Example 9: Newton-Raphson method is used to find the pth root of a positive real number R. Set up the iteration scheme. Perform three iterations of the method for R = 16.0, p = 3, starting with the initial approximation 2.0.
Solution: Let us denote the pth root of R by x. We have R = x^p. The Newton-Raphson Iteration scheme is given by x_(k+1) = x_k - (f(x_k))/(f'(x_k)), where f(x) = x^p - R.
On simplification, we get x_(k+1) = x_k - (x_k^p - R)/(p*x_k^(p-1)).