Question

Restructure Newton's method (Case Study: Approximating Square Roots) by decomposing it into three cooperating functions: newton, limitReached, and improveEstimate.

The newton function can use either the recursive strategy of Project 2 or the iterative strategy of the Approximating Square Roots Case Study. The task of testing for the limit is assigned to a function named limitReached, whereas the task of computing a new approximation is assigned to a function named improveEstimate. Each function expects the relevant arguments and returns an appropriate value.

Enter a positive number or enter/return to quit:
The program's estimate is 1.4142135623746899

Python's estimate is 1.4142135623730951

import math 6
while True: 7
while True:
x = input("Enter a positive number or press enter/return to quit: ")
if not x.replace('.', '', 1).isdigit():
break
X = float(x)
break

tolerance = 0.000001
estimate = 1.0

while True:
estimate = (estimate + x / estimate) / 2
difference = abs(x if difference <= tolerance
break
if difference > tolerance: print("Please input a number.")
break

print("The program's estimate is", estimate)
print("Python's estimate is math.sqrt(x))
if x == "enter":
break

Answer 1

Newton's method for approximating square roots can be decomposed into three functions: newton, limitReached, and improveEstimate. This method provides an expedient approach to find square roots without solving quadratic equations, and it can be implemented either recursively or iteratively.

Step-by-step explanation:

Newton's method is a powerful algorithm for finding successively better approximations to the roots (or zeroes) of a real-valued function. When approximating square roots, it avoids the more complicated process of solving quadratic equations by using a simple iterative formula. The task can be decomposed into three functions: newton, which can be either recursive or iterative; limitReached for determining when the approximation is sufficiently close to the actual root; and improveEstimate for calculating a better estimation using the iterative formula.

Here's an example of how these functions might work together in Python:

def improveEstimate(x, estimate):
return (estimate + x / estimate) / 2

def limitReached(x, estimate, tolerance):
return abs(x - estimate * estimate) <= tolerance

def newton(x, tolerance=0.000001, estimate=1.0):
while not limitReached(x, estimate, tolerance):
estimate = improveEstimate(x, estimate)
return estimate

You would call the newton function and pass in the number whose square root you wish to approximate. The function uses the other two functions to iteratively improve the estimate until the difference is within the given tolerance level. This method is an example of an expedient mathematical approach to approximating square roots.