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<H1><A NAME="SECTION03440000000000000000"></A><A NAME="secAx_b"></A>
<BR>
Error Bounds for Linear Equation Solving
</H1>
<P>
<P>
Let <B><I>Ax</I>=<I>b</I></B> be the system to be solved, and <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
the computed
solution. Let <B><I>n</I></B> be the dimension of <B><I>A</I></B>.
An approximate error bound<A NAME="10532"></A>
for <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
may be obtained in one of the following two ways,
depending on whether the solution is computed by a simple driver or
an expert driver:
<P>
<DL COMPACT>
<DT>1.
<DD>Suppose that <B><I>Ax</I>=<I>b</I></B> is solved using the simple driver SGESV
<A NAME="10535"></A>
(subsection <A HREF="node26.html#subsecdrivelineq">2.3.1</A>).
Then the approximate error bound<A NAME="tex2html2046"
HREF="footnode.html#foot13155"><SUP>4.10</SUP></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\frac{\| \hat{x} - x \|_{\infty}}{\|x\|_{\infty}} \leq {\tt ERRBD}
\end{displaymath}
-->
<IMG
WIDTH="144" HEIGHT="48" BORDER="0"
SRC="img396.png"
ALT="\begin{displaymath}
\frac{\Vert \hat{x} - x \Vert _{\infty}}{\Vert x\Vert _{\infty}} \leq {\tt ERRBD}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
can be computed by the following code fragment.
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Get infinity-norm of A
ANORM = SLANGE( 'I', N, N, A, LDA, WORK )
* Solve system; The solution X overwrites B
CALL SGESV( N, 1, A, LDA, IPIV, B, LDB, INFO )
IF( INFO.GT.0 ) THEN
PRINT *,'Singular Matrix'
ELSE IF (N .GT. 0) THEN
* Get reciprocal condition number RCOND of A
CALL SGECON( 'I', N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO )
RCOND = MAX( RCOND, EPSMCH )
ERRBD = EPSMCH / RCOND
END IF
</PRE>
<A NAME="10546"></A>
<P>
For example, suppose<A NAME="tex2html2047"
HREF="footnode.html#foot13159"><SUP>4.11</SUP></A>
<!-- 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{array}{ccc} 4 & 16000 & 17000 \\2 & 5 & 8 \\3 & 6 & 10 \end{array} \right)
\; \; {\rm and} \; \;
b = \left( \begin{array}{c} 100.1 \\.1 \\.01 \end{array} \right) \; . \; \;
\end{displaymath}
-->
<IMG
WIDTH="384" HEIGHT="73" BORDER="0"
SRC="img398.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 4 & 16000 & 17000 \\ 2 & 5 & 8...
...in{array}{c} 100.1 \\ .1 \\ .01 \end{array} \right) \; . \; \;
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Then (to 4 decimal places)
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x = \left( \begin{array}{c} -.3974 \\-.3349 \\.3211 \end{array} \right) \; , \; \;
\hat{x} = \left( \begin{array}{c} -.3968 \\-.3344 \\.3207 \end{array} \right) \; ,
\end{displaymath}
-->
<IMG
WIDTH="297" HEIGHT="73" BORDER="0"
SRC="img399.png"
ALT="\begin{displaymath}
x = \left( \begin{array}{c} -.3974 \\ -.3349 \\ .3211 \end{a...
...n{array}{c} -.3968 \\ -.3344 \\ .3207 \end{array} \right) \; ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<!-- MATH
${\tt ANORM} = 3.300 \cdot 10^4$
-->
<IMG
WIDTH="151" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img400.png"
ALT="${\tt ANORM} = 3.300 \cdot 10^4$">,
<!-- MATH
${\tt RCOND} = 3.907 \cdot 10^{-6}$
-->
<IMG
WIDTH="161" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img401.png"
ALT="${\tt RCOND} = 3.907 \cdot 10^{-6}$">,
the true reciprocal condition number
<!-- MATH
$= 3.902 \cdot 10^{-6}$
-->
<IMG
WIDTH="111" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img402.png"
ALT="$= 3.902 \cdot 10^{-6}$">,
<!-- MATH
${\tt ERRBD} = 1.5 \cdot 10^{-2}$
-->
<IMG
WIDTH="144" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img403.png"
ALT="${\tt ERRBD} = 1.5 \cdot 10^{-2}$">,
and the true error
<!-- MATH
$= 1.5 \cdot 10^{-3}$
-->
<IMG
WIDTH="93" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img404.png"
ALT="$= 1.5 \cdot 10^{-3}$">.
<A NAME="10566"></A>
<P>
<DT>2.
<DD>Suppose that <B><I>Ax</I>=<I>b</I></B> is solved using the expert driver SGESVX
(subsection <A HREF="node26.html#subsecdrivelineq">2.3.1</A>).
<A NAME="10568"></A>
This routine provides an explicit error bound <TT>FERR</TT>, measured
with the infinity-norm:
<A NAME="10570"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\frac{\| \hat{x} - x \|_{\infty}}{\|x\|_{\infty}} \leq {\tt FERR}
\end{displaymath}
-->
<IMG
WIDTH="135" HEIGHT="48" BORDER="0"
SRC="img405.png"
ALT="\begin{displaymath}
\frac{\Vert \hat{x} - x \Vert _{\infty}}{\Vert x\Vert _{\infty}} \leq {\tt FERR}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
For example, the following code fragment solves
<B><I>Ax</I>=<I>b</I></B> and computes an approximate error bound <TT>FERR</TT>:
<P>
<PRE>
CALL SGESVX( 'E', 'N', N, 1, A, LDA, AF, LDAF, IPIV,
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
$ WORK, IWORK, INFO )
IF( INFO.GT.0 ) PRINT *,'(Nearly) Singular Matrix'
</PRE>
<P>
For the same <TT>A</TT> and <TT>b</TT> as above,
<!-- MATH
$\hat{x} = \left( \begin{array}{c} -.3974 \\-.3349 \\.3211 \end{array} \right)$
-->
<IMG
WIDTH="135" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
SRC="img406.png"
ALT="$\hat{x} = \left( \begin{array}{c} -.3974 \\ -.3349 \\ .3211 \end{array} \right) $">,
<!-- MATH
${\tt FERR} = 3.0 \cdot 10^{-5}$
-->
<IMG
WIDTH="135" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img407.png"
ALT="${\tt FERR} = 3.0 \cdot 10^{-5}$">,
and the actual error is
<!-- MATH
$4.3 \cdot 10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img408.png"
ALT="$4.3 \cdot 10^{-7}$">.
<P>
</DL>
<P>
This example illustrates
that the expert driver provides an error bound with less programming
effort than the simple driver, and also that it may produce a significantly
more accurate answer.
<P>
Similar code fragments, with obvious adaptations,
may be used with all the driver routines for linear
equations listed in Table <A HREF="node26.html#tabdrivelineq">2.2</A>.
For example, if a symmetric system is solved using the simple driver xSYSV,
then xLANSY must be used to compute <TT>ANORM</TT>, and xSYCON must be used
to compute <TT>RCOND</TT>.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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