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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 Newton polynomial ： ウィキペディア英語版 Newton polynomial In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using divided differences. For any given finite set of data points, there is only one polynomial, of least possible degree, that passes through all of them. Thus, it is more appropriate to speak of "the Newton form of the interpolation polynomial" rather than of "the Newton interpolation polynomial". Like the Lagrange form, it is merely another way to write the same polynomial. ==Definition== Given a set of ''k'' + 1 data points :$(x\_0,\; y\_0),\backslash ldots,(x\_k,\; y\_k)$ where no two ''x''''j'' are the same, the interpolation polynomial in the Newton form is a linear combination of Newton basis polynomials :$N(x)\; :=\; \backslash sum\_^\; a\_\; n\_(x)$ with the Newton basis polynomials defined as :$n\_j(x)\; :=\; \backslash prod\_^\; (x\; -\; x\_i)$ for ''j'' > 0 and $n\_0(x)\; \backslash equiv\; 1$. The coefficients are defined as :$a\_j\; :=\; ()$ where :$()$ is the notation for divided differences. Thus the Newton polynomial can be written as :$N(x)\; =\; ()\; +\; ()(x-x\_0)\; +\; \backslash cdots\; +\; ()(x-x\_0)(x-x\_1)\backslash cdots(x-x\_).$ The Newton Polynomial above can be expressed in a simplified form when $x\_0,\; x\_1,\; \backslash dots,\; x\_k$ are arranged consecutively with equal space. Introducing the notation $h\; =\; x\_-x\_i$ for each $i=0,1,\backslash dots,k-1$ and $x=x\_0+sh$, the difference $x-x\_i$ can be written as $(s-i)h$. So the Newton Polynomial above becomes: :$\backslash begin$ N(x) &= () + ()sh + \cdots + () s (s-1) \cdots (s-k+1)^ \\ &= \sum_^s(s-1) \cdots (s-i+1)^() \\ &= \sum_^i!^() \end is called the Newton Forward Divided Difference Formula. If the nodes are reordered as $\_,\_,\backslash dots,\_$, the Newton Polynomial becomes: :$N(x)=()+(\_)(x-\_)+\backslash cdots+()(x-\_)(x-\_)\backslash cdots(x-\_)$ If $\_,\backslash ;\_,\backslash ;\backslash dots,\backslash ;\_$ are equally spaced with x=$\_+sh$ and $\_=\_-(k-i)h$ for ''i'' = 0, 1, ..., ''k'', then, :$\backslash begin$ N(x) &= ()+ (_ )sh+\cdots+()s(s+1)\cdots(s+k-1)^ \\ &=\sum_^^i!^() \end is called the Newton Backward Divided Difference Formula. 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「Newton polynomial」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.