

\subsection{\bf Why is M Poorly Conditioned?}

\begin{slide}
\begin{itemize}
\item I note that in using this method, I got to look at values of
the $\lambda_j$.  It was not uncommon for
\[
\frac{\lambda_{\rm max}}{\lambda_{\rm min}} =
\frac{\lambda_1}{\lambda_n} \sim 10^7 \ \ \mbox{for} \ \ n \sim 30
\]

\item You can ask why one gets such large ratios.  

\item Intuitively, the eigenvectors of $M$ with small eigenvalue
correspond to linear combinations of $\alpha_j$ that are badly
determined.

\item So very small $\lambda$'s say that your theoretical formalism is
redundant or alternatively the data inadequate.

\item For instance, in fitting a cross section $\frac{d\sigma}{d(\cos
\theta)}$ to ${\displaystyle\sum^n_{j=1}} \alpha_j P_{j-1}
(\cos\theta)$ 

\begin{description}
\item{$\bullet$} then often the coefficients of the higher order
Legendre functions are poorly determined, and
\item{$\bullet$} if you choose $n$ to be ``too big,'' then small
eigenvalues will develop.
\end{description}

\item In this case, one can cure
the problem by reducing the top index $n$ but in general no such simple
solution exists.
\end{itemize}
\end{slide}

