The Newton-Raphson Method is an iterative algorithm for finding a zero of a function given the estimate of the zero. The method uses the derivative of the function and iterates the current value to find the next value using the formula:
Then, the absolute relative approximate error is found using:
While the Newton-Raphson method is a powerful root-finding algorithm , it has some shortcomings the most obvious of which is when the derivative of a particular iterate in the process equals zero.The pre-specified relative error tolerance is taken as 0.00001 in my calculations.
Starting with an initial guess of 6, we get: x_(1)=x_(0)-f(x_0)/(f'(x_0))=6-215/108=4.00 bar(925)
The subsequent estimates are found as:
x_(8)=x_(7)-f(x_7)/(f'(x_7))$\approx$1.000000 leading us to the correct root, 1 , after eight iterations.