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

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++)

Solution

// 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

*/