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    Programming Assignment 3}
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\begin{center}
%{\large\sf The University of Texas as San Antonio \\
%Division of Mathematics, Computer Science, and Statistics \\
%San Antonio, Texas  78285} \\
%
%\vspace{0.4in}

{\large\bf
   CS 2073, Computer Programming with Engineering Applications\\
    Spring Semester, 1992\longst  \\
    Programming Assignment 3\longst } \\
 {\Large\it  Root of an Equation}
\end{center}

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The objective of this programming assignment is to find
a real number $x$ satisfying $\cos(x) = x$, where
$0 \leq x \leq \pi/2$.
(From a graph you can see that the line $y = x$ crosses
$y = \cos(x)$, and the $x$-coordinate of the point
where they cross is what we want.)

You are to try out two different root-finding
methods for finding this $x$ value,
applying them to the equation $f(x) = \cos(x) - x$.
(A root is a place where $f(x) = 0$, or $\cos(x) - x = 0$,
or $\cos(x) = x$.)

\noindent {\bf 1. Newton's method.}
The first approach will use {\em Newton's
method}.  For this method,
one also needs the derivative of $f$,
$f'(x) = -\sin(x) - 1$.  Newton's method
starts with a guess at the answer, call it $x_0$,
and produces a new, hopefully better guess $x_1$ using
the ``magic'' formula

\vspace{-0.2in}
\begin{displaymath}
 x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}
\end{displaymath}

Replace the old guess with the new one and repeat
in a loop until the old guess and the new guess are within
some small value $\epsilon$ of one another.
In this assignment we will take $\epsilon$ to be $0.00001$ .
You should use the new
guess as your final answer.
Your program must use two Pascal {\tt FUNCTION}s named
``{\tt f}'' and ``{\tt fprime}'' as shown below, so that
the magic formula actually uses these functions.
%Finding the root must be done with a separate Pascal
%{\tt PROCEDURE} named ``{\tt newton}'' with two input parameters, the
%initial guess $x_0$, and the tolerance $\epsilon$.
%Thus the header for this procedure should look like:
%
%    \hspace{1in} {\tt PROCEDURE newton (x0, epsilon: real);}

\noindent Use $x_0 = 0$ for the initial guess.

\vspace{-0.2in}
\begin{verbatim}
     function f(x: real): real;
     begin
        f := cos(x) - x
     end; (* f *)
     
     function fprime(x: real): real;
     begin
        fprime := -sin(x) - 1.0
     end; (* fprime *)
\end{verbatim}

\noindent {\bf 2. Bisection method.}
The second approach will use the
{\em bisection method}.  Here one starts with two values
$x_1$ and $x_2$  (with $x_1 < x_2$) for which $f(x_1)$ and $f(x_2)$
have opposite signs (i.e., one is positive and
the other is negative, or $f(x_1) * f(x_2) < 0$).
If $f$ is a continuous function
(no jumps or breaks), then there must be an $x$
between $x_1$ and $x_2$ for which $f(x) = 0$.
To get closer to that value, let $x_{mid} = 
(x_1 + x_2)/2$, the midpoint.
Consider $f(x_{mid})$ and replace either $x_1$ or $x_2$
by $x_{mid}$, so that after the replacement we
still have $f(x_1)$ and $f(x_2)$ with opposite signs,
but now $x_1$ and $x_2$ are half as far apart as they
were before.
Repeat this process until the distance between $x_1$ and
$x_2$ is less than $\epsilon$, again taken to be $0.00001$
in this assignment.
Your final answer should be $x_{mid}$.
Again you must use a Pascal {\tt FUNCTION} named ``{\tt f}''.
%Finding the root must be done with a separate Pascal
%{\tt PROCEDURE} named ``{\tt bisection}'' with three input
%parameters, the initial numbers $x_1$ and $x_2$ and the
%tolerance $\epsilon$.
%This time the header for the procedure should look like:
%
%    \hspace{1in} {\tt PROCEDURE bisection (x1, x2, epsilon: real);}


\noindent Use the numbers $0$ and $\pi/2$
as the initial values.

\noindent{\bf Directions for both approaches.}
As before your program should first print
``{\tt This program written by:}'' followed by your name.
Your program should next read the value $0.00001$ for the
number $\epsilon$ used to determine how close an
approximation to the root one gets.  (This number
can be read from the standard file ``{\tt input}'',
i.e., the terminal.)
Then print ``{\tt Tolerance value:}'' and the
value for $\epsilon$.
For each of the methods the program must {\em count} the iterations
and must terminate the loop
in case the iteration count gets larger than 30.
First you should print ``{\tt Newton's method:}''
and a short list of approximate values obtained at each iteration,
ending with a final approximation.
Next print ``{\tt Bisection method}''
and a somewhat longer list of {\em pairs} of values obtained
at each stage of
the second method, again ending with the final
approximation.
Notice that you should get (almost) the same answer by
the two methods.
All real number must be printed out with 7 digits to the
right of the decimal place.

\noindent{\bf Extras.} If you are interested but for no
extra credit, you could
play around with other values for $\epsilon$.
You could also use the type ``{\tt double}'' in place
of ``{\tt real}'' everywhere in your program, to get much
more accurate answers.  (In Turbo Pascal, ``{\tt double}''
is a non-standard way to specify about 16 digits of accuracy
rather than the (about) 11 digits of accuracy that one gets
with ``{\tt real}''.)  Try $\epsilon = 0.000000001$ and
print numbers with 15 digits to the right of the decimal place.
 
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