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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<TITLE>Linear Equality Constrained Least Squares Problem</TITLE>
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<H2><A NAME="SECTION03461000000000000000">
Linear Equality Constrained Least Squares Problem</A>
</H2>
<P>
The linear equality constrained least squares (LSE) problem is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\min_x \| Ax - b \| \quad \mbox{subject to} \quad Bx = d
\end{displaymath}
-->
<IMG
WIDTH="268" HEIGHT="38" BORDER="0"
SRC="img465.png"
ALT="\begin{displaymath}\min_x \Vert Ax - b \Vert \quad \mbox{subject to} \quad Bx = d \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <B><I>A</I></B> is an <B><I>m</I></B>-by-<B><I>n</I></B> matrix, <B><I>B</I></B> is a <B><I>p</I></B>-by-<B><I>n</I></B> matrix,
<B><I>b</I></B> is an <B><I>m</I></B> vector, and <B><I>d</I></B> is a <B><I>p</I></B> vector, with
<!-- MATH
$p \leq n \leq p+m$
-->
<IMG
WIDTH="116" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img466.png"
ALT="$p \leq n \leq p+m$">.
<BR>
<P>
The LSE problem is solved by the driver routine xGGLSE
(see section <A HREF="node84.html#sec_lseglm_drivers">4.6</A>).
Let <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$\widehat {x}$">
be the value of <B><I>x</I></B> computed by xGGLSE.
The approximate error bound
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="439" HEIGHT="130" BORDER="0"
SRC="img468.png"
ALT="\begin{eqnarray*}
% latex2html id marker 10929\frac{ \Vert x - \widehat {x} \...
... ...">
</DIV><P></P>
<BR CLEAR="ALL">
can be computed by the following code fragment
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Get the 2-norm of the right hand side C
CNORM = SNRM2( M, C, 1 )
print*,'CNORM = ',CNORM
* Solve the least squares problem with equality constraints
CALL SGGLSE( M, N, P, A, LDA, B, LDA, C, D, Xc, WORK, LWORK, IWORK, INFO )
* Get the Frobenius norm of A and B
ANORM = SLANTR( 'F', 'U', 'N', N, N, A, LDA, WORK )
BNORM = SLANTR( 'F', 'U', 'N', P, P, B( 1, N-P+1 ), LDA, WORK )
MN = MIN( M, N )
IF( N.EQ.P ) THEN
APPSNM = ZERO
RNORM = SLANTR( '1', 'U', 'N', N, N, B, LDB, WORK(P+MN+N+1) )
CALL STRCON( '1', 'U', 'N', N, B, LDB, RCOND, WORK( P+MN+N+1 ),
$ IWORK, INFO )
BAPSNM = ONE/ (RCOND * RNORM )
ELSE
* Estimate norm of (AP)^+
RNORM = SLANTR( '1', 'U', 'N', N-P, N-P, A, LDA, WORK(P+MN+1) )
CALL STRCON( '1', 'U', 'N', N-P, A, LDA, RCOND, WORK( P+MN+1 ),
$ IWORK, INFO )
APPSNM = ONE/ (RCOND * RNORM )
* Estimate norm of B^+_A
KASE = 0
CALL SLACON( P, WORK( P+MN+1 ), WORK( P+MN+N+1 ), IWORK, EST, KASE )
30 CONTINUE
CALL STRSV( 'Upper', 'No trans', 'Non unit', P, B( 1, N-P+1 ),
$ LDB, WORK( P+MN+N+1 ), 1 )
CALL SGEMV( 'No trans', N-P, P, -ONE, A( 1, N-P+1 ), LDA,
$ WORK( P+MN+N+1 ), 1, ZERO, WORK( P+MN+P+1 ), 1 )
CALL STRSV( 'Upper', 'No transpose', 'Non unit', N-P, A, LDA,
$ WORK( P+MN+P+1 ), 1 )
DO I = 1, P
WORK( P+MN+I ) = WORK( P+MN+N+I )
END DO
CALL SLACON( N, WORK( P+MN+N+1 ), WORK( P+MN+1 ), IWORK, EST, KASE )
*
IF( KASE.EQ.0 ) GOTO 40
DO I = 1, P
WORK( P+MN+N+I ) = WORK( MN+N+I )
END DO
CALL STRSV( 'Upper', 'Trans', 'Non unit', N-P, A, LDA,
$ WORK( P+MN+1 ), 1 )
CALL SGEMV( 'Trans', N-P, P, -ONE, A( 1, N-P+1 ), LDA,
$ WORK( P+MN+1 ), 1, ONE, WORK( P+MN+N+1 ), 1 )
CALL STRSV( 'Upper', 'Trans', 'Non unit', P, B( 1, N-P+1 ),
$ LDB, WORK( P+MN+N+1 ), 1 )
CALL SLACON( P, WORK( P+MN+1 ), WORK( P+MN+N+1 ), IWORK, EST, KASE )
*
IF( KASE.EQ.0 ) GOTO 40
GOTO 30
40 CONTINUE
BAPSNM = EST
*
END IF
* Estimate norm of A*B^+_A
IF( P+M.EQ.N ) THEN
EST = ZERO
ELSE
R22RS = MIN( P, M-N+P )
KASE = 0
CALL SLACON( P, WORK( P+MN+P+1 ), WORK( P+MN+1 ), IWORK, EST, KASE )
50 CONTINUE
CALL STRSV( 'Upper', 'No trans', 'Non unit', P, B( 1, N-P+1 ),
$ LDB, WORK( P+MN+1 ), 1 )
DO I = 1, R22RS
WORK( P+MN+P+I ) = WORK( P+MN+I )
END DO
CALL STRMV( 'Upper', 'No trans', 'Non unit', R22RS,
$ A( N-P+1, N-P+1 ), LDA, WORK( P+MN+P+1 ), 1 )
IF( M.LT.N ) THEN
CALL SGEMV( 'No trans', R22RS, N-M, ONE, A( N-P+1, M+1 ), LDA,
$ WORK( P+MN+R22RS+1 ), 1, ONE, WORK( P+MN+P+1 ), 1 )
END IF
CALL SLACON( R22RS, WORK( P+MN+1 ), WORK( P+MN+P+1 ), IWORK, EST,
$ KASE )
*
IF( KASE.EQ.0 ) GOTO 60
DO I = 1, R22RS
WORK( P+MN+I ) = WORK( P+MN+P+I )
END DO
CALL STRMV( 'Upper', 'Trans', 'Non Unit', R22RS,
$ A( N-P+1, N-P+1 ), LDA, WORK( P+MN+1 ), 1 )
IF( M.LT.N ) THEN
CALL SGEMV( 'Trans', R22RS, N-M, ONE, A( N-P+1, M+1 ), LDA,
$ WORK( P+MN+P+1 ), 1, ZERO, WORK( P+MN+R22RS+1 ), 1 )
END IF
CALL STRSV( 'Upper', 'Trans', 'Non unit', P, B( 1, N-P+1 ), LDB,
$ WORK( P+MN+1 ), 1 )
CALL SLACON( P, WORK( P+MN+P+1 ), WORK( P+MN+1 ), IWORK, EST, KASE )
*
IF( KASE.EQ.0 ) GOTO 60
GOTO 50
60 CONTINUE
END IF
ABAPSN = EST
* Get the 2-norm of Xc
XNORM = SNRM2( N, Xc, 1 )
IF( APPSNM.EQ.0.0E0 ) THEN
* B is square and nonsingular
ERRBD = EPSMCH*BNORM*BAPSNM
ELSE
* Get the 2-norm of the residual A*Xc - C
RNORM = SNRM2( M-N+P, C( N-P+1 ), 1 )
* Get the 2-norm of Xc
XNORM = SNRM2( N, Xc, 1 )
* Get the condition numbers
CNDBA = BNORM*BAPSNM
CNDAB = ANORM*APPSNM
* Get the approximate error bound
ERRBD = EPSMCH*( (1.0E0 + CNORM/(ANORM*XNORM))*CNDAB +
$ RNORM/(ANORM*XNORM)*(1.0E0 + BNORM*ABAPSN/ANORM)*
$ (CNDAB*CNDAB) + 2.0E0*CNDBA )
END IF
</PRE>
<P>
For example, if
<!-- MATH
${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
-->
<IMG
WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img397.png"
ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = \left( \begin{tabular}{rrrr}
1 & 1 & 1 & 1 \\
1 & 3 & 1 & 1 \\
1 & -1 & 3 & 1 \\
1 & 1 & 1 & 3 \\
1 & 1 & 1 & -1
\end{tabular} \right),
b = \left( \begin{array}{r}
2 \\
1 \\
6 \\
3 \\
1
\end{array} \right),
B = \left( \begin{tabular}{rrrr}
1 & 1 & 1 & -1 \\
1 & -1 & 1 & 1\\
1 & 1 & -1 & 1
\end{tabular} \right)
\quad \mbox{and} \quad
d = \left( \begin{array}{r}
1 \\
3 \\
-1
\end{array} \right),
\end{displaymath}
-->
<IMG
WIDTH="656" HEIGHT="115" BORDER="0"
SRC="img469.png"
ALT="\begin{displaymath}A = \left( \begin{tabular}{rrrr}
1 & 1 & 1 & 1 \\
1 & 3 & ...
...left( \begin{array}{r}
1 \\
3 \\
-1
\end{array} \right),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then (to 7 decimal places),
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\widehat {x} = \left( \begin{array}{r}
0.5000000 \\
-0.5000001 \\
1.4999999 \\
0.4999998
\end{array} \right).
\end{displaymath}
-->
<IMG
WIDTH="172" HEIGHT="93" BORDER="0"
SRC="img470.png"
ALT="\begin{displaymath}\widehat {x} = \left( \begin{array}{r}
0.5000000 \\
-0.5000001 \\
1.4999999 \\
0.4999998
\end{array} \right).\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The computed error bound
<!-- MATH
${\tt ERRBD} = 5.7\cdot10^{-7}$
-->
<IMG
WIDTH="144" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img471.png"
ALT="${\tt ERRBD} = 5.7\cdot10^{-7}$">,
where
<!-- MATH
${\tt CNDAB} = 2.09$
-->
<IMG
WIDTH="104" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img472.png"
ALT="${\tt CNDAB} = 2.09$">,
<!-- MATH
${\tt CNDBA} = 3.12$
-->
<IMG
WIDTH="104" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img473.png"
ALT="${\tt CNDBA} = 3.12$">.
The true error
<!-- MATH
$= 1.2\cdot10^{-7}$
-->
<IMG
WIDTH="93" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img474.png"
ALT="$= 1.2\cdot10^{-7}$">.
The exact solution is
<!-- MATH
$x = [0.5, -0.5, 1.5, 0.5]^T$
-->
<B><I>x</I> = [0.5, -0.5, 1.5, 0.5]<SUP><I>T</I></SUP></B>.
<P>
<BR><HR>
<!--Table of Child-Links-->
<A NAME="CHILD_LINKS"></A>
<UL>
<LI><A NAME="tex2html5387"
HREF="node86.html">Further Details:
Error Bounds for Linear Equality Constrained Least Squares Problems</A>
</UL>
<!--End of Table of Child-Links-->
<BR><HR>
<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
</ADDRESS>
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