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\begin{document}
\title{Perspectives of Mathematics II}
\author{Homework 2}
\date{Due in class Monday, March 15}
\maketitle

\begin{enumerate}
\item Listen to the excellent lecture by Gilbert Strang of MIT on the
singular value decomposition, which is available at
\begin{center}
  \scriptsize
  \url{http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture29.htm}
\end{center}
At the end of his first example (16--27 minutes into the lecture),
Strang notices a sign problem which he cannot resolve on the spot.  It
is not due to an accidental miscalculation.  Can you help Prof.\
Strang and explain the wrong sign?

\item Consider the Tychonov regularization of an ill-conditioned
system of linear equations, see the class notes for details.  Show
that the Tychonov regularized solution $\x_\alpha$ minimizes the
function
\[
  f_\alpha (\x) 
  = \lVert A \x - \b \rVert^2 + \alpha \, \lVert \x \rVert^2
\]
over all $\x \in \R^n$.  

\item Replace the range condition used in class for the error analysis
of the Tychonov regularization of an ill-conditioned system of linear
equations by the stronger condition
\[
  \x_{\operatorname{true}} = A^TA \z
\]
for some $\z$ whose norm is comparable to the norm of
$\x_{\operatorname{true}}$.  Redo the error analysis to find an
expression for the optimal choice of $\alpha$ as a function of $\lVert
\Eta \rVert$ and $\lVert \z \rVert$ under this stronger range
condition.

\item Consider the matrix $A \in M(n \times n)$ with components
\[
  a_{ij} = \frac{n - \lvert i-j \rvert}{n^2} \,.
\]
Use a computer program to compute its singular values $\sigma_1,
\dots, \sigma_n$ when $n$ is large.  Formulate a conjecture how the
singular value $\sigma_i$ depends on $i$ over large ranges of the
index.

\emph{Hint:} Cf.\ Homework 1, Problem 3.

\item The data set located at
\begin{center}
  \scriptsize
  \url{http://math.jacobs-university.de/oliver/teaching/jacobs/spring2010/math292/handouts/noisyblurred.txt}
\end{center}
was generated by applying the blur matrix $A$ of the previous problem
(with $n=200$), then adding Gaussian random noise $\Eta$ with $\lVert
\Eta \rVert = 0.001$.  Obtain a reconstruction of the original data
set using \emph{a posteriori} selection of the regularization
parameter.  You should send in your program and the reconstructed data
in plain ASCII format by email.

\end{enumerate}

\end{document}
