- Adaptive Simpson's method
Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of
numerical integration proposed by William M. McKeeman in 1962.William M. McKeeman: Algorithm 145: Adaptive numerical integration by Simpson's rule. Commun. ACM 5(12): 604 (1962). doi|10.1145/355580.369102] It is perhaps the first adaptive algorithm for numerical integration to appear in print, although more modern adaptive methods based on Gauss–Kronrod quadrature andClenshaw–Curtis quadrature are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral usingSimpson'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 aquadratic function .The criterion for determining when to stop subdividing an interval is
:
where is an interval with midpoint , , , and are the estimates given by Simpson's rule on the corresponding intervals and is the desired tolerance for the interval.Numerical Analysis. Mathematics of Scientific Computing Third Edition. by David Kincaid and Ward Cheney. Published by Brooks/Cole Third Edition, 2002]
Simpson's rule is a special case of
Romberg's method . The theory of this method shows that Simpson's rule is exact when the integrand is a polynomial of degree three or lower. According toRomberg's method , the more accurate Simpson estimate for six function values is combined with the less accurate estimate for three function values by applying the correction . Here the constant 15 is chosen to ensure that after applying the correction, an estimate is obtained that is exact for polynomials of degree five or less.Here is an implementation of adaptive Simpson's method in Python. Note that this is explanatory code, without regard for efficiency. Every call to recursive_asr entails six function evaluations. For actual use, one will want to modify it so that the minimum of two function evaluations are performed.
Bibliography
External links
* [http://math.fullerton.edu/mathews/n2003/AdaptiveQuadMod.html Module for Adaptive Simpson's Rule]
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