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In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. : The Newton–Raphson method in one variable is implemented as follows: Given a function ''ƒ'' defined over the reals ''x'', and its derivative ''ƒ, we begin with a first guess ''x''0 for a root of the function ''f''. Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation ''x''1 is : Geometrically, (''x''1, 0) is the intersection with the ''x''-axis of the tangent to the graph of ''f'' at (''x''0, ''f'' (''x''0)). The process is repeated as : until a sufficiently accurate value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations. ==Description== The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the ''x''-intercept of this tangent line (which is easily done with elementary algebra). This ''x''-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated. Suppose ''ƒ'' : () → R is a differentiable function defined on the interval () with values in the real numbers R. The formula for converging on the root can be easily derived. Suppose we have some current approximation ''x''''n''. Then we can derive the formula for a better approximation, ''x''''n''+1 by referring to the diagram on the right. The equation of the tangent line to the curve ''y'' = ''ƒ''(''x'') at the point ''x=x''''n'' is : where, ''ƒ denotes the derivative of the function ''ƒ''. The ''x''-intercept of this line (the value of ''x'' such that ''y''=0) is then used as the next approximation to the root, ''x''''n''+1. In other words, setting ''y'' to zero and ''x'' to ''x''''n''+1 gives : Solving for ''x''''n''+1 gives : We start the process off with some arbitrary initial value ''x''0. (The closer to the zero, the better. 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 ''ƒ(''x''0) ≠ 0. 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 at least doubles in every step. More details can be found in the analysis section below. The Householder's methods are similar but have higher order for even faster convergence. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Newton's method」の詳細全文を読む スポンサード リンク
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