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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 Lattice reduction ： ウィキペディア英語版 Lattice reduction In mathematics, the goal of lattice basis reduction is given an integer lattice basis as input, to find a basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice. ==Nearly Orthogonal== One measure of ''nearly orthogonal'' is the orthogonality defect. This compares the product of the lengths of the basis vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same. Any particular basis of $n$ vectors may be represented by a matrix $B$, whose columns are the basis vectors $b\_i,\; i\; =\; 1,\; \backslash ldots,\; n$. In the fully dimensional case where the number of basis vectors is equal to the dimension of the space they occupy, this matrix is square, and the volume of the fundamental parallelepiped is simply the absolute value of the determinant of this matrix $\backslash det(B)$. If the number of vectors is less than the dimension of the underlying space, then volume is $\backslash sqrt$. For a given lattice $\backslash Lambda$, this volume is the same (up to sign) for any basis, and hence is referred to as the determinant of the lattice $\backslash det(\backslash Lambda)$ or lattice constant $d(\backslash Lambda)$. The orthogonality defect is the product of the basis vector lengths divided by the parallelepiped volume; :$\backslash delta(B)\; =\; \backslash frac^n\; ||b\_i||\}$ From the geometric definition it may be appreciated that $\backslash delta(B)\; \backslash ge\; 1$ with equality if and only if the basis is orthogonal. If the lattice reduction problem is defined as finding the basis with the smallest possible defect, then the problem is NP complete. However, there exist polynomial time algorithms to find a basis with defect $\backslash delta(B)\; \backslash le\; c$ where ''c'' is some constant depending only on the number of basis vectors and the dimension of the underlying space (if different). This is a good enough solution in many practical applications. 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「Lattice reduction」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.