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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<TITLE>Further Details: Error Bounds for Linear Least Squares
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<H2><A NAME="SECTION03451000000000000000"></A><A NAME="secbackgroundlsq"></A>
<BR>
Further Details: Error Bounds for Linear Least Squares
Problems
</H2>
<P>
The conventional error analysis of linear least squares problems goes
as follows<A NAME="10832"></A>.
As above, let <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
be the solution to minimizing
<IMG
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img437.png"
ALT="$\Vert Ax-b \Vert _2$">
computed by
LAPACK using one of the least squares drivers
xGELS, xGELSX, xGELSY, xGELSS or xGELSD
(see subsection <A HREF="node27.html#subsecdrivellsq">2.3.2</A>).
We discuss the most common case, where <B><I>A</I></B> is
overdetermined<A NAME="10834"></A>
(i.e., has more rows than columns) and has full rank
[<A
HREF="node151.html#bjorck3">16</A>,<A
HREF="node151.html#demmelMA221">25</A>,<A
HREF="node151.html#GVL2">55</A>,<A
HREF="node151.html#higham96">67</A>]:
<A NAME="10836"></A><A NAME="10837"></A><A NAME="10838"></A><A NAME="10839"></A>
<A NAME="10840"></A><A NAME="10841"></A><A NAME="10842"></A><A NAME="10843"></A>
<A NAME="10844"></A><A NAME="10845"></A><A NAME="10846"></A><A NAME="10847"></A>
<A NAME="10848"></A><A NAME="10849"></A><A NAME="10850"></A><A NAME="10851"></A>
<A NAME="10852"></A><A NAME="10853"></A><A NAME="10854"></A><A NAME="10855"></A>
<P>
<BLOCKQUOTE>
The computed solution <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
has a small normwise backward error.
In other words <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
minimizes
<!-- MATH
$\|(A+E) \hat{x}- (b+f)\|_2$
-->
<IMG
WIDTH="175" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img449.png"
ALT="$\Vert(A+E) \hat{x}- (b+f)\Vert _2$">,
where
<B><I>E</I></B> and <B><I>f</I></B> satisfy
<A NAME="10857"></A>
<A NAME="10858"></A>
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\max \left( \frac{\| E \|_2}{\| A \|_2} ,
\frac{\| f \|_2}{\| b \|_2} \right) \leq p(n) \epsilon
\end{displaymath}
-->
<IMG
WIDTH="211" HEIGHT="48" BORDER="0"
SRC="img450.png"
ALT="\begin{displaymath}
\max \left( \frac{\Vert E \Vert _2}{\Vert A \Vert _2} ,
\frac{\Vert f \Vert _2}{\Vert b \Vert _2} \right) \leq p(n) \epsilon
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
and <B><I>p</I>(<I>n</I>)</B> is a modestly growing function of <B><I>n</I></B>. We take <B><I>p</I>(<I>n</I>)=1</B> in
the code fragments above.
Let
<!-- MATH
$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$
-->
<IMG
WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img451.png"
ALT="$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$">
(approximated by
1/<TT>RCOND</TT> in the above code fragments),
<!-- MATH
$\rho = \|A \hat{x} -b\|_2$
-->
<IMG
WIDTH="113" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img452.png"
ALT="$\rho = \Vert A \hat{x} -b\Vert _2$">
(= <TT>RNORM</TT> above), and
<!-- MATH
$\sin ( \theta ) = \rho / \|b\|_2$
-->
<IMG
WIDTH="122" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img453.png"
ALT="$\sin ( \theta ) = \rho / \Vert b\Vert _2$">
(<TT>SINT = RNORM / BNORM</TT> above). Here, <IMG
WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img328.png"
ALT="$\theta$">
is the acute angle between
the vectors <IMG
WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img454.png"
ALT="$A \hat{x}$">
and <B><I>b</I></B>.
<A NAME="10870"></A>
<A NAME="10871"></A>
Then when <IMG
WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img455.png"
ALT="$p(n) \epsilon$">
is small, the error <IMG
WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img456.png"
ALT="$\hat{x}- x$">
is bounded by
</BLOCKQUOTE>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\frac{\|x-\hat{x}\|_2}{\|x\|_2} \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}p(n) \epsilon
\left\{ \frac{2 \kappa_2 (A)}{\cos ( \theta )} + \tan ( \theta ) \kappa_2^2 (A)
\right\},
\end{displaymath}
-->
<IMG
WIDTH="334" HEIGHT="48" BORDER="0"
SRC="img457.png"
ALT="\begin{displaymath}
\frac{\Vert x-\hat{x}\Vert _2}{\Vert x\Vert _2} \mathrel{\ra...
...)}{\cos ( \theta )} + \tan ( \theta ) \kappa_2^2 (A)
\right\},
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P><BLOCKQUOTE>
where
<!-- MATH
$\cos ( \theta )$
-->
<IMG
WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img458.png"
ALT="$\cos ( \theta ) $">
= <TT>COST</TT> and
<!-- MATH
$\tan ( \theta )$
-->
<IMG
WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img459.png"
ALT="$\tan ( \theta )$">
= <TT>TANT</TT> in the code fragments
above.
</BLOCKQUOTE>
<P>
We avoid overflow by making sure <TT>RCOND</TT> and <TT>COST</TT> are both at least
<IMG
WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img420.png"
ALT="$\epsilon =$">
<TT>EPSMCH</TT>, and by handling the case of a zero <TT>B</TT> matrix
separately (<TT>BNORM = 0</TT>).
<A NAME="10884"></A>
<A NAME="10885"></A>
<P>
<!-- MATH
$\kappa_2 (A) = \sigma_{\max} (A) / \sigma_{\min} (A)$
-->
<IMG
WIDTH="203" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img451.png"
ALT="$\kappa_2 (A) = \sigma_{\max} (A)/\sigma_{\min} (A)$">
may be computed directly
from the singular values of <B><I>A</I></B> returned by xGELSS or xGELSD (as in the code fragment)
or by xGESVD or xGESDD. It may also be approximated by using xTRCON following calls to
xGELS, xGELSX or xGELSY. xTRCON estimates
<!-- MATH
$\kappa_{\infty}$
-->
<IMG
WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img460.png"
ALT="$\kappa_{\infty}$">
or <IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img461.png"
ALT="$\kappa_1$">
instead
of <IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img462.png"
ALT="$\kappa_2$">,
but these can differ from <IMG
WIDTH="22" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img462.png"
ALT="$\kappa_2$">
by at most a factor of <B><I>n</I></B>.
<A NAME="10889"></A><A NAME="10890"></A><A NAME="10891"></A><A NAME="10892"></A>
<A NAME="10893"></A><A NAME="10894"></A><A NAME="10895"></A><A NAME="10896"></A>
<A NAME="10897"></A><A NAME="10898"></A><A NAME="10899"></A><A NAME="10900"></A>
<P>
If <B><I>A</I></B> is rank-deficient, xGELSS, xGELSD, xGELSY and xGELSX can be used to
<B>regularize</B> the
problem<A NAME="10902"></A><A NAME="10903"></A>
by treating all singular values
less than a user-specified threshold
(
<!-- MATH
${\tt RCND} \cdot \sigma_{\max} (A)$
-->
<IMG
WIDTH="116" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="${\tt RCND} \cdot \sigma_{\max} (A)$">)
as
exactly zero. The number of singular values treated as nonzero is returned
in <TT>RANK</TT>. See [<A
HREF="node151.html#bjorck3">16</A>,<A
HREF="node151.html#GVL2">55</A>,<A
HREF="node151.html#higham96">67</A>]
for error bounds in this case, as well as
<A NAME="10908"></A>
[<A
HREF="node151.html#demmelhigham1">28</A>] for the
underdetermined<A NAME="10910"></A>
<A NAME="10911"></A> case.
The ability to deal with rank-deficient matrices is the principal attraction
of these four drivers, which are more expensive than the simple driver xGELS.
<P>
The solution of the overdetermined,
<A NAME="10912"></A>
<A NAME="10913"></A>
full-rank problem may also be
characterized as the solution of the linear system of equations
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\left( \begin{array}{cc} I & A \\A^T & 0 \end{array} \right) \left( \begin{array}{c} r \\x \end{array} \right) =
\left( \begin{array}{c} b \\0 \end{array} \right) .
\end{displaymath}
-->
<IMG
WIDTH="225" HEIGHT="54" BORDER="0"
SRC="img464.png"
ALT="\begin{displaymath}
\left( \begin{array}{cc} I & A \\ A^T & 0 \end{array} \right...
...\right) =
\left( \begin{array}{c} b \\ 0 \end{array} \right) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
By solving this linear system using xyyRFS or xyySVX (see section
<A HREF="node80.html#secAx_b">4.4</A>) componentwise error bounds can also be obtained
[<A
HREF="node151.html#arioliduffderijk">7</A>].
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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