Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Simpson's rule
Summary
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761).
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). The approximate equality in the rule becomes exact if f is a polynomial up to and including 3rd degree.
If the 1/3 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 1/3 rule. Points inside the integration range are given alternating weights 4/3 and 2/3.
Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error.
If the 3/8 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 3/8 rule.
Simpson's 1/3 and 3/8 rules are two special cases of closed Newton–Cotes formulas.
In naval architecture and ship stability estimation, there also exists Simpson's third rule, which has no special importance in general numerical analysis, see Simpson's rules (ship stability).
Simpson's 1/3 rule, also simply called Simpson's rule, is a method for numerical integration proposed by Thomas Simpson. It is based upon a quadratic interpolation. Simpson's 1/3 rule is as follows:
where is the step size.
The error in approximating an integral by Simpson's rule for is
where (the Greek letter xi) is some number between and .
The error is asymptotically proportional to . However, the above derivations suggest an error proportional to . Simpson's rule gains an extra order because the points at which the integrand is evaluated are distributed symmetrically in the interval .
Since the error term is proportional to the fourth derivative of at , this shows that Simpson's rule provides exact results for any polynomial of degree three or less, since the fourth derivative of such a polynomial is zero at all points.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (4)
MATH-459: Numerical methods for conservation laws
Introduction to the development, analysis, and application of computational methods for solving conservation laws with an emphasis on finite volume, limiter based schemes, high-order essentially non-o
ChE-312: Numerical methods
This course introduces students to modern computational and mathematical techniques for solving problems in chemistry and chemical engineering. The use of introduced numerical methods will be demonstr
MATH-250: Numerical analysis
Construction et analyse de méthodes numériques pour la solution de problèmes d'approximation, d'algèbre linéaire et d'analyse
Related concepts (7)
Trapezoidal rule
In calculus, the trapezoidal rule (also known as the trapezoid rule or trapezium rule; see Trapezoid for more information on terminology) is a technique for approximating the definite integral. The trapezoidal rule works by approximating the region under the graph of the function as a trapezoid and calculating its area. It follows that The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way.
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together.
Gaussian quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes x_i and weights w_i for i = 1, ..., n.