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In numerical analysis, the Newton–Cotes formulae, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulae for numerical integration (also called ''quadrature'') based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes. Newton–Cotes formulae can be useful if the value of the integrand at equally spaced points is given. If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are probably more suitable. ==Description== It is assumed that the value of a function ''ƒ'' defined on () is known at equally spaced points ''x''''i'', for ''i'' = 0, …, ''n'', where ''x''0 = ''a'' and ''x''''n'' = ''b''. There are two types of Newton–Cotes formulae, the "closed" type which uses the function value at all points, and the "open" type which does not use the function values at the endpoints. The closed Newton–Cotes formula of degree ''n'' is stated as : where , with ''h'' (called the ''step size'') equal to . The ''w''''i'' are called ''weights''. As can be seen in the following derivation the weights are derived from the Lagrange basis polynomials. They depend only on the ''x''''i'' and not on the function ''ƒ''. Let ''L''(''x'') be the interpolation polynomial in the Lagrange form for the given data points , then : The open Newton–Cotes formula of degree ''n'' is stated as : The weights are found in a manner similar to the closed 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–Cotes formulas」の詳細全文を読む スポンサード リンク
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