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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 Fourier–Motzkin elimination ： ウィキペディア英語版 Fourier–Motzkin elimination Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier and Theodore Motzkin. ==Elimination== The elimination of a set of variables, say ''V'', from a system of relations (here linear inequalities) refers to the creation of another system of the same sort, but without the variables in ''V'', such that both systems have the same solutions over the remaining variables. If all variables are eliminated from a system of linear inequalities, then one obtains a system of constant inequalities. It is then trivial to decide whether the resulting system is true or false. It is true if and only if the original system has solutions. As a consequence, elimination of all variables can be used to detect whether a system of inequalities has solutions or not. Consider a system $S$ of $n$ inequalities with $r$ variables $x\_1$ to $x\_r$, with $x\_r$ the variable to be eliminated. The linear inequalities in the system can be grouped into three classes depending on the sign (positive, negative or null) of the coefficient for $x\_r$. * those inequalities that are of the form $x\_r\; \backslash geq\; b\_i-\backslash sum\_^\; a\_\; x\_k$; denote these by $x\_r\; \backslash geq\; A\_j(x\_1,\; \backslash dots,\; x\_)$, for $j$ ranging from 1 to $n\_A$ where $n\_A$ is the number of such inequalities; * those inequalities that are of the form $x\_r\; \backslash leq\; b\_i-\backslash sum\_^\; a\_\; x\_k$; denote these by $x\_r\; \backslash leq\; B\_j(x\_1,\; \backslash dots,\; x\_)$, for $j$ ranging from 1 to $n\_B$ where $n\_B$ is the number of such inequalities; * those inequalities in which $x\_r$ plays no role, grouped into a single conjunction $\backslash phi$. The original system is thus equivalent to :$\backslash max(A\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; A\_(x\_1,\; \backslash dots,\; x\_))\; \backslash leq\; x\_r\; \backslash leq\; \backslash min(B\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; B\_(x\_1,\; \backslash dots,\; x\_))\; \backslash wedge\; \backslash phi$. Elimination consists in producing a system equivalent to $\backslash exists\; x\_r~S$. Obviously, this formula is equivalent to :$\backslash max(A\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; A\_(x\_1,\; \backslash dots,\; x\_))\; \backslash leq\; \backslash min(B\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; B\_(x\_1,\; \backslash dots,\; x\_))\; \backslash wedge\; \backslash phi$. The inequality :$\backslash max(A\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; A\_(x\_1,\; \backslash dots,\; x\_))\; \backslash leq\; \backslash min(B\_1(x\_1,\; \backslash dots,\; x\_),\; \backslash dots,\; B\_(x\_1,\; \backslash dots,\; x\_))$ is equivalent to $n\_A\; n\_B$ inequalities $A\_i(x\_1,\; \backslash dots,\; x\_)\; \backslash leq\; B\_j(x\_1,\; \backslash dots,\; x\_)$, for $1\; \backslash leq\; i\; \backslash leq\; n\_A$ and $1\; \backslash leq\; j\; \backslash leq\; n\_B$. We have therefore transformed the original system into another system where $x\_r$ is eliminated. Note that the output system has $(n-n\_A-n\_B)+n\_A\; n\_B$ inequalities. In particular, if $n\_A\; =\; n\_B\; =\; n/2$, then the number of output inequalities is $n^2/4$. 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「Fourier–Motzkin elimination」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.