Exercise 5

In this exercise, you will explore an alternate way to use recursion in computing powers of integers.

First, start by reading the Supplement to Module 4 section that describes the use of BigInteger's. Nothing below will make sense until you read that first.

Next, download and examine BigPower.java, in which you will write all your code. You will see that the power() method computes powers in the standard way as described in the Module 4 supplement. You will also see calls to a method called power2(), which you are to write. This recursive method will use a slightly different approach to computing powers:

• To explain how, let's first consider a number a raised to the power x, where x is a real number.
• Then, we can write a^x = (a²)^x/2.
• This works fine when the exponent (x, in this case) is a real number. But what about the case when the exponent is an integer?
• For integers, we are interested in a raised to the power n, where n is an integer.
• The following is true: if n is even, then aⁿ = (a²)^n/2. If n is odd, then aⁿ = a(a²)^(n-1)/2.
• This gives us a way to use recursion because we have decomposed the problem into a smaller problem size.

Write your code in power2() to perform the power computation recursively in the manner described above. Note: you won't need to understand any more of BigInteger's than is used in the examples shown in the Module 4 supplement. Indeed, you might try to get it working for regular int's before creating a version with BigInteger's.

Now, it's not clear that we've gained anything in this new approach. To see why it's useful, do the following:

• Count the number of arithmetic operations used for each of the methods.
• Print the "counts" for each of the two in your code.
• How many arithmetic operations (how many add's, multiply's etc) were carried out in each method for the computation of of 2¹⁰⁰⁰?

Submission:

• As usual, submit your file BigPower.java using these submission instructions.

What else is due:

• Read through the supplemental material for Module 4 and try to do the exercises therein. Do NOT submit these exercises.