However, the extra computations required for each step can slow down the overall performance relative to Newton's method, particularly if f or its derivatives are computationally expensive to evaluate. Householder's methods are similar but have higher order for even faster convergence. More details can be found in the analysis section below. Furthermore, for a zero of multiplicity 1, the convergence is at least quadratic (see rate of convergence) in a neighbourhood of the zero, which intuitively means that the number of correct digits roughly doubles in every step. But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that f ′( x 0) ≠ 0. We start the process with some arbitrary initial value x 0. If the function satisfies sufficient assumptions and the initial guess is close, then The most basic version starts with a single-variable function f defined for a real variable x, the function's derivative f′, and an initial guess x 0 for a root of f.
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. For Newton's method for finding minima, see Newton's method in optimization.
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