Simpson's Rule


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Simpson's rule

[′sim·sənz ‚rül]
(mathematics)
Also known as parabolic rule.
A basic approximation formula for definite integrals which states that the integral of a real-valued function ƒ on an interval [a,b ] is approximated by h [ƒ(a) + 4ƒ(g + h) + ƒ(b)]/3, where h = (b-a)/2; this is the area under a parabola which coincides with the graph of ƒ at the abscissas a, a + h, and b.
A method of approximating a definite integral over an interval which is equivalent to dividing the interval into equal subintervals and applying the formula in the first definition to each subinterval.
(petroleum engineering)
A mathematical relationship for calculating the oil- or gas-bearing net-pay volume of a reservoir; uses the contour lines from a subsurface geological map of the reservoir, including gas-oil and gas-water contacts.

Simpson’s Rule

 

a formula for approximating definite integrals. In has the form

where h = (b - a)/2n and fi = f(a + ih), i = 0, 1, 2,…, 2n.

The derivation of Simpson’s rule is based on the replacement of the integrand f(x) on each of the closed intervals [a + 2hk, a + 2h(k + 1)], k = 0, 1,…, n - 1, by a corresponding interpolation polynomial of the second degree (seeINTERPOLATION FORMULAS). In geometric terms, the curve described by the equation y = f(x) is replaced by an approximating curve consisting of segments of parabolas. The error resulting from the use of Simpson’s rule is

where aξb. If f(x) is a polynomial of degree m ≤ 3, then Simpson’s rule is not approximate but exact, since in this case f(4)(x) = 0.

Simpson’s rule is named after T. Simpson, who devised it in 1743. The rule was, however, known earlier. For example, it was given by J. Gregory in 1668.

Other formulas for the approximate calculation of definite integrals are discussed in the article APPROXIMATE INTEGRATION.

References in periodicals archive ?
To test different numerical integration methods, needed for Dirlik's method, the Midpoint Rule, Trapezoidal Rule and Simpson's Rule were compared by integrating over a Rayleigh distribution for various probabilities from .
01%, the minimum number of steps needed using the Midpoint of Trapezoidal Rule is 256 and when using Simpson's Rule was 32 steps.
During preliminary tests integrating the Rayleigh probability density function, the trapezoidal rule method was found to converge faster than the midpoint rule and to be easier than Simpson's rule to set up the formulas.
In the numerical analysis, one method for numerical integration is Simpson's rule or Simpson's method.
If the interval of integration is "small", then Simpson's rule will provide an adequate approximation of the exact integral.
How much support Simpson's rule changes will draw, though, remains unclear.
Use of Simpson's Rule requires determination of half-breadth distances (or distance from centerline to the hull of the boat) and the area of complex shapes if this calculation is to be performed manually, Figure 5.
In particular, students would be familiar with the midpoint rule, the elementary trapezoidal rule and Simpson's rule.
Simpson's rule integrates quadratics as well as cubics exactly.
These results demonstrated that the vertical motions were similar for the two finite differencing methods, and that the Simpson's Rule integrations produced the most robust vertical motions.
Simpson's rule is a fairly standard introduction to the fundamental theorem of Calculus, but teachers are equally comfortable with using it as a postscript to integration, justifying its inclusion as an escape clause for functions whose integrals have difficult or non-existent closed forms.
One of the problems with using Simpson's rule as an introduction to integral calculus is that the proof of the remarkably simple formula depends on knowledge of integration of quadratic functions.