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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 Direct multiple shooting method ： ウィキペディア英語版 Direct multiple shooting method In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods. == Single shooting methods == Shooting methods can be used to solve boundary value problems (BVP) like :$y\text{'}\text{'}(t)\; =\; f(t,\; y(t),\; y\text{'}(t)),\; \backslash quad\; y(t\_a)\; =\; y\_a,\; \backslash quad\; y(t\_b)\; =\; y\_b,$ in which the time points ''t''a and ''t''b are known and we seek :$y(t),\backslash quad\; t\; \backslash in\; (t\_a,t\_b).$ Single shooting methods proceed as follows. Let ''y''(''t''; ''t''0, ''y''0) denote the solution of the initial value problem (IVP) :$y\text{'}\text{'}(t)\; =\; f(t,\; y(t),\; y\text{'}(t)),\; \backslash quad\; y(t\_0)\; =\; y\_0,\; \backslash quad\; y\text{'}(t\_0)\; =\; p$ Define the function ''F''(''p'') as the difference between ''y''(''t''''b''; ''p'') and the specified boundary value ''y''''b'': ''F''(''p'') = ''y''(''t''''b''; ''p'') − ''y''''b''. Then for every solution (''y''a, ''y''b) of the boundary value problem we have ''y''a=''y''0 while ''y''b corresponds to a root of ''F''. This root can be solved by any root-finding method given that certain method-dependent prerequisites are satisfied. This often will require initial guesses to ''y''a and ''y''b. Typically, analytic root finding is impossible and iterative methods such as Newton's method are used for this task. The application of single shooting for the numerical solution of boundary value problems suffers from several drawbacks. * For a given initial value ''y''0 the solution of the IVP obviously must exist on the interval () so that we can evaluate the function ''F'' whose root is sought. For highly nonlinear or unstable ODEs, this requires the initial guess ''y''0 to be extremely close to an actual but unknown solution ''y''a. Initial values that are chosen slightly off the true solution may lead to singularities or breakdown of the ODE solver method. Choosing such solutions is inevitable in an iterative root-finding method, however. * Finite precision numerics may make it impossible at all to find initial values that allow for the solution of the ODE on the whole time interval. * The nonlinearity of the ODE effectively becomes a nonlinearity of ''F'', and requires a root-finding technique capable of solving nonlinear systems. Such methods typically converge slower as nonlinearities become more severe. The boundary value problem solver's performance suffers from this. * Even stable and well-conditioned ODEs may make for unstable and ill-conditioned BVPs. A slight alteration of the initial value guess ''y''0 may generate an extremely large step in the ODEs solution ''y''(''t''b; ''t''a, ''y''0) and thus in the values of the function ''F'' whose root is sought. Non-analytic root-finding methods can seldom cope with this behaviour. 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「Direct multiple shooting 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.