The Newton-Raphson method requires an initial estimate of the location of the root. Following the evaluation of the function and its derivative at the estimated root, the algorithm extrapolates to an improved estimate.
Iteration leads to rapid convergence provided the initial estimate is suitably located.
A more definitive convergence check might be to test the function value slightly above and below the estimated root for a change of sign. Newton's method for determining the root of an equation. These shortcuts save three multiplication steps in the function evaluation and one multiplication in the derivative calculation compared with Equations 40 and 41. Situations exist in which the Newton-Raphson algorithm will be led astray or converge slowly. The major cause of this difficulty is an unfortunate or poor initial estimate of the root.