CS1101C Lab 6

Preliminary — Finding Roots of the sinc Function

Question 1 (Guessing the Location of Roots)

• Use the M_PI constant (which is available from the math.h header file) as the value of pi.
• Set the tolerance at 0 to be EPSILON, a constant of the value 1.0e-8. That is to say, if the value of x is within a tolerance of EPSILON to the right or left of 0, then sinc(x) evaluates to the value of 1.
• For consistency, replace all hard-coded 0.1e-4 within the program ch4_7.c with EPSILON.

Test your program with the following sample run.

Enter interval limits a, b (a<b): -5.5 5.5 Enter step size: 1 Root detected at -5.000 Root detected at -4.000 Root detected at -3.000 Root detected at -2.000 Root detected at -1.000 Root detected at 1.000 Root detected at 2.000 Root detected at 3.000 Root detected at 4.000 Root detected at 5.000

Enter interval limits a, b (a<b): -5.3 5.3 Enter step size: 1 Root detected at -4.800 Root detected at -3.800 Root detected at -2.800 Root detected at -1.800 Root detected at -0.800 Root detected at 1.200 Root detected at 2.200 Root detected at 3.200 Root detected at 4.200 Root detected at 5.000

The name of your C program file must be called lab6q1.c, files with any other name will not be marked.

Question 2 (Finding Roots Using the Bisection Method)

The above method can be refined to give a close approximation to the root. This method is called the Bisection method. For a given interval [a,b] and a function f,

1. Find the mid-point p of the two intervals a and b;
2. If f(p) is close to 0 (again we use the tolerance of EPSILON), then p is the approximate root.
3. Otherwise, if f(a) and f(p) have different signs, set b to be p and repeat step 1;
4. Else, set a to be p and repeat step 1.

Implement the Bisection method in a function find_root so that given the interval limits a and b, the function returns the value of the approximate root. From the check_roots function of lab6q1.c, if a root exists in an interval, the function find_root is invoked to output a more accurate value of the root. The root values should be printed to 12 decimal places. The sample run is as follows:

Enter interval limits a, b (a<b): -5.5 5.5 Enter step size: 1 Root detected at -5.000000000000 Root detected at -4.000000000000 Root detected at -3.000000000000 Root detected at -2.000000000000 Root detected at -1.000000000000 Root detected at 1.000000000000 Root detected at 2.000000000000 Root detected at 3.000000000000 Root detected at 4.000000000000 Root detected at 5.000000000000

Enter interval limits a, b (a<b): -5.3 5.3 Enter step size: 1 Root detected at -5.000000047684 Root detected at -3.999999988079 Root detected at -2.999999988079 Root detected at -1.999999988079 Root detected at -1.000000002980 Root detected at 0.999999997020 Root detected at 2.000000011921 Root detected at 3.000000011921 Root detected at 4.000000011921 Root detected at 5.000000000000

The name of your C program file must be called lab6q2.c, files with any other name will not be marked.

Question 3 (Finding Roots Using the Secant Method)

In the Bisection method above, we use the mid-point p of the interval [a,b] as an approximate to the root, and repeat with subsequently smaller intervals until f(p) is close to 0. As such the value of p converges closer and closer to the true value of the root.

A generally faster way of convergence is to use the Secant method (see problem 30 of Etter), in which a more accurate estimate of the root location is usually the intersection of a straight line through the function values with the x-axis. In this respect p is now the intersection point and can be computed as

(a*f(b) - b*f(a)) / (f(b) - f(a)).

Modify the find_root function of program lab6q2.c such that the Secant method is now used to find the roots. You should modularize the computation of the intersection point. The sample output is given below.

Enter interval limits a, b (a<b): -5.5 5.5 Enter step size: 1 Root detected at -5.000000001940 Root detected at -4.000000006141 Root detected at -3.000000019529 Root detected at -2.000000010457 Root detected at -1.000000004709 Root detected at 1.000000004709 Root detected at 2.000000010457 Root detected at 3.000000019529 Root detected at 4.000000006141 Root detected at 5.000000001940

Enter interval limits a, b (a<b): -5.3 5.3 Enter step size: 1 Root detected at -5.000000000001 Root detected at -4.000000003786 Root detected at -3.000000002464 Root detected at -2.000000006970 Root detected at -1.000000007891 Root detected at 1.000000009108 Root detected at 2.000000001602 Root detected at 3.000000002385 Root detected at 4.000000000917 Root detected at 5.000000001890

The name of your C program file must be called lab6q3.c, files with any other name will not be marked.

Question 4 (Finding Roots Using Newton-Raphson Method)

Begin by setting p to be the initial intersection point from the Secant Method. The approximation is given by the following method.

1. If f(p) is close to 0, then p is the approximate root.
2. Otherwise, set the new p value to be

   p - [f(p)/f_dev(p)] where f_dev is the first derivative of f

   and repeat step 1.

Enter interval limits a, b (a<b): -5.5 5.5 Enter step size: 1 Root detected at -5.000000000000 Root detected at -4.000000000000 Root detected at -2.999999999987 Root detected at -1.999999994066 Root detected at -1.000000000000 Root detected at 1.000000000000 Root detected at 1.999999994066 Root detected at 2.999999999987 Root detected at 4.000000000000 Root detected at 5.000000000000

Enter interval limits a, b (a<b): -5.3 5.3 Enter step size: 1 Root detected at -4.999999999982 Root detected at -3.999999999998 Root detected at -2.999999977538 Root detected at -2.000000000000 Root detected at -1.000000000000 Root detected at -2.999999999188 Root detected at 2.000000000000 Root detected at 2.999999999993 Root detected at 4.000000000000 Root detected at 4.999999998586

The name of your C program file must be called lab6q4.c, files with any other name will not be marked.

Ponder: Inspect the root values carefully. Why does such a "phenomenon" happen?

Homework (No need to submit)

Compare the three root finding methods and find out which has a generally faster convergence.

Note and Ponder

1. Assume that all user input is valid unless otherwise stated.

2. If your program does not work as you expect (logical errors), use extra printf statements to print out all the values of your variables to aid in your debugging.

3. Since this is not a Math course, you may post your queries on finding the derivative of sinc in the IVLE forum.

4. Most importantly, have lots of fun programming!

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