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\begin{document}
\title{Perspectives of Mathematics II}
\author{Homework 1}
\date{Due in class Wednesday, February 24}
\maketitle

Each problem is worth 5 points.  Except for the first two problems,
the tasks are computational.  There is no need to write programs
bottom-up; it is recommended to use any of the standard high level
mathematical software packages.

\begin{enumerate}
\item Let $\V=\R^n$ and $A \in M(n \times n)$ be a symmetric matrix.
Show that the maximum of 
\[
  f(\v) = \v^T A \v
\]
on the unit ball of $\V$ is attained when $\v$ is a principal
eigenvector of $A$.  (A principal eigenvector is an eigenvector
corresponding to the largest eigenvalue.)

\item Let $\V = C^2([-1,1])$.  Find the minimum of
\[
  f(u) = \int_{-1}^1 u'(x)^2 \, \d x
\]
on $\V$ subject to the constraint $g(u)=0$ with
\[
  g(u) = \int_{-1}^1 u(x) \, \d x - 2 \,.
\]


\item You know from Calculus that for a smooth function on the reals,
\[
  f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}h
        = \lim_{h \to 0} \frac{f(x+h/2)-f(x-h/2)}h \,.
\]
Thus, we expect these fractions, for finite values of $h$, to
approximate $f'(x)$.  Test out the quality of these approximations by 
considering the errors
\[
  E_1(h) = \biggl\vert f'(0) - \frac{f(h)-f(0)}h \biggr\vert
\]
and 
\[
  E_2(h) = \biggl\vert f'(0) - \frac{f(h/2)-f(-h/2)}h \biggr\vert
\]
for the function $f(x)=\exp(x)$ within a reasonable range of values
for $h$.  Fit the model
\[
  E(h) = c \, h^p
\]
to the data obtained for each $E_1(h)$ and $E_2(h)$.  What values for
$c$ and $p$ do you obtain in each case?  Which approximation is
better?

\item The United States National Institute of Standards and Technology
(NIST) publishes reference data sets for use in validating numerical
software.  Use the software of your choice to reproduce the linear
least square fit to a polynomial of degree two of the \emph{Pontius}
data set, available at
\begin{quote}
  \url{http://www.itl.nist.gov/div898/strd/lls/data/Pontius.shtml}
\end{quote}
Compute the polynomial coefficients and compare with the published
values.  Plot the data and the graph of your fitted model.

\item Repeat the previous task with the \emph{Filip} data set,
available at
\begin{quote}
  \url{http://www.itl.nist.gov/div898/strd/lls/data/Filip.shtml}
\end{quote}
Why is this set considered difficult?

\item Perform a nonlinear least square fit on the \emph{Misra1a} data
set, available at
\begin{quote}
  \url{http://www.itl.nist.gov/div898/strd/nls/data/misra1a.shtml}
\end{quote}
Compute the coefficients of the exponential model described therein
and compare with the published values.  Plot the data and the graph of
your fitted model.

\item \emph{(Up to 15 points extra credit.)}  Consider the model for
the rise in sea level due to the melting of small ice caps and
glaciers, equation (1) in the 2008 MCM paper by Evans and Stepien.
Track the estimation of the model coefficients $\tau$ and $\beta$
though the literature.  Can you reproduce these estimates or validate
with old or even newer data?
\end{enumerate}

\end{document}
