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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 Polynomial SOS ： ウィキペディア英語版 Polynomial SOS In mathematics, a form (i.e. a homogeneous polynomial) ''h''(''x'') of degree 2''m'' in the real ''n''-dimensional vector ''x'' is sum of squares of forms (SOS) if and only if there exist forms $g\_1(x),\backslash ldots,g\_k(x)$ of degree ''m'' such that :$$ h(x)=\sum_^k g_i(x)^2 . Explicit sufficient conditions for a form to be SOS have been found. However every real nonnegative form can be approximated as closely as desired (in the $l\_1$-norm of its coefficient vector) by a sequence of forms $\backslash$ that are SOS. == Square matricial representation (SMR) == To establish whether a form ''h''(''x'') is SOS amounts to solving a convex optimization problem. Indeed, any ''h''(''x'') can be written as :$$ h(x)=x^\left(H+L(\alpha)\right)x^} is a vector containing a base for the forms of degree ''m'' in ''x'' (such as all monomials of degree ''m'' in ''x''), the prime ′ denotes the transpose, ''H'' is any symmetric matrix satisfying :$$ h(x)=x^Hx^=\left\ L x^. The dimension of the vector $x^$ whereas the dimension of the vector $\backslash alpha$ is given by :$$ \omega(n,2m)=\frac\sigma(n,m)\left(1+\sigma(n,m)\right)-\sigma(n,2m). Then, ''h''(''x'') is SOS if and only if there exists a vector $\backslash alpha$ such that :$$ H + L(\alpha) \ge 0, meaning that the matrix $H\; +\; L(\backslash alpha)$ is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression $h(x)=x^\backslash left(H+L(\backslash alpha)\backslash right)x^\{\backslash \{m\backslash \}\}$ was introduced in with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI. This representation is also known as Gram matrix. 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「Polynomial SOS」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.