Simpson’s 3/8 Rule for Numerical Integrations
by admin in Math, Statistics, and Optimization , MATLAB Family , Numerical Integration on April 29, 2019This code calculates the Numerical Integration of a given equation on a predefined interval using Simpson’s 3/8 Rule. This codes uses variable step size so it gives accurate results. This method called the second Simpson’s rule and more accurate than 1/3 rule.
Code Outputs:
- Chart of Integration History
- Chart of Residuals
- Printed values of Final Iteration value, Accuracy and Number of Iterations.
Result is: 0.97609, Accuracy: 9.42e-05, NO Iterations: 19
Input Requirements:
- The equation you intended to integrate
- Integration interval [a b].
- predefined error (default 0.0001).
- Maximum number of iterations (default 50000).
About Simpson’s Method:
rules (ship stability). Simpson’s rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red).
Simpson’s 3/8 Rule:
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:
![{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\tfrac {3h}{8}}\left[f(a)+3f\left({\tfrac {2a+b}{3}}\right)+3f\left({\tfrac {a+2b}{3}}\right)+f(b)\right]={\tfrac {(b-a)}{8}}\left[f(a)+3f\left({\tfrac {2a+b}{3}}\right)+3f\left({\tfrac {a+2b}{3}}\right)+f(b)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a671bf11cf72d7813e6b05820b649863fae741b)
where b − a = 3h. The error of this method is:
where xi is some number between a and b. 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.
A further generalization of this concept for interpolation with arbitrary-degree polynomials are the Newton–Cotes formulas.
Composite Simpson’s 3/8 Rule:
Dividing the interval [a, b] into n subintervals of length h = (b-a)/n and introducing the nodes xi = a + ih we have
![{\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx {\tfrac {3h}{8}}\left[f(x_{0})+3f(x_{1})+3f(x_{2})+2f(x_{3})+3f(x_{4})+3f(x_{5})+2f(x_{6})+\cdots +f(x_{n})\right].\\&={\frac {3h}{8}}\left[f(x_{0})+3\sum _{i\neq 3k}^{n-1}f(x_{i})+2\sum _{j=1}^{n/3-1}f(x_{3j})+f(x_{n})\right]\qquad {\text{For: }}k\in \mathbb {N} _{0}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c003a02545bdf22b79ea653cd4afbe9c36202357)
While the remainder for the rule is shown as:
Note, we can only use this if n is a multiple of three. The 3/8th rule is also called Simpson’s second rule.
References:
[1] Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis (2nd ed.). John Wiley & Sons. ISBN 0-471-50023-2.
[2] Burden, Richard L.; Faires, J. Douglas (2000). Numerical Analysis (7th ed.). Brooks/Cole. ISBN 0-534-38216-9.
[3] Pate, McCall (1918). The naval artificer’s manual: (The naval artificer’s handbook revised) text, questions and general information for deck. United States. Bureau of Reconstruction and Repair. p. 198.
[4] Matthews, John H. (2004). ‘Simpson’s 3/8 Rule for Numerical Integration’. Numerical Analysis – Numerical Methods Project. California State University, Fullerton. Archived from the original on 4 December 2008. Retrieved 11 November 2008.
[5] Press, William H.; Flannery, Brian P.; Vetterling, William T.; Teukolsky, Saul A. (1989). Numerical Recipes in Pascal: The Art of Scientific Computing. Cambridge University Press. ISBN 0-521-37516-9.
[6] Süli, Endre; Mayers, David (2003). An Introduction to Numerical Analysis. Cambridge University Press. ISBN 0-521-00794-1.
[7] Kaw, Autar; Kalu, Egwu; Nguyen, Duc (2008). ‘Numerical Methods with Applications’.
[8] Weisstein, Eric W. (2010). ‘Newton-Cotes Formulas’. MathWorld–A Wolframtite Web Resource. MathWorld. Retrieved 2 August 2010.
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