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<HTML>
<HEAD><TITLE>SB09MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="SB09MD">SB09MD</A></H2>
<H3>
Evaluation of closeness of two multivariable sequences
</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 compare two multivariable sequences M1(k) and M2(k) for
k = 1,2,...,N, and evaluate their closeness. Each of the
parameters M1(k) and M2(k) is an NC by NB matrix.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE SB09MD( N, NC, NB, H1, LDH1, H2, LDH2, SS, LDSS, SE,
$ LDSE, PRE, LDPRE, TOL, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDH1, LDH2, LDPRE, LDSE, LDSS, N, NB, NC
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION H1(LDH1,*), H2(LDH2,*), PRE(LDPRE,*),
$ SE(LDSE,*), SS(LDSS,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The number of parameters. N >= 0.
NC (input) INTEGER
The number of rows in M1(k) and M2(k). NC >= 0.
NB (input) INTEGER
The number of columns in M1(k) and M2(k). NB >= 0.
H1 (input) DOUBLE PRECISION array, dimension (LDH1,N*NB)
The leading NC-by-N*NB part of this array must contain
the multivariable sequence M1(k), where k = 1,2,...,N.
Each parameter M1(k) is an NC-by-NB matrix, whose
(i,j)-th element must be stored in H1(i,(k-1)*NB+j) for
i = 1,2,...,NC and j = 1,2,...,NB.
LDH1 INTEGER
The leading dimension of array H1. LDH1 >= MAX(1,NC).
H2 (input) DOUBLE PRECISION array, dimension (LDH2,N*NB)
The leading NC-by-N*NB part of this array must contain
the multivariable sequence M2(k), where k = 1,2,...,N.
Each parameter M2(k) is an NC-by-NB matrix, whose
(i,j)-th element must be stored in H2(i,(k-1)*NB+j) for
i = 1,2,...,NC and j = 1,2,...,NB.
LDH2 INTEGER
The leading dimension of array H2. LDH2 >= MAX(1,NC).
SS (output) DOUBLE PRECISION array, dimension (LDSS,NB)
The leading NC-by-NB part of this array contains the
matrix SS.
LDSS INTEGER
The leading dimension of array SS. LDSS >= MAX(1,NC).
SE (output) DOUBLE PRECISION array, dimension (LDSE,NB)
The leading NC-by-NB part of this array contains the
quadratic error matrix SE.
LDSE INTEGER
The leading dimension of array SE. LDSE >= MAX(1,NC).
PRE (output) DOUBLE PRECISION array, dimension (LDPRE,NB)
The leading NC-by-NB part of this array contains the
percentage relative error matrix PRE.
LDPRE INTEGER
The leading dimension of array PRE. LDPRE >= MAX(1,NC).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
The tolerance to be used in the computation of the error
matrices SE and PRE. If the user sets TOL to be less than
EPS then the tolerance is taken as EPS, where EPS is the
machine precision (see LAPACK Library routine DLAMCH).
</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>
The (i,j)-th element of the matrix SS is defined by:
N 2
SS = SUM M1 (k) . (1)
ij k=1 ij
The (i,j)-th element of the quadratic error matrix SE is defined
by:
N 2
SE = SUM (M1 (k) - M2 (k)) . (2)
ij k=1 ij ij
The (i,j)-th element of the percentage relative error matrix PRE
is defined by:
PRE = 100 x SQRT( SE / SS ). (3)
ij ij ij
The following precautions are taken by the routine to guard
against underflow and overflow:
(i) if ABS( M1 (k) ) > 1/TOL or ABS( M1 (k) - M2 (k) ) > 1/TOL,
ij ij ij
then SE and SS are set to 1/TOL and PRE is set to 1; and
ij ij ij
(ii) if ABS( SS ) <= TOL, then PRE is set to 100.
ij ij
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires approximately
2xNBxNCx(N+1) multiplications/divisions,
4xNBxNCxN additions/subtractions and
NBxNC square roots.
</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>
* SB09MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, NCMAX, NBMAX
PARAMETER ( NMAX = 20, NCMAX = 20, NBMAX = 20 )
INTEGER LDH1, LDH2, LDSS, LDSE, LDPRE
PARAMETER ( LDH1 = NCMAX, LDH2 = NCMAX, LDSS = NCMAX,
$ LDSE = NCMAX, LDPRE = NCMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NB, NC
* .. Local Arrays ..
DOUBLE PRECISION H1(LDH1,NMAX*NBMAX), H2(LDH2,NMAX*NBMAX),
$ PRE(LDPRE,NBMAX), SE(LDSE,NBMAX), SS(LDSS,NBMAX)
* .. External Subroutines ..
EXTERNAL SB09MD
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, NC, NB, TOL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE IF ( NB.LT.0 .OR. NB.GT.NBMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) NB
ELSE IF ( NC.LT.0 .OR. NC.GT.NCMAX ) THEN
WRITE ( NOUT, FMT = 99991 ) NC
ELSE
READ ( NIN, FMT = * ) ( ( H1(I,J), I = 1,NC ), J = 1,N*NB )
READ ( NIN, FMT = * ) ( ( H2(I,J), I = 1,NC ), J = 1,N*NB )
* Compare the given sequences and evaluate their closeness.
CALL SB09MD( N, NC, NB, H1, LDH1, H2, LDH2, SS, LDSS, SE, LDSE,
$ PRE, LDPRE, TOL, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 )
DO 20 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( SS(I,J), J = 1,NB )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( SE(I,J), J = 1,NB )
40 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 60 I = 1, NC
WRITE ( NOUT, FMT = 99996 ) ( PRE(I,J), J = 1,NB )
60 CONTINUE
END IF
END IF
STOP
*
99999 FORMAT (' SB09MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB09MD = ',I2)
99997 FORMAT (' The sum-of-squares matrix SS is ')
99996 FORMAT (20(1X,F8.4))
99995 FORMAT (/' The quadratic error matrix SE is ')
99994 FORMAT (/' The percentage relative error matrix PRE is ')
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' NB is out of range.',/' NB = ',I5)
99991 FORMAT (/' NC is out of range.',/' NC = ',I5)
END
</PRE>
<B>Program Data</B>
<PRE>
SB09MD EXAMPLE PROGRAM DATA
2 2 2 0.0
1.3373 0.1205 0.6618 -0.3372
-0.4062 1.6120 0.9299 0.7429
1.1480 -0.1837 0.8843 -0.4947
-0.4616 1.4674 0.6028 0.9524
</PRE>
<B>Program Results</B>
<PRE>
SB09MD EXAMPLE PROGRAM RESULTS
The sum-of-squares matrix SS is
1.9534 1.3027
2.6131 0.6656
The quadratic error matrix SE is
0.0389 0.1565
0.1134 0.0687
The percentage relative error matrix PRE is
14.1125 34.6607
20.8363 32.1262
</PRE>
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