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\centerline{Mathematics 171--Numerical Analysis I}
\centerline{Review Sheet--Midterm 1}
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\noindent
{\bf General Information}
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 The first midterm will be given in class {\bf Thursday,
October 8}.  This will be a 75 minute in-class exam. 
You may (and certainly should) use a hand calculator.
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\noindent
{\bf Topics to be Covered}
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The exam will cover the material from the beginning of the semester
through class on Thursday, October 1.  This includes:
\item{1.}  Floating point numbers, floating point arithmetic,
and some of their effects in numerical calculations.
\item{2.}  Maple's floating point number systems and elementary
Maple commands for numerical computation.
\item{3.} The bisection, Newton-Raphson iteration, and 
fixed point iteration methods to solve $f(x) = 0$.
Be prepared to 
apply 3 or 4 iteration steps of bisection, fixed-point iteration or Newton's
Method on a ``reasonable'' example, by hand.
\item{4.}  Criterion for convergence of fixed-point iteration.  Linear
and quadratic convergence.

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{\bf Proofs to Know}
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The following are somewhat loose statements of the major theoretical
points in the material we have studied.  You should review and 
know exact statements and proofs of these results.  
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\item{1.} If $L$ is a fixed point of $g$ satisfying $|g'(L)| < 1$ then 
iteration $x_n = g(x_{n-1})$ will converge to $L$ 
starting from any $x_0$ in a suitable interval $I$ containing $L$.  
\item{2.}  Newton's Method converges quadratically to a {\it simple\/}
root of $f(x) = 0$, starting from $x_0$ sufficiently close to 
the root.  
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{\bf Review Problems}
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From Burden and Faires, questions like p. 26/4,5,25, p. 53/3 (how many 
steps could each of these take, in theory?), 11, 12, p. 63/1,2, p. 75/6.
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\item{1.} 
\itemitem{a.} Find a rearrangement $x = g(x)$ of the equation 
$x^3 - 9x^2 + 18x - 6 = 0$
for which fixed point iteration will converge to a root near $x = 2$.
(Just show that your rearrangement will ``work.'')
Will this iteration give a linearly or a quadratically convergent
sequence of approximations to the root?
\itemitem{b.} Perform three iteration steps.
\item{2.}  As methods for computing approximate values of $\sqrt{A}$,
the two iterations:
$$x_n = g_1(x_{n-1}) = {1\over 2} x_{n-1} + {A \over 2 x_{n-1}}$$
\noindent
and
$$x_n = g_2(x_{n-1}) = {2\over 3} x_{n-1} + {A \over 3 x_{n-1}}$$
are proposed.
\itemitem{a)} Give Maple commands that could be used to compute
the 10th term in each of these sequences with $A = 7.0$,
starting from $x_0 = 2.0$. 
\itemitem{b)} Show that both methods will yield sequences converging to $\sqrt{A}$ 
(at least if $x_0$ is chosen suitably).
\itemitem{c)}  Which iteration would tend to converge {\it faster\/} to
$\sqrt{A}$.  Why?
\item{3.}   The series
$$\sum_{n=2}^\infty {1\over n \ln n}$$
is divergent.  (The partial sums exceed every positive real
number if enough terms are included.) 
\itemitem{a)}  Give a Maple input command or commands that could be
used to compute the partial sum 
$$\sum_{n=2}^{20} {1\over n \ln n}$$
(Hint: use a variable {\tt psum} to represent the running total as
you add in terms one at a time.)
\itemitem{b)}  (A ``thought'' question.) Suppose you write a computer program
to compute the partial sum 
 $$\sum_{n=2}^k {1\over n \ln n}$$
using the floating point arithmetic in a 10-decimal digit floating
point number system, with exponents $e$, $-128 \le e \le 127$, 
using $k$ as an input parameter.  If you call it supplying 
huge values for $k$, say $k = 10^{120}, \ldots
10^{125}, 10^{126}, 10^{127}, \ldots $, discuss
how the results would compare with what you would expect mathematically.
\item{4.}  (Another ``thought'' question.)  The
function $f(x) = \sin(1/x) - x$ has infinitely many different roots
in the interval $(0,1]$.  Can {\it any\/}
root-finding method, implemented in a given floating point number
system, find {\it all of these\/} these roots?  (By this, I 
mean could you solve {\it every one\/} of the problems of the form:
find the root closest to 1, 2nd closest to 1, ... , $n$th closest to 1, ... .)
Why or why not? 
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