
In
numerical integration, Simpson's rules are several
approximation
An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...
s for
definite integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
s, 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
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
, who derived it in 1615 after seeing it used for wine barrels (barrel rule, ). The approximate equality in the rule becomes exact if 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 rules are a set of rules used in ship stability and naval architecture, to calculate the areas and volumes of irregular figures. This is an application of Simpson's rule for finding the values of an integral, here interpreted as the a ...
.
Simpson's 1/3 rule
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