Step by Step solution about How to resolve the algorithm Nth root step by step in the C# programming language

The provided C# code defines a method called NthRoot that calculates the nth root of a number A using the Newton-Raphson method, with a specified precision p.

Here's a detailed explanation of the code:

1. Main Method:

   • The program starts with a Main method that calls the NthRoot method twice to calculate the square root of 81 and the cube root of 1000, with a precision of 0.001.

2. NthRoot Method:

   • The NthRoot method takes three parameters:
     • A: The number to find the nth root of.
     • n: The nth root to calculate.
     • p: The precision or tolerance for the calculation.
   • It calculates the nth root of A using the Newton-Raphson method, which is an iterative process that starts with an initial guess and refines it until it meets the specified precision.

3. Newton-Raphson Iteration:

   • The method initializes two arrays of doubles, x, with two elements, x[0] and x[1].
   • It sets x[0] to the input A and x[1] to A divided by n, which is the initial guess for the nth root.
   • The method enters a loop that continues until the absolute difference between x[0] and x[1] is less than or equal to the specified precision p.
   • Inside the loop, it updates x[1] with the value of x[0], then calculates x[0] using the Newton-Raphson formula:
     • The formula is: x[0] = (1/_n)*(((_n-1)*x[1]) + (A/Math.Pow(x[1],_n-1)))
     • This formula is a recursive definition of the nth root and it involves the previous approximation x[1].

4. Convergence:

   • The loop continues until the difference between x[0] and x[1] becomes smaller than the specified precision, at which point the method considers the current value of x[0] to be the nth root of A.

5. Return Value:

   • The method returns x[0], which is the calculated nth root of A to the specified precision p.

In summary, this code provides a function to calculate the nth root of a number using the Newton-Raphson method, which gives progressively better approximations until reaching the desired precision.

static void Main(string[] args)
{
	Console.WriteLine(NthRoot(81,2,.001));
        Console.WriteLine(NthRoot(1000,3,.001));
        Console.ReadLine();
}

public static double NthRoot(double A,int n,  double p)
{
	double _n= (double) n;
	double[] x = new double[2];		
	x[0] = A;
	x[1] = A/_n;
	while(Math.Abs(x[0] -x[1] ) > p)
	{
		x[1] = x[0];
		x[0] = (1/_n)*(((_n-1)*x[1]) + (A/Math.Pow(x[1],_n-1)));
			
	}
	return x[0];
}