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<HTML>
<HEAD><TITLE>MB04DS - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04DS">MB04DS</A></H2>
<H3>
Balancing a real skew-Hamiltonian matrix
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
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.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Benner, P.
Symplectic balancing of Hamiltonian matrices.
SIAM J. Sci. Comput., 22 (5), pp. 1885-1904, 2001.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* 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
</PRE>
<B>Program Data</B>
<PRE>
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
</PRE>
<B>Program Results</B>
<PRE>
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
</PRE>
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