Final answer:
The question involves writing iterative and recursive functions to compute the power of a number xn for n ≥ 0. Functions have been provided for an iterative approach (power1), a simple recursive approach (power2), and an optimized recursive approach using squaring (power3).
Step-by-step explanation:
Iterative and Recursive Functions to Compute xn
To compute xn for n ≥ 0, there are different approaches one can take programmatically. Below are iterative and recursive solutions to calculate the power of a number.
Part A: Iterative Function (power1)
An iterative function to compute xn can be written as follows:
def power1(x, n):
result = 1
for i in range(n):
result *= x
return result
This function uses a loop that multiplies the base x to the result for n times.
Part B: Recursive Function (power2)
A recursive function to compute xn using the given formulation is:
def power2(x, n):
if n == 0:
return 1
else:
return x * power2(x, n-1)
This is a direct implementation of the recursive formula provided where the function calls itself with a decremented value of n until n is zero.
Part C: Recursive Function Optimized (power3)
The following is a more optimized recursive function which uses the fact that powers can be reduced by squaring the number when the exponent is even:
def power3(x, n):
if n == 0:
return 1
elif n % 2 == 0:
half_power = power3(x, n // 2)
return half_power * half_power
else:
half_power = power3(x, (n-1) // 2)
return x * half_power * half_power
This function checks if n is even or odd and applies the recursive formulation accordingly to reduce the number of multiplications required, which can be vastly more efficient, especially for large values of n.