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Newton–Cotes formulas, also called the Newton–Cotes quadrature rules
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Bog'liq AL MIInstability for high degree[edit]
A Newton–Cotes formula of any degree n can be constructed. However, for large n a Newton–Cotes rule can sometimes suffer from catastrophic Runge's phenomenon[2] where the error grows exponentially for large n. Methods such as Gaussian quadrature and Clenshaw–Curtis quadrature with unequally spaced points (clustered at the endpoints of the integration interval) are stable and much more accurate, and are normally preferred to Newton–Cotes. If these methods cannot be used, because the integrand is only given at the fixed equidistributed grid, then Runge's phenomenon can be avoided by using a composite rule, as explained below.
Alternatively, stable Newton–Cotes formulas can be constructed using least-squares approximation instead of interpolation. This allows building numerically stable formulas even for high degrees.[3][4]
Closed Newton–Cotes formulas[edit]
This table lists some of the Newton–Cotes formulas of the closed type. For , let where , and .
Closed Newton–Cotes Formulas
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n
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Step size h
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Common name
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Formula
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Error term
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1
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Trapezoidal rule
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2
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Simpson's rule
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3
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Simpson's 3/8 rule
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4
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Boole's rule
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Boole's rule is sometimes mistakenly called Bode's rule, as a result of the propagation of a typographical error in Abramowitz and Stegun, an early reference book.[5]
The exponent of the step size h in the error term gives the rate at which the approximation error decreases. The order of the derivative of f in the error term gives the lowest degree of a polynomial which can no longer be integrated exactly (i.e. with error equal to zero) with this rule. The number must be taken from the interval (a,b), therefore, the error bound is equal to the error term when .
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