CS 2073, Spring 2006
 Program 4
 Root of an Equation
    Week 4: Feb 6-10
 Due (on time): 2006-02-17  23:59:59
 Due (late):        2006-02-20  23:59:59

Program 4 must be emailed to: nrwagner@cs.utsa.edu
following directions for: running and submitting a C program, with deadlines:
  • 2006-02-17  23:59:59 (that's Friday, 17 February 2006, 11:59:59 pm) for full credit.
  • 2006-02-20  23:59:59 (that's Monday, 20 February 2006, 11:59:59 pm) for 75% credit.

Introduction: The objective of this programming assignment is to find a real number x satisfying cos(x) = x, where 0 <= x <= pi/2, given in radians. (All C trig functions use radians, rather than degrees.) Here is a graph of the function y = cos(x):

Now draw the graph of the line y = x, which is just a line through the origin making an angle of pi/4 (45 degrees). These two will cross at a point with x-coordinate between 0 and pi/2, and this is the root we want.

It's interesting that this simple equation has no algebraic or analytic solution. This means that there does not exist an expression for the root in terms of elementary functions. The best we can do is to calculate an approximation to the solution.

This is perhaps surprising behavior, since the equation sin(x) = x has solution x = 0, and the equation cos(x) = 0.5 has solution x = acos(0.5) = 1.047197551, which is just 60 in degrees.

Two Different Programs: You are to write two different root-finding programs 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 in this case cos(x) - x = 0, or cos(x) = x.)

First Program: Newton's method: The first program will use Newton's method, which is taught in calculus courses. Don't worry if you forgotten your calculus, because you just need to use the formulas given below. A write-up about Newtons's Method: here.

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 x0, and produces a new, hopefully better guess x1 using the ``magic'' formula

In this particular case the formula looks like:

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. (You should ask if the absolute value of their difference is less than epsilon, using the function fabs.) In this assignment we will take epsilon to be 0.000000000001 . You should use the last new guess as your final answer.

Second Program: the Bisection Method: The second program will use the bisection method. Here one starts with two values x1 and x2 (with x1 < x2) for which f(x1) and f(x2) have opposite signs (i.e., one is positive and the other is negative, or f(x1) * f(x2) < 0). If f is a continuous function (no jumps or breaks), then there must be an x between x1 and x2 for which f(x) = 0. To get closer to that value, let xmid = (x1 + x2)/2, the midpoint. Consider f(xmid) and replace either x1 or x2 by xmid, so that after the replacement we still have f(x1) and f(x2) with opposite signs, but now x1 and x2 are half as far apart as they were before. Repeat this process until the distance between x1 and x2 is less than epsilon, again taken to be 0.000000000001 in this assignment. Your final answer should be xmid.

For the particular example here, you should take x1 = 0.0 , x2 = PI/2.0 , and f(x) = cos(x) - x.

Directions for both programs: Your programs set epsilon = 0.000000000001 for the margin of error. Then print "Tolerance value:" and the value for epsilon. Each program must count the iterations and must terminate the loop in case the iteration count gets larger than 50, since it is easy to produce an infinite printing loop if you make a mistake. (See the last example in the Week 4 lectures for a way to do this counter.) The first program should print "Newton's method:" and a short list of approximate values obtained at each iteration, ending with a final approximation. The second program should print "Bisection method" and a somewhat longer list of 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 numbers must be printed out with 15 digits to the right of the decimal place.
What you should email: Refer to the submissions directions and to deadlines at the top of this page. The text file that you submit should first have Your Name, the Course Number, and the Program Number. The rest of the file should have the following in it, in the order below, and clearly labeled, including at the beginning the appropriate item letters: a, b, c, etc.

 Contents of email submission for Program 4:

Last Name, First Name; Course Number; Program Number.

  1. C source for the first program that uses Newton's Method, along with a run of this program.
  2. C source for the second program that uses the Bisection Method, along with a run of this program.

Revision date: 2006-02-13. (Please use ISO 8601, the International Standard.)