The function sqrt from the header file cmath can be used to find the square root

Question

The function sqrt from the header file cmath can be used to find the square root of a nonnegative real number. Using Newton’s method, you can also write an algorithm to find the square root of a nonnegative real number within a given tolerance as follows: Suppose x is a nonnegative real number, a is the approximate square root of x, and epsilon is the tolerance. Start with a = x.

If |a2– x| <= epsilon, then a is the square root of x within the tolerance, otherwise:

Replace a with (a2 + x) / (2a) and repeat Step a in which |a2– x| denotes the absolute value of a2– x.

Write a recursive function to implement this algorithm to find the square root of a nonnegative real number. Also, write a program to test your function.(C++)

Explanation / Answer

// C++ to determine square root recursively using newton method

#include <iostream>
#include <stdlib.h>
#include <string.h>
#include <algorithm>
#include <fstream>
#include <vector>
#include <math.h>
#include <complex.h>
#include <cmath>        // std::abs

using namespace std;

// Returns the square root of x.
float squareRoot(float x, float e, float a)
{
// base case
if (abs(a * a - x) <= e)
{
        return a;
}
else
{
      // updating square root value
        a = (a * a + x) / (2 * a);
        // recursively calling function with updated a
        return 1.0 *(squareRoot(x, e, a));
}
}

int main()
{
int x;
cout << "Enter x: ";
cin >> x;

float e;
cout << "Enter tolerance: ";
cin >> e;
/* e decides the accuracy level*/

cout << "Square root of " << x << " is " << squareRoot(x,e,x) << endl;

return 0;
}

/*
output:

Enter x: 25
Enter tolerance: 0.000001
Square root of 25 is 5

Enter x: 101
Enter tolerance: 0.001
Square root of 101 is 10.0499

*/

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