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\centerline{Mathematics 171 -- Numerical Analysis 1}
\centerline{Discussion 2 -- Newton-Raphson Iteration}
\centerline{September 17, 1998}
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\noindent
{\bf Background}
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Last time at the end of class we introduced a second root-finding
method called the {\it Newton-Raphson iteration method\/}.  The 
idea was this:  To generate approximate solutions of an equation
$f(x)$, starting from some initial approximation $x_0$ (usually
chosen ``close'' to the desired root), we
can generate a sequence $x_n$, $n \ge 1$ of better and 
better approximations (we hope!) by computing:
$$x_n = x_{n-1} - {f(x_{n-1}) \over f'(x_{n-1})}$$
for as many $n \ge 1$ as we like.  $f'(x)$ is the 
ordinary derivative of $f(x)$; the formula above will
fail if $f'(x_{n-1}) = 0$ for some $n$.  
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\noindent
{\bf Discussion Questions}
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\item{A)}  Apply Newton-Raphson iteration to try to determine
approximate values of the roots of the following equations
in the indicated interval.  In parts 1,2,3, for $x_0$ you 
can take the midpoint of the interval.  Starting from 
$x_0$, compute $x_1,x_2,x_3,x_4,x_5$:
\itemitem{1)}  $x^3-2x^2-5 = 0$, root in $[1,4]$.
\itemitem{2)}  $x-0.8-0.2\sin(x) = 0$, root in $[0,\pi/2]$.
\itemitem{3)}  ${1\over 2}+{x^2\over 4}-x\sin(x)-{\cos(2x)\over 2}=0$,
root in $[0,3.6]$.
\itemitem{4)}  $x^3-{x\over 2}=0$, starting from 
$x_0=\sqrt{.1}$
\item{} Discuss your results.  Is there a difference between
what happens in 3 and 4 and what happens in 1 and 2? 
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\item{B)}  In most root-finding situations, of course, we do not
actually know the value of the root we are looking for.  Suppose
we wanted to determine an approximation to a root that
was (at least ``probably'') good to a given tolerance, that
is:
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\centerline{$|$exact root - approximate root$|$ $<$ tolerance}
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\item{}
Suggest
``stopping condition(s)'' we could use to decide when to 
terminate the Newton-Raphson iteration.  Think about part
4 of A again.  How could you keep from repeating 
the computation of new $x_n$'s for ever?
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\item{C)}  Look at the Maple code for Newton-Raphson from 
the handout given out in class on Tuesday, September 15.  
What ``stopping conditions'' are used there?  How does this
relate to what you said in B?
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\noindent
{\bf Assignment}
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Group write-ups for questions A, B, and C due in class
Tuesday, September 22.
\bye     
