Required Reading: Chapters 3,6,7 [IACU]
 

1. using makefile
2. recursion

1. Exercise (30 points)
• (a) Write a RECURSIVE function Power(x,n) that computes xⁿ for double X and a non-negative integer n >=0.
• (b) Now improve the running time of this function - Write the a version of Power(x,n) that works by breaking n down to halfs (where half of n = n/2), squaring Power(x,n/2), and multiplying by x again if n was odd.

For example, x¹¹ = (x⁵)*(x⁵)*x, whereas x¹⁰ = (x⁵)*(x⁵).
Find a suitable base case to stop the recursion.
Write your main function in a file "p1.c".
Note: Do not use this function on very large numbers.
What is the largest number a double type can hold?
 
 
2. Exercise (40 points)

 

 

The function sin(x) can be approximated for values of x which are close to zero. This is done by a Tailor expansion - an infinite sum:


Write a function ,Sin_Approx(double angle, int M); that computes the Tailor expansion up to the Mth element (this means that the summation goes up until n = M, instead of infinity).
Your function should use the recursive factorial function (n! studied in class) and the recursive function Power(x,n) from the previous exercise (you can use a or b). Note, however, that it is time consuming to compute a power of -1, so don't use the regular power function for that!
In order to test your function, you should write a program that computes the sine of x using the sin(x) function from "math.h", and compares it to the approximated Sin_Approx function. The format of the output should be the following:

Sin(0) is 0.00, Sin_Approx(0) is 0.00

Your program should prompt the user for "x" (read and use as double) and "M" (integer >=0).

What value should M have so that | Approx_Sin(0.333,M)-sin(0.333) | < 0.0000000001 ?
Run your program to test it, and write the answer in the README file you submit.
Write your main function in a file "p2.c".
 
 

3. Exercise (30 points)



Create a makefile that handles the following targets:

% make p1 (creates an executable p1 for exercise #1)
% make p2  (creates an executable p2 for exercise #2)
           (the last two should create appropriate object files)
% make test1 (runs exercise #1 with "1.inp" as input file
              and "1.out" as output file)
% make clean (deletes the object files, the executables and 1.out)

(Note, CVN students: this is a UNIX makefile, creating makefiles for compilers on PC is only an option)