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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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<H1><A NAME="SECTION03450000000000000000"></A><A NAME="seclsq"></A>
<BR>
Error Bounds for Linear Least Squares Problems
</H1>
<P>
<P>
The linear least squares problem is to find <B><I>x</I></B> that minimizes
<IMG
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img437.png"
ALT="$\Vert Ax-b \Vert _2$">.
We discuss error bounds for the most common case where <B><I>A</I></B> is <B><I>m</I></B>-by-<B><I>n</I></B>
with <B><I>m</I> > <I>n</I></B>, and <B><I>A</I></B> has full rank<A NAME="10758"></A>;
this is called an <EM>overdetermined least squares problem</EM>
<A NAME="10760"></A>
(the following code fragments deal with <B><I>m</I>=<I>n</I></B> as well).
<P>
Let <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
be the solution computed by one of the driver routines
xGELS, xGELSX, xGELSY, xGELSS, or xGELSD (see section <A HREF="node27.html#subsecdrivellsq">2.3.2</A>).
An approximate error
bound<A NAME="footfnm 0"><SUP>4.10</SUP></A><A NAME="10764"></A><A NAME="10765"></A><A NAME="10766"></A><A NAME="10767"></A>
<A NAME="10768"></A><A NAME="10769"></A><A NAME="10770"></A><A NAME="10771"></A>
<A NAME="10772"></A><A NAME="10773"></A><A NAME="10774"></A><A NAME="10775"></A>
<A NAME="10776"></A><A NAME="10777"></A><A NAME="10778"></A><A NAME="10779"></A>
<A NAME="10780"></A><A NAME="10781"></A><A NAME="10782"></A><A NAME="10783"></A>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\frac{\| \hat{x} - x \|_2}{\| x \|_2} \mathrel{\raisebox{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}{\tt ERRBD}
\end{displaymath}
-->
<IMG
WIDTH="137" HEIGHT="48" BORDER="0"
SRC="img438.png"
ALT="\begin{displaymath}
\frac{\Vert \hat{x} - x \Vert _2}{\Vert x \Vert _2} \mathrel...
...box{-.75ex}{$\mathop{\sim}\limits^{\textstyle <}$}}{\tt ERRBD}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
may be computed in one of the following ways, depending on which type
of driver routine is used:
<P>
<DL COMPACT>
<DT>1.
<DD>Suppose the simple driver SGELS is used:
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Get the 2-norm of the right hand side B
BNORM = SNRM2( M, B, 1 )
* Solve the least squares problem; the solution X overwrites B
CALL SGELS( 'N', M, N, 1, A, LDA, B, LDB, WORK, LWORK, INFO )
IF ( MIN(M,N) .GT. 0 ) THEN
* Get the 2-norm of the residual A*X-B
RNORM = SNRM2( M-N, B( N+1 ), 1 )
* Get the reciprocal condition number RCOND of A
CALL STRCON('I', 'U', 'N', N, A, LDA, RCOND, WORK, IWORK, INFO)
RCOND = MAX( RCOND, EPSMCH )
IF ( BNORM .GT. 0.0 ) THEN
SINT = RNORM / BNORM
ELSE
SINT = 0.0
ENDIF
COST = MAX( SQRT( (1.0E0 - SINT)*(1.0E0 + SINT) ), EPSMCH )
TANT = SINT / COST
ERRBD = EPSMCH*( 2.0E0/(RCOND*COST) + TANT / RCOND**2 )
ENDIF
</PRE>
<A NAME="10790"></A>
<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>,
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{array}{ccc} 4 & 3 & 5 \\2 & 5 & 8 \\3 & 6 & 10 \\4 & 5 & 11 \end{array} \right)
\; {\rm and} \;
b = \left( \begin{array}{c} 100.1 \\.1 \\.01 \\.01 \end{array} \right) \; ,
\end{displaymath}
-->
<IMG
WIDTH="313" HEIGHT="93" BORDER="0"
SRC="img439.png"
ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 4 & 3 & 5 \\ 2 & 5 & 8 \\ 3 & ...
...n{array}{c} 100.1 \\ .1 \\ .01 \\ .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 = \hat{x} = \left( \begin{array}{c} 38.49 \\21.59 \\-23.88 \end{array} \right) \; \; ,
\end{displaymath}
-->
<IMG
WIDTH="180" HEIGHT="73" BORDER="0"
SRC="img440.png"
ALT="\begin{displaymath}
x = \hat{x} = \left( \begin{array}{c} 38.49 \\ 21.59 \\ -23.88 \end{array} \right) \; \; ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<!-- MATH
${\tt BNORM} = 100.1$
-->
<IMG
WIDTH="113" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img441.png"
ALT="${\tt BNORM} = 100.1$">,
<!-- MATH
${\tt RNORM} = 8.843$
-->
<IMG
WIDTH="113" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img442.png"
ALT="${\tt RNORM} = 8.843$">,
<!-- MATH
${\tt RCOND} = 4.712 \cdot 10^{-2}$
-->
<IMG
WIDTH="161" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img443.png"
ALT="${\tt RCOND} = 4.712 \cdot 10^{-2}$">,
<!-- MATH
${\tt ERRBD} = 4.9 \cdot 10^{-6}$
-->
<IMG
WIDTH="144" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img444.png"
ALT="${\tt ERRBD} = 4.9 \cdot 10^{-6}$">,
and the true error
is
<!-- MATH
$4.6 \cdot 10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img445.png"
ALT="$4.6 \cdot 10^{-7}$">.
<P>
<DT>2.
<DD>Suppose the expert driver SGELSX or SGELSY is used.
<A NAME="10807"></A>
<A NAME="10808"></A>
This routine has an input argument <TT>RCND</TT>,
which is used to determine the rank of the input matrix (briefly,
<A NAME="10810"></A>
the matrix is considered not to have full rank if its condition
number exceeds <TT>1/RCND</TT>).
<A NAME="10812"></A>
The code fragment below only computes error bounds
if the matrix has been determined to have full rank.
When the matrix does not have full rank,
computing and interpreting error bounds is more complicated, and
the reader is referred to the next section.
<P>
<PRE>
EPSMCH = SLAMCH( 'E' )
* Get the 2-norm of the right hand side B
BNORM = SNRM2( M, B, 1 )
* Solve the least squares problem; the solution X overwrites B
RCND = 0
CALL SGELSX( M, N, 1, A, LDA, B, LDB, JPVT, RCND, RANK, WORK,
$ INFO )
IF ( RANK.LT.N ) THEN
PRINT *,'Matrix less than full rank'
ELSE IF ( MIN( M,N ) .GT. 0 ) THEN
* Get the 2-norm of the residual A*X-B
RNORM = SNRM2( M-N, B( N+1 ), 1 )
* Get the reciprocal condition number RCOND of A
CALL STRCON('I', 'U', 'N', N, A, LDA, RCOND, WORK, IWORK, INFO)
RCOND = MAX( RCOND, EPSMCH )
IF ( BNORM .GT. 0.0 ) THEN
SINT = RNORM / BNORM
ELSE
SINT = 0.0
ENDIF
COST = MAX( SQRT( (1.0E0 - SINT)*(1.0E0 + SINT) ), EPSMCH )
TANT = SINT / COST
ERRBD = EPSMCH*( 2.0E0/(RCOND*COST) + TANT / RCOND**2 )
END IF
</PRE>
The numerical results of this code fragment on the above <B><I>A</I></B> and <B><I>b</I></B> are
the same as for the first code fragment.
<P>
<DT>3.
<DD>Suppose the other type of expert driver SGELSS or SGELSD is
used<A NAME="10815"></A><A NAME="10816"></A>.
This routine also has an input argument <TT>RCND</TT>, which is used to
determine the rank of the matrix <B><I>A</I></B>. The same code fragment can be used
to compute error bounds as for SGELSX or SGELSY,
except that the call to SGELSX must
be replaced by:
<P>
<PRE>
CALL SGELSD( M, N, 1, A, LDA, B, LDB, S, RCND, RANK, WORK, LWORK,
$ IWORK, INFO )
</PRE>
<P>
and the call to STRCON must be replaced by:
<P>
<PRE>
RCOND = S( N ) / S( 1 )
</PRE>
<A NAME="10822"></A>
<P>
Applied to the same <B><I>A</I></B> and <B><I>b</I></B> as above, the computed <IMG
WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img295.png"
ALT="$\hat{x}$">
is
nearly the same,
<!-- MATH
${\tt RCOND} = 5.428 \cdot 10^{-2}$
-->
<IMG
WIDTH="161" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img446.png"
ALT="${\tt RCOND} = 5.428 \cdot 10^{-2}$">,
<!-- MATH
${\tt ERRBD} = 4.0 \cdot 10^{-6}$
-->
<IMG
WIDTH="144" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img447.png"
ALT="${\tt ERRBD} = 4.0 \cdot 10^{-6}$">,
and the true error is
<!-- MATH
$6.6 \cdot 10^{-7}$
-->
<IMG
WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
SRC="img448.png"
ALT="$6.6 \cdot 10^{-7}$">.
<P>
</DL>
<P>
<BR><HR>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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