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The Simpsons Updated 10 months ago 0 Comment. The Simpsons 7 Updated 10 months ago 0 Comment. In numerical analysis , Simpson's rule is a method for numerical integration , the numerical approximation of definite integrals.

Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule. In English, the method is credited to the mathematician Thomas Simpson — of Leicestershire, England.

However, Johannes Kepler used similar formulas over years prior, and for this reason, the method is sometimes called Kepler's rule , or Keplersche Fassregel Kepler's barrel rule in German.

One can use Lagrange polynomial interpolation to find an expression for this polynomial,. Using integration by substitution one can show that [1].

Another derivation constructs Simpson's rule from two simpler approximations: the midpoint rule. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the weighted average.

Using another approximation for example, the trapezoidal rule with twice as many points , it is possible to take a suitable weighted average and eliminate another error term.

This is Romberg's method. This yields Simpson's rule. For such a function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

However, it is often the case that the function we are trying to integrate is not smooth over the interval.

Typically, this means that either the function is highly oscillatory, or it lacks derivatives at certain points. In these cases, Simpson's rule may give very poor results.

Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval.

This sort of approach is termed the composite Simpson's rule. Then, the composite Simpson's rule is given by. In practice, it is often advantageous to use subintervals of different lengths, and concentrate the efforts on the places where the integrand is less well-behaved.

This leads to the adaptive Simpson's method. 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].

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