Adaptive Simpson's method

Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration proposed by G.F. Kuncir in 1962. It is probably the first recursive adaptive algorithm for numerical integration to appear in print, although more modern adaptive methods based on Gauss–Kronrod quadrature and Clenshaw–Curtis quadrature are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than composite Simpson's rule since it uses fewer function evaluations in places where the function is well-approximated by a cubic function.

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A criterion for determining when to stop subdividing an interval, suggested by J.N. Lyness, is

where [ a , b ] is an interval with midpoint c , S ( a , b ) , S ( a , c ) , and S ( c , b ) are the estimates given by Simpson's rule on the corresponding intervals and ϵ is the desired tolerance for the interval.

Simpson's rule is an interpolatory quadrature rule which is exact when the integrand is a polynomial of degree three or lower. Using Richardson extrapolation, the more accurate Simpson estimate S ( a , c ) + S ( c , b ) for six function values is combined with the less accurate estimate S ( a , b ) for three function values by applying the correction [ S ( a , c ) + S ( c , b ) − S ( a , b ) ] / 15 . The thus obtained estimate is exact for polynomials of degree five or less.