PlanetMath: Simpson's rule

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Simpson's rule

(Theorem)

Simpson's rule is a method of (approximate) numerical definite integration (or quadrature). Simpson's rule is based on a parabolic model of the function to be integrated (in contrast to the trapezoidal model of the trapezoidal rule). Thus, a minimum of three points and three function values are required. Here we take three equidistant points: the interval endpoints, the midpoint, and let $h=\vert b-a\vert/2$ the distance between each. The definite integral is then approximated by:

$\displaystyle \int_{x_0}^{x_2} f(x) dx \approx I = \frac{h}{3} (f(x_0) + 4f(x_1) + f(x_2))$

We can extend this to greater precision by breaking our target domain into equal-length fragments. The quadrature is then the weighted sum of the above formula for every pair of adjacent regions, which works out for even to

$\displaystyle I = \frac{h}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-3}) + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n))$

"Simpson's rule" is owned by drini. [ full author list (2) | owner history (2) ]

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Attachments:: proof of Simpson's rule (Proof) by drini
code for Simpson's rule (Algorithm) by drini
bound on error of Simpson's rule (Result) by Wkbj79

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Cross-references: even, regions, adjacent, sum, domain, definite integral, distance, midpoint, endpoints, interval, points, trapezoidal rule, function, quadrature
There are 5 references to this entry.

This is version 6 of Simpson's rule, born on 2003-06-08, modified 2006-06-15.
Object id is 4335, canonical name is SimpsonsRule.
Accessed 25434 times total.

Classification:

AMS MSC:	65D32 (Numerical analysis :: Numerical approximation and computational geometry :: Quadrature and cubature formulas)
	41A55 (Approximations and expansions :: Approximate quadratures)
	26A06 (Real functions :: Functions of one variable :: One-variable calculus)
	28-00 (Measure and integration :: General reference works )

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