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<HTML>
<HEAD><TITLE>MB04MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04MD">MB04MD</A></H2>
<H3>
Balancing a general real 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 reduce the 1-norm of a general real matrix A by balancing.
This involves diagonal similarity transformations applied
iteratively to A to make the rows and columns as close in norm as
possible.
This routine can be used instead LAPACK Library routine DGEBAL,
when no reduction of the 1-norm of the matrix is possible with
DGEBAL, as for upper triangular matrices. LAPACK Library routine
DGEBAK, with parameters ILO = 1, IHI = N, and JOB = 'S', should
be used to apply the backward transformation.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, N
DOUBLE PRECISION MAXRED
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix A. N >= 0.
MAXRED (input/output) DOUBLE PRECISION
On entry, the maximum allowed reduction in the 1-norm of
A (in an iteration) if zero rows or columns are
encountered.
If MAXRED > 0.0, MAXRED must be larger than one (to enable
the norm reduction).
If MAXRED <= 0.0, then the value 10.0 for MAXRED is
used.
On exit, if the 1-norm of the given matrix A is non-zero,
the ratio between the 1-norm of the given matrix and the
1-norm of the balanced matrix. Usually, this ratio will be
larger than one, but it can sometimes be one, or even less
than one (for instance, for some companion matrices).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the input matrix A.
On exit, the leading N-by-N part of this array contains
the balanced matrix.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SCALE (output) DOUBLE PRECISION array, dimension (N)
The scaling factors applied to A. If D(j) is the scaling
factor applied to row and column j, then SCALE(j) = D(j),
for j = 1,...,N.
</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="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Balancing consists of applying a diagonal similarity
transformation inv(D) * A * D to make the 1-norms of each row
of A and its corresponding column nearly equal.
Information about the diagonal matrix D is returned in the vector
SCALE.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
None.
</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>
* MB04MD EXAMPLE PROGRAM TEXT.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA
PARAMETER ( LDA = NMAX )
* .. Local Scalars ..
INTEGER I, INFO, J, N
DOUBLE PRECISION MAXRED
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), SCALE(NMAX)
* .. External Subroutines ..
EXTERNAL MB04MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, MAXRED
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
* Balance matrix A.
CALL MB04MD( N, MAXRED, A, LDA, SCALE, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, N
WRITE ( NOUT, FMT = 99996 ) ( A(I,J), J = 1,N )
20 CONTINUE
WRITE ( NOUT, FMT = 99994 ) ( SCALE(I), I = 1,N )
END IF
END IF
STOP
*
99999 FORMAT (' MB04MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB04MD = ',I2)
99997 FORMAT (' The balanced matrix is ')
99996 FORMAT (20(1X,F10.4))
99994 FORMAT (/' SCALE is ',/20(1X,F10.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
MB04MD EXAMPLE PROGRAM DATA
4 0.0
1.0 0.0 0.0 0.0
300.0 400.0 500.0 600.0
1.0 2.0 0.0 0.0
1.0 1.0 1.0 1.0
</PRE>
<B>Program Results</B>
<PRE>
MB04MD EXAMPLE PROGRAM RESULTS
The balanced matrix is
1.0000 0.0000 0.0000 0.0000
30.0000 400.0000 50.0000 60.0000
1.0000 20.0000 0.0000 0.0000
1.0000 10.0000 1.0000 1.0000
SCALE is
1.0000 10.0000 1.0000 1.0000
</PRE>
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