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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 Bairstow's method ： ウィキペディア英語版 Bairstow's method In numerical analysis, Bairstow's method is an efficient algorithm for finding the roots of a real polynomial of arbitrary degree. The algorithm first appeared in the appendix of the 1920 book "Applied Aerodynamics" by Leonard Bairstow. The algorithm finds the roots in complex conjugate pairs using only real arithmetic. See root-finding algorithm for other algorithms. ==Description of the method== Bairstow's approach is to use Newton's method to adjust the coefficients ''u'' and ''v'' in the quadratic $x^2\; +\; ux\; +\; v$ until its roots are also roots of the polynomial being solved. The roots of the quadratic may then be determined, and the polynomial may be divided by the quadratic to eliminate those roots. This process is then iterated until the polynomial becomes quadratic or linear, and all the roots have been determined. Long division of the polynomial to be solved :$P(x)=\backslash sum\_^n\; a\_i\; x^i$ by $x^2\; +\; ux\; +\; v$ yields a quotient $Q(x)=\backslash sum\_^\; b\_i\; x^i$ and a remainder $cx+d$ such that :$P(x)=(x^2+ux+v)\backslash left(\backslash sum\_^\; b\_i\; x^i\backslash right)\; +\; (cx+d).$ A second division of $Q(x)$ by $x^2\; +\; ux\; +\; v$ is performed to yield a quotient $R(x)=\backslash sum\_^\; f\_i\; x^i$ and remainder $gx+h$ with :$Q(x)=(x^2+ux+v)\backslash left(\backslash sum\_^\; f\_i\; x^i\backslash right)\; +\; (gx+h).$ The variables $c,\backslash ,d,\backslash ,g,\backslash ,h$, and the $\backslash ,\backslash ;\backslash$ are functions of $u$ and $v$. They can be found recursively as follows. :$\backslash begin$ b_n &= b_ = 0,& f_n &= f_ = 0,\\ b_i &= a_-ub_-vb_&f_i &= b_-uf_-vf_ \qquad (i=n-2,\ldots,0),\\ c &= a_1-ub_0-vb_1,& g &= b_1-uf_0-vf_1,\\ d & =a_0-vb_0,& h & =b_0-vf_0. \end The quadratic evenly divides the polynomial when :$c(u,v)=d(u,v)=0.\; \backslash ,$ Values of $u$ and $v$ for which this occurs can be discovered by picking starting values and iterating Newton's method in two dimensions :$$ \beginu\\ v\end := \beginu\\ v\end - \begin \frac&\frac\\() \frac &\frac \end^ \beginc\\ d\end := \beginu\\ v\end - \frac \begin -h & g\\() -gv & gu-h \end \beginc\\ d\end until convergence occurs. This method to find the zeroes of polynomials can thus be easily implemented with a programming language or even a spreadsheet. 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「Bairstow's method」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.