Simpson’s 1/48 Rule for Numerical Integration

by admin in , , on April 29, 2019

This 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:

For Simpson’s voting rule, see Minimax Condorcet. For Simpson’s rules used in ship stability, see Simpson’s rules (ship stability). Simpson’s rule can be derived by approximating the integrand (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]

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.


[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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  • Released

    April 29, 2019

  • Last Updated

    May 28, 2019

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