how to use newton-raphson to find reciprocal

To use Newton-Raphson method to find the reciprocal of a number, you can follow these steps:

1. Let the number whose reciprocal you want to find be denoted by x.

2. Define a function f(y) = 1/y - x, where y is the reciprocal of x.

3. Set an initial guess for the reciprocal, say y0.

4. Apply the Newton-Raphson formula to iteratively improve your estimate for y:

y_n+1 = y_n - f(y_n)/f'(y_n)

where f'(y) is the derivative of f(y).

5. Repeat step 4 until the estimate for y converges to a desired level of accuracy.

6. The reciprocal of x is equal to the final estimate for y.

Here's some Python code to implement the Newton-Raphson method for finding the reciprocal of a number:

def reciprocal(x, y0, tol=1e-6, max_iter=100):
    def f(y):
        return 1/y - x
    
    def df(y):
        return -1/y**2
    
    y = y0
    for i in range(max_iter):
        y_new = y - f(y)/df(y)
        if abs(y_new - y) < tol:
            return y_new
        y = y_new
    return y

You can call this function with the value of x and an initial guess for the reciprocal y0, and it will return the reciprocal of x to within a tolerance of tol (default is 1e-6) after at most max_iter iterations (default is 100).