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It is based upon a cubic interpolation rather than a quadratic interpolation. A further generalization of this concept for interpolation with arbitrary-degree polynomials are the Newton—Cotes formulas.
This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding: [5].
The result is then obtained by taking the mean of the two formulas. In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule.
No wonder that error of the sum corresponds to less accurate term. Those rules are very much similar to Press's alternative extended Simpson's rule.
Coefficients within the major part of the region being integrated equal one, differences are only at the edges.
These three rules can be associated with Euler-MacLaurin formula with the first derivative term and named Euler-MacLaurin integration rules. Then the composite Simpson's rule is given by [7] [8].
From Wikipedia, the free encyclopedia. Method for numerical integration. For Simpson's voting rule, see Minimax Condorcet. For the application to ship stability, see Simpson's rules ship stability.
Chemometrics and Intelligent Laboratory Systems. Computational Physics course. Tampere University.
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