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<H3><A NAME="SECTION03462100000000000000"></A>
<A NAME="sec_lseglm_glmdetails"></A>
<BR>
Further Details:  Error Bounds for General Linear Model Problems
</H3>

<P>
In this subsection, we will summarize the available error bounds.
The reader may also refer to [<A
 HREF="node151.html#lawn31">2</A>,<A
 HREF="node151.html#baifahey97">13</A>,<A
 HREF="node151.html#elden">50</A>,<A
 HREF="node151.html#paige79b">80</A>]
for further details.  
<BR>

<P>
Let <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img467.png"
 ALT="$\widehat {x}$">
and <IMG
 WIDTH="14" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img492.png"
 ALT="$\widehat {y}$">
be the solutions
by the driver routine xGGGLM (see subsection
<A HREF="node84.html#sec_lseglm_drivers">4.6</A>). Then <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img467.png"
 ALT="$\widehat {x}$">
is normwise
backward stable and <IMG
 WIDTH="14" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img492.png"
 ALT="$\widehat {y}$">
is stable
in a mixed forward/backward sense [<A
 HREF="node151.html#baifahey97">13</A>]. Specifically,
we have <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img467.png"
 ALT="$\widehat {x}$">
and 
<!-- MATH
 $\widehat {y} = \bar {y} + \Delta \bar {y}$
 -->
<IMG
 WIDTH="91" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img504.png"
 ALT="$\widehat {y} = \bar {y} + \Delta \bar {y}$">,
where <IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img467.png"
 ALT="$\widehat {x}$">
and <IMG
 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img505.png"
 ALT="$\bar {y}$">
solve
  
<!-- MATH
 $\min\{\| y \|_2: \; (A + \Delta A)x + (B + \Delta B)y= d + \Delta d \}$
 -->
<IMG
 WIDTH="375" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img506.png"
 ALT="$\min\{\Vert y \Vert _2: \; (A + \Delta A)x + (B + \Delta B)y= d + \Delta d \} $">,
and
<DIV ALIGN="CENTER">
<IMG
 WIDTH="234" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="img507.png"
 ALT="$\textstyle \parbox{2in}{
\begin{eqnarray*}
\Vert \Delta \bar {y} \Vert _2 &amp; \...
... \Delta A \Vert _F &amp; \leq &amp; q(m,n,p)\epsilon\Vert A\Vert _F,
\end{eqnarray*} }$">&nbsp;&nbsp;&nbsp;&nbsp;
<IMG
 WIDTH="234" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="img508.png"
 ALT="$\textstyle \parbox{2in}{
\begin{eqnarray*}
\Vert \Delta d \Vert _2 &amp; \leq &amp; q...
...t \Delta B \Vert _F &amp; \leq &amp; q(m,n,p)\epsilon\Vert B\Vert _F,
\end{eqnarray*}}$">
</DIV>
and <B><I>q</I>(<I>m</I>,<I>n</I>,<I>p</I>)</B> is a modestly growing function of <B><I>m</I></B>, <B><I>n</I></B>, and <B><I>p</I></B>.
We take <B><I>q</I>(<I>m</I>,<I>n</I>,<I>p</I>) = 1</B> in the code fragment above.
Let <IMG
 WIDTH="27" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img480.png"
 ALT="$X^{\dagger}$">
denote the Moore-Penrose pseudo-inverse of <B><I>X</I></B>.
Let 
<!-- MATH
 $\kappa_B(A) =  \| A \|_F  \| (A^\dagger_B) \|_2$
 -->
<IMG
 WIDTH="181" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img509.png"
 ALT="$\kappa_B(A) = \Vert A \Vert _F \Vert (A^\dagger_B) \Vert _2 $">( = <TT>CNDAB</TT> above) and
    
<!-- MATH
 $\kappa_A(B) =  \| B \|_F  \| (GB)^\dagger \|_2$
 -->
<IMG
 WIDTH="192" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img510.png"
 ALT="$\kappa_A(B) = \Vert B \Vert _F \Vert (GB)^\dagger \Vert _2 $">( = <TT>CNDBA</TT> above)
where 
<!-- MATH
 $G = I - AA^\dagger$
 -->
<IMG
 WIDTH="105" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img511.png"
 ALT="$G = I - AA^\dagger$">
and 
<!-- MATH
 $A^\dagger_B = A^\dagger[I-B(GB)^\dagger]$
 -->
<IMG
 WIDTH="174" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="img512.png"
 ALT="$A^\dagger_B = A^\dagger[I-B(GB)^\dagger]$">.
When 
<!-- MATH
 $q(m,n,p) \epsilon$
 -->
<IMG
 WIDTH="84" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img485.png"
 ALT="$q(m,n,p)\epsilon$">
is small, the errors 
<!-- MATH
 $x-\widehat {x}$
 -->
<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img513.png"
 ALT="$x-\widehat {x}$">
and 
<!-- MATH
 $y - \widehat {y}$
 -->
<IMG
 WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img514.png"
 ALT="$y - \widehat {y}$">
are bounded by
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
 WIDTH="647" HEIGHT="222" BORDER="0"
 SRC="img515.png"
 ALT="\begin{eqnarray*}
\frac{ \Vert x-\widehat {x} \Vert _2 }{ \Vert x \Vert _2 } &amp; ...
...ght)
+ \Vert B \Vert _F \Vert (GB)^\dagger \Vert _2 ^2 \bigg] .
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">

<P>
<BR>
<BR>

<P>
When <B><I>B</I> = <I>I</I></B>, the GLM problem is the standard LS problem.
<B><I>y</I></B> is the residual vector <B><I>y</I> = <I>d</I> - <I>Ax</I></B>, and we have
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{ \| x-\widehat {x} \|_2 }{ \| x \|_2 } \leq q(m,n)\epsilon
      \kappa(A)\Bigg( 1 + \frac{ \| d \|_2 }{ \| A \|_F  \| x \|_2 }
         + \kappa(A)\frac{ \| y \|_2 }{ \| A \|_F  \| x \|_2 } \Bigg),
\end{displaymath}
 -->


<IMG
 WIDTH="469" HEIGHT="54" BORDER="0"
 SRC="img516.png"
 ALT="\begin{displaymath}
\frac{ \Vert x-\widehat {x} \Vert _2 }{ \Vert x \Vert _2 } ...
...Vert y \Vert _2 }{ \Vert A \Vert _F \Vert x \Vert _2 } \Bigg),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\frac{ \| y - \widehat {y} \|_2 }{ \| y \|_2 } \leq q(m,n)\epsilon
         \kappa(A)\frac{ \| y \|_2 }{ \| d \|_2 } .
\end{displaymath}
 -->


<IMG
 WIDTH="231" HEIGHT="48" BORDER="0"
 SRC="img517.png"
 ALT="\begin{displaymath}
\frac{ \Vert y - \widehat {y} \Vert _2 }{ \Vert y \Vert _2 ...
...ilon
\kappa(A)\frac{ \Vert y \Vert _2 }{ \Vert d \Vert _2 } .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where 
<!-- MATH
 $\kappa(A) = \kappa_B(A) =  \| A \|_F  \| A^\dagger \|_2$
 -->
<IMG
 WIDTH="222" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img518.png"
 ALT="$\kappa(A) = \kappa_B(A) = \Vert A \Vert _F \Vert A^\dagger \Vert _2 $">
and

<!-- MATH
 $\kappa_A(B) = 1$
 -->
<IMG
 WIDTH="85" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img519.png"
 ALT="$\kappa_A(B) = 1$">.
The error bound of 
<!-- MATH
 $x-\widehat {x}$
 -->
<IMG
 WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img513.png"
 ALT="$x-\widehat {x}$">
is the same
as in the LSE problem (see section&nbsp;<A HREF="node86.html#sec_lseglm_lsedetails">4.6.1.1</A>),
which is essentially the same as given in section 4.5.1.
The bound on the error in <IMG
 WIDTH="14" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img492.png"
 ALT="$\widehat {y}$">
is the same as that provided
in [<A
 HREF="node151.html#GVL2">55</A>, section 5.3.7]. 
<BR>

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