Question1 : (JAVA) Write a public class named MyMath. (So the filename must be MyMath.java.) The class MyMath...

(JAVA) Write a public class named MyMath. (So the filename must be MyMath.java.) The class MyMath should have exactly 2 member functions: sqrt and main.

Problem 1: (Square root) In this homework assignment, write a function that computes the square root of a given double up to a certain precision. This function should be a member function of of class MyMath and should have signature

public static double sqrt ( double d )

You will use a technique called bisection or binary search for its implementation. First assume the input, which we call d, is between 0 and 1. Then we know that l d u, where l = 0 and u = 1. Here, l represents a lower bound for the (yet unknown) value of d, and u represents an upper bound.

To perform bisection, take the midpoint m = (l +u)/2 and check whether d is within the interval [l, m] or [m, u]. To do so, check whether d m2 , since if d m2 then d m. If d m2 then d is in the interval [l, m]. So perform the update u = m. If m2 < d then d is in the interval [m, u]. So perform the update l = m.

By iterating this process, we get a successively tighter interval [l, u] that contains d. If the length of this interval is small enough, i.e., if ul is small enough, we terminate this process; the midpoint m = (l + u)/2 will be very close to the true value of d. For the purpose of this assignment, you can terminate when the interval is smaller than 1010 .

Finally, we have to deal with the case when d > 1. One way to handle this case is by normalizing the input. The approach is based on the simple insight

d = 2 ( d/4 )

d = (2^2) ( d/4 ^2)

d = (2^3) ( d/4^3)

. . .

So loop and divide d by 4 until you reach a value less than or equal to 1. Compute the square root of the dividend, and make sure to multiply back the appropriate power of 2. You may not use Math.sqrt in the implementation of MyMath.sqrt.

Problem 2: (JAVA) Write code that compares the speed of your MyMath.sqrt with Math.sqrt in the way specified below. This code should go into the public static member function named main with return type void.

Using a loop, evaluate Math.sqrt on 10, 000, 000 random numbers between 0 and 100. These random numbers should be generated by calling 100*Math.random(). Measure the time it takes to complete this task using System.currentTimeMillis. Do the same with MyMath.sqrt. You may want to consult: https://docs.oracle.com/javase/7/docs/api/java/lang/System.html#currentTimeMillis()

This time measurement includes the time it takes to generate the random numbers. With a separate loop, measure the time it takes to generate the random numbers without evaluating the square root. Putting these measurement together, output, to the command line, the average execution time per evaluation of the two square root functions with the execution time of the random number generation subtracted out.

You will see that Math.sqrt is far superior over MyMath.sqrt. This is to be expected since standard math functions like Math.sqrt are implemented and optimized by a group of experts over many hours.