Question

Give a recursive algorithm whose input is a real number r and a non-negative integer n, and whose output is r(2n) . Note that the exponent of r is 2n. Your algorithm should only use addition and multiplication operations and should not contain any loops.

Answer 1

Answer:

Explanation:

Let's use python for this:

recursive_exponent(n, r):

    if n == 0:

         return 1

    else:

         return r*recursive_exponent(n - 0.5, r)

To use this function, simply call it:

print(recursive_exponent(n, r))

The way it works is that n would start from top, and for each step it would get reduced by 0.5 until it gets to 0. Hence there will be 2n steps. At each step, r gets multiplied by itself. In the end r will multiplied by itself 2n times. Therefore, $r^{2n}$

Answer 2

Recursions are simply functions that calls itself.

The recursive algorithm, where comments are used to explain each line is as follows:

#This defines the function

recursive_algorithm(r, n):

#If n is negative, this returns 0

   if n<0:

       return 0

#If n is 0, this returns 1

   elif n == 0:

       return 1

#For every other value of n, this calculates the exponent $\mathbf{r^{2n}}$, recursively.

   else:

       return r*recursive_algorithm(r,n - 0.5)

The above recursion does not use loops or iterations.

Read more about recursive algorithm at:

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