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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 Backward Euler method ： ウィキペディア英語版 Backward Euler method In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the (standard) Euler method, but differs in that it is an implicit method. The backward Euler method has order one and is A-stable. == Description == Consider the ordinary differential equation :$\backslash frac\; =\; f(t,y)$ with initial value $y(t\_0)\; =\; y\_0.$ Here the function $f$ and the initial data $t\_0$ and $y\_0$ are known; the function $y$ depends on the real variable $t$ and is unknown. A numerical method produces a sequence $y\_0,\; y\_1,\; y\_2,\; \backslash ldots$ such that $y\_k$ approximates $y(t\_0+kh)$, where $h$ is called the step size. The backward Euler method computes the approximations using :$y\_\; =\; y\_k\; +\; h\; f(t\_,\; y\_).$ This differs from the (forward) Euler method in that the latter uses $f(t\_k,\; y\_k)$ in place of $f(t\_,\; y\_)$. The backward Euler method is an implicit method: the new approximation $y\_$ appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown $y\_$. Sometimes, this can be done by fixed-point iteration: :$y\_^\; =\; y\_k,\; \backslash quad\; y\_^\; =\; y\_k\; +\; h\; f(t\_,\; y\_^).$ If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation $y\_$. Alternatively, one can use (some modification of) the Newton–Raphson method to solve the algebraic equation. 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「Backward Euler method」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.