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<H2><A NAME="SECTION03441000000000000000"></A><A NAME="secbackgroundAx_b"></A>
<BR>
Further Details:  Error Bounds for Linear Equation Solving
</H2>

<P>
The conventional error analysis of linear
equation<A NAME="10591"></A> solving goes as follows.
Let <B><I>Ax</I>=<I>b</I></B> be the system to be solved. Let <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
be the solution
computed by LAPACK (or LINPACK) using any of their linear equation solvers.
Let <B><I>r</I></B> be
the residual 
<!-- MATH
 $r = b - A \hat{x}$
 -->
<IMG
 WIDTH="88" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img409.png"
 ALT="$r = b - A \hat{x}$">.
In the absence of rounding error <B><I>r</I></B>
would be zero and <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
would equal <B><I>x</I></B>; with rounding error one can
only say the following:

<P>
<BLOCKQUOTE>
The normwise backward error of the computed solution <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">,
<A NAME="10597"></A>
<A NAME="10598"></A>
with respect to the infinity norm,
is the pair <B><I>E</I>,<I>f</I></B> which minimizes
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\max \left( \frac{\| E \|_{\infty}}{\| A \|_{\infty}} ,
            \frac{\| f \|_{\infty}}{\| b \|_{\infty}} \right)
\end{displaymath}
 -->


<IMG
 WIDTH="159" HEIGHT="48" BORDER="0"
 SRC="img410.png"
 ALT="\begin{displaymath}
\max \left( \frac{\Vert E \Vert _{\infty}}{\Vert A \Vert _{\...
...frac{\Vert f \Vert _{\infty}}{\Vert b \Vert _{\infty}} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
subject to the constraint 
<!-- MATH
 $(A+E) \hat{x} = b+f$
 -->
<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img379.png"
 ALT="$(A+E)\hat{x}=b+f$">.
The minimal value of

<!-- MATH
 $\max \left( \frac{\| E \|_{\infty}}{\| A \|_{\infty}} ,
\frac{\| f \|_{\infty}}{\| b \|_{\infty}} \right)$
 -->
<IMG
 WIDTH="145" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="img411.png"
 ALT="$\max \left( \frac{\Vert E \Vert _{\infty}}{\Vert A \Vert _{\infty}} ,
\frac{\Vert f \Vert _{\infty}}{\Vert b \Vert _{\infty}} \right)$">
is given by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\omega_{\infty}=
\frac{\| r \|_{\infty}}{\| A \|_{\infty}\cdot \| \hat{x} \|_{\infty}+ \| b \|_{\infty}} \; .
\end{displaymath}
 -->


<IMG
 WIDTH="220" HEIGHT="48" BORDER="0"
 SRC="img412.png"
 ALT="\begin{displaymath}
\omega_{\infty}=
\frac{\Vert r \Vert _{\infty}}{\Vert A \Ver...
...t \Vert \hat{x} \Vert _{\infty}+ \Vert b \Vert _{\infty}} \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
One can show that the computed solution <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
satisfies 
<!-- MATH
 $\omega_{\infty}\leq p(n) \cdot \epsilon$
 -->
<IMG
 WIDTH="105" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img413.png"
 ALT="$\omega_{\infty}\leq p(n) \cdot \epsilon$">,
where <B><I>p</I>(<I>n</I>)</B> is a modestly growing function of <B><I>n</I></B>.
The corresponding condition number is

<!-- MATH
 $\kappa_{\infty}(A) \equiv \|A\|_{\infty}\cdot \|A^{-1}\|_{\infty}$
 -->
<IMG
 WIDTH="199" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img414.png"
 ALT="$\kappa_{\infty}(A) \equiv \Vert A\Vert _{\infty}\cdot \Vert A^{-1}\Vert _{\infty}$">.
<A NAME="10612"></A>
The error <IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img415.png"
 ALT="$x-\hat{x}$">
is bounded by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{\|x- \hat{x} \|_{\infty}}{\| x \|_{\infty}}
\mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}2 \cdot \omega_{\infty}\cdot \kappa_{\infty}(A) = {\tt ERRBD} \; .
\end{displaymath}
 -->


<IMG
 WIDTH="286" HEIGHT="48" BORDER="0"
 SRC="img416.png"
 ALT="\begin{displaymath}
\frac{\Vert x- \hat{x} \Vert _{\infty}}{\Vert x \Vert _{\inf...
...dot \omega_{\infty}\cdot \kappa_{\infty}(A) = {\tt ERRBD} \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
In the first code fragment in the last section, 
<!-- MATH
 $2 \cdot \omega_{\infty}$
 -->
<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img417.png"
 ALT="$2 \cdot \omega_{\infty}$">,
which is 
<!-- MATH
 $4.504 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="92" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img418.png"
 ALT="$4.504 \cdot 10^{-8}$">
in the numerical example,
is approximated by 
<!-- MATH
 $\epsilon  = 2^{-24} = 5.960 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="179" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img419.png"
 ALT="$\epsilon = 2^{-24} = 5.960 \cdot 10^{-8}$">.
Approximations<A NAME="10621"></A>
of  
<!-- MATH
 $\kappa_{\infty}(A)$
 -->
<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img388.png"
 ALT="$\kappa_{\infty}(A)$">
-- or, strictly speaking, its reciprocal <TT>RCOND</TT> --
are returned by computational routines
xyyCON (subsection&nbsp;<A HREF="node38.html#subseccomplineq">2.4.1</A>) or driver routines
xyySVX (subsection&nbsp;<A HREF="node26.html#subsecdrivelineq">2.3.1</A>). The code fragment
makes sure <TT>RCOND</TT> is at least <IMG
 WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img420.png"
 ALT="$\epsilon =$">
<TT>EPSMCH</TT> to
avoid overflow in computing
<TT>ERRBD</TT>.<A NAME="10628"></A><A NAME="10629"></A>
This limits
<TT>ERRBD</TT> to a maximum of 1, which is no loss of generality since
a relative error of 1 or more indicates the same thing:
<A NAME="10631"></A><A NAME="10632"></A>
a complete loss of accuracy. <A NAME="10633"></A>
Note that the
value of <TT>RCOND</TT> returned by xyySVX may apply to a linear
system obtained from <B><I>Ax</I>=<I>b</I></B> by <EM>equilibration</EM>, i.e.
scaling the rows and columns of <B><I>A</I></B> in order to make the
condition number smaller. This is the case in the second
code fragment in the last section, where the program
chose to scale the rows by the factors returned in

<!-- MATH
 ${\tt R} = (5.882 \cdot 10^{-5}, .125, .1 )$
 -->
<IMG
 WIDTH="198" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img421.png"
 ALT="${\tt R} = (5.882 \cdot 10^{-5}, .125, .1 )$">
and scale the columns by the factors returned in

<!-- MATH
 ${\tt C} = (3.333, 1.063, 1. )$
 -->
<IMG
 WIDTH="159" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img422.png"
 ALT="${\tt C} = (3.333, 1.063, 1. )$">,
resulting in 
<!-- MATH
 ${\tt RCOND} = 3.454 \cdot 10^{-3}$
 -->
<IMG
 WIDTH="161" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img423.png"
 ALT="${\tt RCOND} = 3.454 \cdot 10^{-3}$">.

</BLOCKQUOTE>

<P>
As stated in section&nbsp;<A HREF="node79.html#seccomponentwise">4.3.2</A>,
this approach does not respect the presence
of zero or tiny entries in <B><I>A</I></B>. In contrast,
the LAPACK computational routines
xyyRFS (subsection&nbsp;<A HREF="node38.html#subseccomplineq">2.4.1</A>) or driver routines xyySVX
(subsection&nbsp;<A HREF="node26.html#subsecdrivelineq">2.3.1</A>) will (except in rare cases)
compute a solution <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
with the following properties:

<P>
<BLOCKQUOTE>
The componentwise backward error
of the computed solution <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
is the pair <B><I>E</I>,<I>f</I></B> which minimizes
<A NAME="10648"></A>
<A NAME="10649"></A>
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\max_{i,j,k} \left( \frac{| e_{ij} |}{|a_{ij}|} ,
            \frac{| f_{k} |}{|b_{k}|} \right)
\end{displaymath}
 -->


<IMG
 WIDTH="129" HEIGHT="54" BORDER="0"
 SRC="img424.png"
 ALT="\begin{displaymath}
\max_{i,j,k} \left( \frac{\vert e_{ij} \vert}{\vert a_{ij}\vert} ,
\frac{\vert f_{k} \vert}{\vert b_{k}\vert} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
(where we interpret <B>0/0</B> as 0)
subject to the constraint 
<!-- MATH
 $(A+E) \hat{x} = b+f$
 -->
<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img379.png"
 ALT="$(A+E)\hat{x}=b+f$">.
The minimal value of 
<!-- MATH
 $\max_{i,j,k} \left( \frac{| e_{ij} |}{|a_{ij}|} ,
\frac{| f_{k} |}{|b_{k}|} \right)$
 -->
<IMG
 WIDTH="146" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="img425.png"
 ALT="$\max_{i,j,k} \left( \frac{\vert e_{ij} \vert}{\vert a_{ij}\vert} ,
\frac{\vert f_{k} \vert}{\vert b_{k}\vert} \right)$">
is given by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\omega_{c}= \max_i \frac{|r_i|}{ (|A| \cdot |\hat{x} | + |b|)_i} \; .
\end{displaymath}
 -->


<IMG
 WIDTH="201" HEIGHT="48" BORDER="0"
 SRC="img426.png"
 ALT="\begin{displaymath}
\omega_{c}= \max_i \frac{\vert r_i\vert}{ (\vert A\vert \cdot \vert\hat{x} \vert + \vert b\vert)_i} \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
One can show that for most problems the <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
computed by xyySVX
satisfies 
<!-- MATH
 $\omega_{c}\leq p(n) \cdot \epsilon$
 -->
<IMG
 WIDTH="98" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img427.png"
 ALT="$\omega_{c}\leq p(n) \cdot \epsilon$">,
where <B><I>p</I>(<I>n</I>)</B> is a modestly growing function of <B><I>n</I></B>.
In other words, <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img295.png"
 ALT="$\hat{x}$">
is the exact solution of the
perturbed problem 
<!-- MATH
 $(A+ E ) \hat{x} = b + f$
 -->
<IMG
 WIDTH="139" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img379.png"
 ALT="$(A+E)\hat{x}=b+f$">
where <B><I>E</I></B> and <B><I>f</I></B> are small relative perturbations in each entry of <B><I>A</I></B> and
<B><I>b</I></B>, respectively.
The corresponding condition number is

<!-- MATH
 $\kappa_{c}(A,b,\hat{x}) \equiv {\| \, |A^{-1}| ( |A| \cdot | \hat{x} | + |b| )
\, \|_{\infty}}/{\| \hat{x} \|_{\infty}}$
 -->
<IMG
 WIDTH="340" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img428.png"
 ALT="$\kappa_{c}(A,b,\hat{x}) \equiv {\Vert \, \vert A^{-1}\vert ( \vert A\vert \cdot...
...t{x} \vert + \vert b\vert )
\, \Vert _{\infty}}/{\Vert \hat{x} \Vert _{\infty}}$">.
<A NAME="10670"></A>
The error <IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img415.png"
 ALT="$x-\hat{x}$">
is bounded by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{\| x- \hat{x} \|_{\infty}}{\| \hat{x} \|_{\infty}}
\leq \omega_{c}\cdot \kappa_{c}(A,b,\hat{x})  .
\end{displaymath}
 -->


<IMG
 WIDTH="208" HEIGHT="48" BORDER="0"
 SRC="img429.png"
 ALT="\begin{displaymath}
\frac{\Vert x- \hat{x} \Vert _{\infty}}{\Vert \hat{x} \Vert _{\infty}}
\leq \omega_{c}\cdot \kappa_{c}(A,b,\hat{x}) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
<BLOCKQUOTE>The routines xyyRFS and xyySVX return
<A NAME="10675"></A>
<A NAME="10676"></A>
<IMG
 WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img430.png"
 ALT="$\omega_{c}$">,
which is called <TT>BERR</TT><A NAME="10678"></A>
(for Backward ERRor),
and a bound on the the actual error

<!-- MATH
 $\|x - \hat{x}\|_{\infty}/ \| \hat{x} \|_{\infty}$
 -->
<IMG
 WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img431.png"
 ALT="$\Vert x - \hat{x}\Vert _{\infty}/ \Vert \hat{x} \Vert _{\infty}$">,
called <TT>FERR</TT>
<A NAME="10682"></A>
(for Forward ERRor), as
in the second code fragment in the last section.
<TT>FERR</TT> is actually calculated by the following formula, which can
be smaller than the bound 
<!-- MATH
 $\omega_{c}\cdot \kappa_{c}(A,b,\hat{x})$
 -->
<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img432.png"
 ALT="$\omega_{c}\cdot \kappa_{c}(A,b,\hat{x})$">
given above:
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{\| x- \hat{x} \|_{\infty}}{\| \hat{x} \|_{\infty}} \leq {\tt FERR} =
\frac{\| \, |A^{-1}| ( |\hat{r}| + n \epsilon (|A| \cdot |\hat{x}| + |b|) )
\|_{\infty}} {\| \hat{x} \|_{\infty}}  \; \; .
\end{displaymath}
 -->


<IMG
 WIDTH="428" HEIGHT="49" BORDER="0"
 SRC="img433.png"
 ALT="\begin{displaymath}
\frac{\Vert x- \hat{x} \Vert _{\infty}}{\Vert \hat{x} \Vert ...
...rt) )
\Vert _{\infty}} {\Vert \hat{x} \Vert _{\infty}} \; \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
Here, <IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img434.png"
 ALT="$\hat{r}$">
is the computed value of the residual <IMG
 WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img435.png"
 ALT="$b-A \hat{x}$">,
and
the norm in the numerator is estimated using the same estimation
subroutine used for <TT>RCOND</TT>.
</BLOCKQUOTE>
<P>
<BLOCKQUOTE>The value of
<TT>BERR</TT> for the example in the last section is 
<!-- MATH
 $4.6 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img436.png"
 ALT="$4.6 \cdot 10^{-8}$">.
</BLOCKQUOTE>
<P>
<BLOCKQUOTE>Even in the rare cases where xyyRFS fails to make
<TT>BERR</TT> close to its minimum <IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img262.png"
 ALT="$\epsilon$">,
the error bound <TT>FERR</TT>
may remain small. See [<A
 HREF="node151.html#ariolidemmelduff">6</A>]
for details.

</BLOCKQUOTE>

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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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