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\begin{center}
{\bf \hfill Seema Nanda\\}
{\bf Numerical Analysis \hfill Assignment 4 \hfill due: February 25, 2010
 in class}
\end{center}
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\begin{center}
Please follow the instructions for assignments and homework as given in the
course web page. You may discuss the problems and solutions with anyone but the work written up and submitted must be done on your own. Also programs must be written by you. (P) indicates programming is required
\end{center}
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% Homework starts here.
\begin{enumerate}
\item \textbf{Norms and Conditioning of a matrix }
\begin{enumerate}
 \item Using a 2-matrix norm, show that in the case of a symmetric matrix 
the condition number is the absolute value of the quotient of the largest and smallest eigenvalue of the matrix.
 \item Prove that all $n \times n$ invertible matrices form a dense set in the set of all $n \times n$ matrices.
That is show that if $A$ is an $n \times n$ matrix then for $\epsilon > 0$ there is an invertible matrix $B$ 
with $\|B-A\| < \epsilon$.
\item Prove that if $\|AB-I\| < 1$, then $2B-BAB$ is a better approximate inverse of $A$ than $B$ in the sense that 
$A(2B-BAB)$ is closer to $I$.

\end{enumerate}

\item  \textbf{(Iterative methods for solving $Ax = b$)}\\
%Quarteroni  pg 181 
let $A \in R^{n \times n}$ be such that $A = (1+\omega)P - (N+\omega P)$ with $P^{-1}N$ non singular and with real eigenvalues $1 > \lambda_1 \ge \lambda_2 \cdots  \ge \lambda_n$. Find the values of $\omega \in R$ for which the following iterative method converges, $\forall x^{(0)}$, to the solution of the system $Ax = b$:
\begin{equation}
(1+\omega)Px^{(k+1)} = (N+\omega P)x^{(k)} + b, ~~ k \ge 0
\end{equation}
Can you determine the value of $\omega$ for which convergence is fastest.

\item {Check that finding the roots of the zeros of a polynomial of deg $\le n$ with real coefficients\\
%\begin{equation}
$P_n(x) = {\sum ^n}_{k=0} {a_k x^k}, ~a_n \ne 0 $\\
%\end{equation}
is equivalent to determining the spectrum of the Frobenius matrix $C$  in $R^{n \times n}$ associated with $P_n$.\\
$ C =
  \left[ \begin{array}{cccc}
-(a_{n-1}/a_n) & -(a_{n-2}/a_n) & ...& -(a_{0}/a_n)\\
1 & 0 & ...& 0\\
0 & 1 & ...& 0\\
. & . & ...& .\\
. & . & ...& .\\
0 & 0 &  ...& 1\\
\end{array} \right] $
}\\
By Abel's theorem  there exists in general no direct method to compute the eigenvalue of a matrix of size $n \ge 5$.
%Quarteroni pg 242 # 8

\item{\bf Interpolation}
\begin{enumerate}
%kincaid cheney
\item What condition  will have to be put on nodes $x_0$ and $x_1$ if the interpolation problem:\\
$p(x_i) = c_{i0}$ , $p''(x_i) = c_{i2}$ for $i = 0,1$\\
is to be solvable by a cubic polynomial for arbirtray $c_{ij}$.

%Determine if this is a quadratic spline:\\

%$ f(x) = 
%\begin{cases}
% x, & \text{if }x \in(\infty,1] \\
% -\frac{1}{2} (2-x)^2+3/2, & \text{if }x \in [1,2]\\
% \frac{3}{2}, & \text{if } x \in [2, \infty)
% \end{cases} $

%Is it a cubic spline?

\item %kincaid cheney
Draw a curve of your liking eg. a spiral or an oval on a sheet of graph paper. Select points in a 
rectangular distribution along the the curve and label points $t_0$, $t_1$, $t_2$,.... etc. Record the $x(t)$ and $y(t)$ 
values in a table. Then fit each of $x$ and $y$ to spline functions $S$ annd $S^*$. 
Hence get $x = S(t)$ and $y=S^*(t)$ as parametric representations of the curve.
Plot the resulting curve for two test cases using matlab. Submit your graph paper.

\end{enumerate}
\end{enumerate}









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