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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 Lady Windermere's Fan (mathematics) ： ウィキペディア英語版 Lady Windermere's Fan (mathematics) In mathematics, Lady Windermere's Fan is a telescopic identity employed to relate global and local error of a numerical algorithm. The name is derived from Oscar Wilde's play Lady Windermere's Fan, A Play About a Good Woman. ==Lady Windermere's Fan for a function of one variable== Let $E(\backslash \; \backslash tau,t\_0,y(t\_0)\backslash \; )$ be the exact solution operator so that: ::$y(t\_0+\backslash tau)\; =\; E(\backslash tau,t\_0,y(t\_0))\backslash \; y(t\_0)$ with $t\_0$ denoting the initial time and $y(t)$ the function to be approximated with a given $y(t\_0)$. Further let $y\_n$, $n\; \backslash in\; \backslash N,\backslash \; n\backslash le\; N$ be the numerical approximation at time $t\_n$, . $y\_n$ can be attained by means of the approximation operator $\backslash Phi(\backslash \; h\_n,t\_n,y(t\_n)\backslash \; )$ so that: ::$y\_n\; =\; \backslash Phi(\backslash \; h\_,t\_,y(t\_)\backslash \; )\backslash \; y\_\backslash quad$ with $h\_n\; =\; t\_\; -\; t\_n$ The approximation operator represents the numerical scheme used. For a simple explicit forward euler scheme with step witdth $h$ this would be: $\backslash Phi\_,y(t\_)\backslash \; )\backslash \; y(t\_)\; =\; (1\; +\; h\; \backslash frac)\backslash \; y(t\_)$ The local error $d\_n$ is then given by: ::$d\_n:=\; D(\backslash \; h\_,t\_,y(t\_\backslash \; )\backslash \; y\_\; :=\; \backslash left(\backslash Phi(\backslash \; h\_,t\_,y(t\_)\backslash \; )\; -\; E(\backslash \; h\_,t\_,y(t\_)\backslash \; )\; \backslash right)\backslash \; y\_$ In abbreviation we write: ::$\backslash Phi(h\_n)\; :=\; \backslash Phi(\backslash \; h\_n,t\_n,y(t\_n)\backslash \; )$ ::$E(h\_n)\; :=\; E(\backslash \; h\_n,t\_n,y(t\_n)\backslash \; )$ ::$D(h\_n)\; :=\; D(\backslash \; h\_n,t\_n,y(t\_n)\backslash \; )$ Then Lady Windermere's Fan for a function of a single variable $t$ writes as: $y\_N-y(t\_N)\; =\; \backslash prod\_^\backslash Phi(h\_j)\backslash \; (y\_0-y(t\_0))\; +\; \backslash sum\_^N\backslash \; \backslash prod\_^\; \backslash Phi(h\_j)\backslash \; d\_n$ with a global error of $y\_N-y(t\_N)$ 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「Lady Windermere's Fan (mathematics)」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.