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
    99993 FORMAT (/' ILO = ',I4)
    99992 FORMAT (/' Norm of subdiagonal blocks: ',G7.2)
          END

Program Data

    MB04DS EXAMPLE PROGRAM DATA
           6       B
        0.0576         0    0.5208         0    0.7275   -0.7839
        0.1901    0.0439    0.1663    0.0928    0.6756   -0.5030
        0.5962         0    0.4418         0   -0.5955    0.7176
        0.5869         0    0.3939    0.0353    0.6992   -0.0147
        0.2222         0   -0.3663         0    0.5548   -0.4608
             0         0         0         0         0    0.1338
             0         0   -0.9862   -0.4544   -0.4733    0.4435         0
             0         0         0   -0.6927    0.6641    0.4453         0
       -0.3676         0         0         0    0.0841    0.3533         0
             0         0         0         0         0    0.0877         0
        0.9561         0    0.4784         0         0         0         0
       -0.0164   -0.4514   -0.8289   -0.6831   -0.1536         0         0

Program Results

     MB04DS EXAMPLE PROGRAM RESULTS

     The balanced matrix A is
        0.1338    0.4514    0.6831    0.8289    0.1536    0.0164
        0.0000    0.0439    0.0928    0.1663    0.6756    0.1901
        0.0000    0.0000    0.0353    0.3939    0.6992    0.5869
        0.0000    0.0000    0.0000    0.4418   -0.5955    0.5962
        0.0000    0.0000    0.0000   -0.3663    0.5548    0.2222
        0.0000    0.0000    0.0000    0.5208    0.7275    0.0576

     The balanced matrix QG is
        0.0000    0.0000    0.5030    0.0147   -0.7176    0.4608    0.7839
        0.0000    0.0000    0.0000    0.6641   -0.6927    0.4453    0.9862
        0.0000    0.0000    0.0000    0.0000   -0.0841    0.0877    0.4733
        0.0000    0.0000    0.0000    0.0000    0.0000    0.3533    0.4544
        0.0000    0.0000    0.0000    0.4784    0.0000    0.0000   -0.4435
        0.0000    0.0000    0.0000    0.3676   -0.9561    0.0000    0.0000

     ILO =    4

     Norm of subdiagonal blocks: 0.0

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