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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Simpson’s Rule

a formula for approximating definite integrals. In has the form

where h = (b - a)/2n and f_i = 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 a ≤ ξ ≤ b. 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.

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References in periodicals archive

He transforms the Volterra integrodifferential equation in a matrix form and solved it by using finite difference method based on Simpson's rule and trapezoidal rule.

Since z(x) in F(x, y(x), z(x)) is the integral term in VIDE and cannot be solved explicitly, therefore, Simpson's rule is adapted for solving the integral part.

Solving Volterra Integrodifferential Equations via Diagonally Implicit Multistep Block Method

Generally Simpson's rule can be considered a suitable numerical integration method.

Dynamic Analysis of an Infinitely Long Beam Resting on a Kelvin Foundation under Moving Random Loads

In the numerical analysis, one method for numerical integration is Simpson's rule or Simpson's method.

Optimization of the inflationary inventory control model under stochastic conditions with Simpson approximation: Particle swarm optimization approach

Another effective application is integration of discrete data, which is the same as Simpson's rule, rather than the conventional Gauss quadrature which is inapplicable owing to its use of inner-range data.

Comprehensive interpretation of a three-point gauss quadrature with variable sampling points and its application to integration for discrete data

Other authors such as [9-12] used quadrature rules like repeated trapezoidal and repeated Simpson's rule to solve linear Volterra integral equations.

Numerical solutions of a class of nonlinear Volterra integral equations

Finally, students are challenged to perform hydrostatic calculations (displacement and centers at the design waterline) by using Simpson's Rule. 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.

Integrating technology and engineering with a marine design contest and 3-D modeling: a significant highlight of the competitive event is that it provides a "real-world" engineering design activity that is free and open to all area high schools

In particular, students would be familiar with the midpoint rule, the elementary trapezoidal rule and Simpson's rule. The following paper derives these techniques by methods which secondary students may not be familiar with and an approach that undergraduate students should be familiar with.

Numerical integration

In order to use the kinematic method, several numerical methods for the integration were tested, including the trapezoidal method and Simpson's Rule. Second and fourth-order finite differencing were used to calculate horizontal divergence.

Atmospheric sciences senior section

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.

Spinners, scroll bars and Simpson's rule