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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Nth root algorithm ： ウィキペディア英語版 Nth root algorithm The principal ''n''th root $\backslash sqrt()$ of a positive real number ''A'', is the positive real solution of the equation :$x^n\; =\; A$ (for integer ''n'' there are ''n'' distinct complex solutions to this equation if $A\; >\; 0$, but only one is positive and real). There is a very fast-converging ''n''th root algorithm for finding $\backslash sqrt()$: #Make an initial guess $x\_0$ #Set . #Repeat step 2 until the desired precision is reached, i.e. . A special case is the familiar square-root algorithm. By setting ''n'' = 2, the ''iteration rule'' in step 2 becomes the square root iteration rule: :$x\_\; =\; \backslash frac\backslash left(x\_k\; +\; \backslash frac\backslash right)$ Several different derivations of this algorithm are possible. One derivation shows it is a special case of Newton's method (also called the Newton-Raphson method) for finding zeros of a function $f(x)$ beginning with an initial guess. Although Newton's method is iterative, meaning it approaches the solution through a series of increasingly accurate guesses, it converges very quickly. The rate of convergence is quadratic, meaning roughly that the number of bits of accuracy doubles on each iteration (so improving a guess from 1 bit to 64 bits of precision requires only 6 iterations). For this reason, this algorithm is often used in computers as a very fast method to calculate square roots. For large ''n'', the ''n''th root algorithm is somewhat less efficient since it requires the computation of $x\_k^$ at each step, but can be efficiently implemented with a good exponentiation algorithm. == Derivation from Newton's method == Newton's method is a method for finding a zero of a function ''f(x)''. The general iteration scheme is: #Make an initial guess $x\_0$ #Set $x\_\; =\; x\_k\; -\; \backslash frac$ #Repeat step 2 until the desired precision is reached. The ''n''th root problem can be viewed as searching for a zero of the function :$f(x)\; =\; x^n\; -\; A$ So the derivative is :$f^\backslash prime(x)\; =\; n\; x^$ and the iteration rule is :$x\_\; =\; x\_k\; -\; \backslash frac$ : ■ウィキペディアで「Nth root algorithm」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.