Simpson’s 1/48 Rule for Numerical Integration
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 1/48 Rule. This codes uses variable step size so it gives accurate results. This method more accurate and faster than Simpson’s 1/3 and 3/8 rules.
Code Outputs:
- Chart of Integration History
- Chart of Residuals
- Printed values of Final Iteration value, Accuracy and Number of Iterations.
Result is: 0.99687, Accuracy: 7.4337e-05, NO Iterations: 7
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 1/48 Rule:
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:[6]
![\int _{a}^{b}f(x)\,dx\approx {\tfrac {h}{48}}{\bigg [}17f(x_{0})+59f(x_{1})+43f(x_{2})+49f(x_{3})+48\sum _{{i=4}}^{{n-4}}f(x_{i})+49f(x_{{n-3}})+43f(x_{{n-2}})+59f(x_{{n-1}})+17f(x_{n}){\bigg ]}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/d01441940dcd7f3bdb06f5c197b8d169662ad3f0)
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.
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.
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