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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 Runge-Kutta ： ウィキペディア英語版 Runge–Kutta methods In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods, which are used in temporal discretization for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M. W. Kutta. See the article on numerical methods for ordinary differential equations for more background and other methods. See also List of Runge–Kutta methods. ==''The'' Runge–Kutta method== One member of the family of Runge–Kutta methods is often referred to as "RK4", "classical Runge–Kutta method" or simply as "''the'' Runge–Kutta method". Let an initial value problem be specified as follows. :$\backslash dot\; =\; f(t,\; y),\; \backslash quad\; y(t\_0)\; =\; y\_0.$ Here, ''y'' is an unknown function (scalar or vector) of time ''t'' which we would like to approximate; we are told that $\backslash dot$, the rate at which ''y'' changes, is a function of ''t'' and of ''y'' itself. At the initial time $t\_0$ the corresponding ''y''-value is $y\_0$. The function ''f'' and the data $t\_0$, $y\_0$ are given. Now pick a step-size ''h''>0 and define :$\backslash begin$ y_ &= y_n + \tfrac\left(k_1 + 2k_2 + 2k_3 + k_4 \right)\\ t_ &= t_n + h \\ \end for ''n'' = 0, 1, 2, 3, . . . , using :$$ \begin k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n + \tfrac, y_n + \tfrac k_1), \\ k_3 &= f(t_n + \tfrac, y_n + \tfrac k_2), \\ k_4 &= f(t_n + h, y_n + hk_3). \end 〔; 〕 :''(Note: the above equations have different but equivalent definitions in different texts).''〔, , and leave out the factor ''h'' in the definition of the stages. , and use the ''y''-values as stages.〕 Here $y\_$ is the RK4 approximation of $y(t\_)$, and the next value ($y\_$) is determined by the present value ($y\_n$) plus the weighted average of four increments, where each increment is the product of the size of the interval, ''h'', and an estimated slope specified by function ''f'' on the right-hand side of the differential equation. * $k\_1$ is the increment based on the slope at the beginning of the interval, using $$, (Euler's method) ; * $k\_2$ is the increment based on the slope at the midpoint of the interval, using $+\; \backslash tfrack\_1$ ; * $k\_3$ is again the increment based on the slope at the midpoint, but now using $+\; \backslash tfrack\_2$ ; * $k\_4$ is the increment based on the slope at the end of the interval, using $+\; hk\_3$ . In averaging the four increments, greater weight is given to the increments at the midpoint. If $f$ is independent of $y$, so that the differential equation is equivalent to a simple integral, then RK4 is Simpson's rule. The RK4 method is a fourth-order method, meaning that the local truncation error is on the order of $O(h^5)$, while the total accumulated error is order $O(h^4)$. 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「Runge–Kutta methods」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.