This 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.
- 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
- 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:
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:
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
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.
 Atkinson, Kendall E. (1989). An Introduction to Numerical Analysis (2nd ed.). John Wiley & Sons. ISBN 0-471-50023-2.
 Burden, Richard L.; Faires, J. Douglas (2000). Numerical Analysis (7th ed.). Brooks/Cole. ISBN 0-534-38216-9.
 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.
 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.
 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.
 Süli, Endre; Mayers, David (2003). An Introduction to Numerical Analysis. Cambridge University Press. ISBN 0-521-00794-1.
 Kaw, Autar; Kalu, Egwu; Nguyen, Duc (2008). ‘Numerical Methods with Applications’.
 Weisstein, Eric W. (2010). ‘Newton-Cotes Formulas’. MathWorld–A Wolframtite Web Resource. MathWorld. Retrieved 2 August 2010.