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{\bf \hfill Seema Nanda\\}
{\bf Numerical Analysis \hfill Assignment 2 \hfill Jan 27, 2010
 in class}
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Show all your work clearly. Please follow the instructions for assignments and homework as given in the
course web page. Late homework will not be accepted. You may discuss problems with anyone but the work in the end should be your own. You MUST give credit to whoever helped you in your homework.
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% Homework starts here.


 {\bf Root finding}
\begin{enumerate}
\item Compute a numerical approximation to the positive solution of $x^2 = 2$ by
Newton's method. What function of $\epsilon_n$ and $\epsilon_{n+1}$ should you tabulate to observe the quadratic convergence? Do it.
\item
Assume that Newton's method converges for a $f(x)$ which has a root of multiplicity $m$.  
Is convergence quadratic for Newton's method for $f(x)$? calculate the order of convergence. 
Suggest a change to Newton's method that will make convergence quadratic in this case. 

\item Suppose you are using Newton's method to find the zeros of a cubic polynomial $f(x)$. 
Let the first guess $p_0$ lie halfway between any two of the three real distinct roots. 
In this case Newton's method converges to the third root in the first iteration. 
Check this and prove it.

\item Same equation, $x^2 = 2$, but this time, use the secant method. Choose $x_0$ and $x_1$ so the $\{x_n\}$ converge. Get your list of approximation errors
$\epsilon_n$ , and find the exponent $s$ so $\frac{|\epsilon_n|}{|(\epsilon_{n+1})^s|} \to$ nonzero constant as $n \to \infty$.


%\item (G and O) 5.2.9. Consider the equation $x^2-2x+2= 0$. What is the behaviour of the Newton
 %iterates for various starting values? Analyse by hand and provide support with %numerics. (use matlab to support any analysis). 

%\item (B and F 8th edition ) pg 72, Number 17.  (problem 13 7th edition)
%use matlab.

\item {Use Newton's method to find reciprocal of a number}

Consider the function $f(x) = a- \frac{1}{x}$

Use Newton to estimate $\frac{1}{a}$.


\item Produce a table of $x ~ versus ~ y$, where $y$ is defined  implicitly as a function of $x$. Use $G(x,y) = 3x^7+2y^5-x^3+y^3 -3$ and start at $x = 0$, proceeding in steps of $0.1$ to $x=10$. 


\end{enumerate}



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