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The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads [() + (+) + ()]. In German and some other languages, it is named after Johannes Kepler , who derived it in 1615 after seeing it used for wine barrels (barrel rule, Keplersche Fassregel ).
Simpson's rules are a set of rules used in ship stability and naval architecture, to calculate the areas and volumes of irregular figures. [1] This is an application of Simpson's rule for finding the values of an integral , here interpreted as the area under a curve.
Simpson's rules (ship stability) Simpson–Kramer method This page was last edited on 29 March 2018, at 20:51 (UTC). Text is available under the Creative Commons ...
Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been found 100 years earlier by Johannes Kepler , and in German it is called Keplersche Fassregel , or ...
Simpson's rule, which is based on a polynomial of order 2, is also a Newton–Cotes formula. Quadrature rules with equally spaced points have the very convenient property of nesting. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used.
Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration proposed by G.F. Kuncir in 1962. [1] It is probably the first recursive adaptive algorithm for numerical integration to appear in print, [ 2 ] although more modern adaptive methods based on Gauss–Kronrod quadrature and Clenshaw–Curtis ...
The zeroeth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2 n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2 n + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2 n + 1 points. The further extrapolations differ from Newton-Cotes formulas.
While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.