Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red).
An animation showing how Simpson's rule approximation improves with more strips.

In numerical analysis, Simpson's method is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation for values bounding equally spaced subdivisions (where is even): (General Form)

,

where and .

Simpson's rule also corresponds to the three-point Newton-Cotes quadrature rule.

In English, the method is credited to the mathematician Thomas Simpson (1710–1761) of Leicestershire, England. However, Johannes Kepler used similar formulas over 100 years prior, and for this reason the method is sometimes called Kepler's rule, or Keplersche Fassregel (Kepler's barrel rule) in German.

Quadratic interpolationEdit

One derivation replaces the integrand   by the quadratic polynomial (i.e. parabola)   which takes the same values as   at the end points a and b and the midpoint m = (a + b) / 2. One can use Lagrange polynomial interpolation to find an expression for this polynomial,

 

Using integration by substitution one can show that[1]

 

Introducing the step size   this is also commonly written as

 

Because of the   factor Simpson's rule is also referred to as Simpson's 1/3 rule (see below for generalization).

The calculation above can be simplified if one observes that (by scaling) there is no loss of generality in assuming that  .

Averaging the midpoint and the trapezoidal rulesEdit

Another derivation constructs Simpson's rule from two simpler approximations: the midpoint rule

 

and the trapezoidal rule

 

The errors in these approximations are

 

respectively, where   denotes a term asymptotically proportional to  . The two   terms are not equal; see Big O notation for more details. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the weighted average

 

This weighted average is exactly Simpson's rule.

Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable weighted average and eliminate another error term. This is Romberg's method.

Undetermined coefficientsEdit

The third derivation starts from the ansatz

 

The coefficients α, β and γ can be fixed by requiring that this approximation be exact for all quadratic polynomials. This yields Simpson's rule.

ErrorEdit

The error in approximating an integral by Simpson's rule for   is

 

where   (the Greek letter xi) is some number between   and  .[2]

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.

If the second derivative   exists and is convex in the interval  :

 

Composite Simpson's ruleEdit

If the interval of integration   is in some sense "small", then Simpson's rule with   subintervals will provide an adequate approximation to the exact integral. By small, what we really mean is that the function being integrated is relatively smooth over the interval  . For such a function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

However, it is often the case that the function we are trying to integrate is not smooth over the interval. Typically, this means that either the function is highly oscillatory, or it lacks derivatives at certain points. In these cases, Simpson's rule may give very poor results. One common way of handling this problem is by breaking up the interval   into   small subintervals. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. This sort of approach is termed the composite Simpson's rule.

Suppose that the interval   is split up into   sub-intervals, with   an even number. Then, the composite Simpson's rule is given by

 

where   for   with  ; in particular,   and  . This composite rule with   corresponds with the regular Simpson's Rule of the preceding section.

The error committed by the composite Simpson's rule is

 

where   is some number between   and   and   is the "step length".[3] The error is bounded (in absolute value) by

 

This formulation splits the interval   in subintervals of equal length. In practice, it is often advantageous to use subintervals of different lengths, and concentrate the efforts on the places where the integrand is less well-behaved. This leads to the adaptive Simpson's method.

Simpson's 3/8 ruleEdit

Simpson's 3/8 rule is another method for numerical integration proposed by Thomas Simpson. It is based upon a cubic interpolation rather than a quadratic interpolation. Simpson's 3/8 rule is as follows:

 

where b − a = 3h. The error of this method is:

 

where   is some number between   and  . Thus, the 3/8 rule is about twice as accurate as the standard method, but it uses one more function value. A composite 3/8 rule also exists, similarly as above.[4]

A further generalization of this concept for interpolation with arbitrary-degree polynomials are the Newton–Cotes formulas.

Composite Simpson's 3/8 ruleEdit

Dividing the interval   into   subintervals of length   and introducing the nodes   we have

 

While the remainder for the rule is shown as:

 [4]

We can only use this if   is a multiple of three.

The 3/8th rule is also called Simpson's second rule.

Alternative extended Simpson's ruleEdit

This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding:[5]

 

The formula above is obtained by combining the original composite Simpson's rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and the standard 3-point rule in the remaining subintervals. The result is then obtained by taking the mean of the two formulas.

Simpson's rules in the case of narrow peaksEdit

In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule. Namely, composite Simpson's 1/3 rule requires 1.8 times more points to achieve the same accuracy[6] as trapezoidal rule. Composite Simpson's 3/8 rule is even less accurate. Integral by Simpson's 1/3 rule can be represented as a sum of 2/3 of integral by trapezoidal rule with step h and 1/3 of integral by rectangle rule with step 2h. No wonder that error of the sum corresponds lo less accurate term. Averaging of Simpson's 1/3 rule composite sums with properly shifted frames produces following rules:

 

where two points outside of integrated region are exploited and

 

Those rules are very much similar to Press's alternative extended Simpson's rule. Coefficients within the major part of the region being integrated equal one, differences are only at the edges. These three rules can be associated with Euler-MacLaurin formula with the first derivative term and named Euler-MacLaurin integration rules.[6] They differ only in how the first derivative at the region end is calculated.

Composite Simpson's rule for irregularly spaced dataEdit

For some applications, the integration interval   needs to be divided into uneven intervals – perhaps due to uneven sampling of data, or missing or corrupted data points. Suppose we divide the interval   into even number   of subintervals of widths  . Then the composite Simpson's rule is given by[7][8]

 

where   are the function values at the  th sampling point on the interval  , and the coefficients   and   are given by

 
 
 

In case of odd number   of subintervals, the above formula are used up to the second to last interval, and the last interval is handled separately by adding the following to the result:

 

where

 
 
 

See alsoEdit

NotesEdit

  1. ^ Atkinson, p. 256; Süli and Mayers, §7.2
  2. ^ Atkinson, equation (5.1.15); Süli and Mayers, Theorem 7.2
  3. ^ Atkinson, pp. 257+258; Süli and Mayers, §7.5
  4. ^ a b Matthews (2004)
  5. ^ Press (1989), p. 122
  6. ^ a b Kalambet, Yuri; Kozmin, Yuri; Samokhin, Andrey (2018). "Comparison of integration rules in the case of very narrow chromatographic peaks". Chemometrics and Intelligent Laboratory Systems. 179: 22–30. doi:10.1016/j.chemolab.2018.06.001. ISSN 0169-7439.
  7. ^ Kylänpää, Ilkka (2019). Computational Physics course. Tampere University.
  8. ^ Cartwright, Kenneth V. (2016). "Simpson's Rule Integration with MS Excel and Irregularly-spaced Data" (PDF). Journal of Mathematical Science and Mathematics Education. 11 (2): 34–42.

ReferencesEdit

External linksEdit

This article incorporates material from Code for Simpson's rule on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.