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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 Carleman matrix ： ウィキペディア英語版 Carleman matrix In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains. ==Definition== The Carleman matrix of a function $f(x)$ is defined as: :$M()\_\; =\; \backslash frac\backslash left((f(x))^j\; \backslash right)\_\; ~,$ so as to satisfy the (Taylor series) equation: :$(f(x))^j\; =\; \backslash sum\_^\; M()\_\; x^k.$ ---- For instance, the computation of $f(x)$ by :$f(x)\; =\; \backslash sum\_^\; M()\_\; x^k.\; ~$ simply amounts to the dot-product of row 1 of $M()$ with a column vector $\backslash left()^\backslash tau$. The entries of $M()$ in the next row give the 2nd power of $f(x)$: :$f(x)^2\; =\; \backslash sum\_^\; M()\_\; x^k\; ~,$ and also, in order to have the zero'th power of $f(x)$ in $M()$, we aadopt the row 0 containing zeros everywhere except the first position, such that :$f(x)^0\; =\; 1\; =\; \backslash sum\_^\; M()\_\; x^k\; =\; 1+\; \backslash sum\_^\; 0$ * x^k ~. Thus, the dot product of $M()$ with the column vector $\backslash left()^\backslash tau$ yields the column vector $\backslash left()^\backslash tau$ :$M()$ * \left(1,x,x^2,x^3,...\right )^\tau = \left(1,f(x),(f(x))^2,(f(x))^3,...\right )^\tau. 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「Carleman matrix」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.