The N-Point Definite Integral Approximation Formula (N-POINT DIAF)
Applied and Computational Mathematics
Volume 6, Issue 1, February 2017, Pages: 1-33
Received: Nov. 20, 2016;
Accepted: Dec. 26, 2016;
Published: Feb. 21, 2017
Views 2896 Downloads 100
Francis Oketch Ochieng’, Mathematics and Informatics Department, Taita Taveta University, Voi, Kenya
Nicholas Muthama Mutua, Mathematics and Informatics Department, Taita Taveta University, Voi, Kenya
Nicholas Mwilu Mutothya, Mathematics and Informatics Department, Taita Taveta University, Voi, Kenya
Follow on us
Various authors have discovered formulae for numerical integration approximation. However these formulae always result to some amount of error which may differ in size depending on the formula. It’s therefore important that a formula with highest precision has been discovered and should be implemented for use in numerical integration approximations problems, especially for the definite integrals which cannot be evaluated by applying the analytical techniques. The present paper therefore explores the derivation of the N-point Definite Integral Approximation Formula (N-point DIAF) which amounts to the discovery of the 2-Point DIAF. This formula will assist in almost accurate evaluation of all definite integrals numerically. The proof of the formula is given, a specific test problem is then solved using the discovered 2-Point DIAF to obtain the solution numerically, which has the highest precision compared to other numerical methods of integration. Further the error terms are obtained and compared with the existing methods. Finally, the effectiveness of the proposed formula is illustrated by means of a numerical example.
Numerical Integration, Approximation, Definite Integrals, Error, Analytical Techniques, Stability
To cite this article
Francis Oketch Ochieng’,
Nicholas Muthama Mutua,
Nicholas Mwilu Mutothya,
The N-Point Definite Integral Approximation Formula (N-POINT DIAF), Applied and Computational Mathematics.
Vol. 6, No. 1,
2017, pp. 1-33.
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Burden, R. L. and Faires, J. D., Numerical Analysis, 9th ed., Brooks/Cole, Pacific Grove, 2011. pp 193-256.
C. -H. Yu, Solving some integrals with Maple, International Journal of Research in Aeronautical and Mechanical Engineering, Vol. 1, Issue. 3, pp. 29-35, 2013.
Dukkipati, R. V., Numerical Methods, New Age International Publishers (P) Ltd., New Delhi, India, 2010. Pp 237-263.
Gordon K. Smyth (1998), Numerical Integration, Encyclopedia of Biostatistics (ISBN 0471 975761), John Wiley & Sons, Ltd, Chichester.
Gupta, C. B and Malik, A. K. (2009), Advanced Mathematics, (New Age International Publishers) Pp 90-101.
Jain, M. K., S. R. K. Iyengar., and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, Sixth Edition, New Age International Publishers, (Formerly Wiley Eastern Limited), New Delhi, 2008. Pp 128-177.
Jain, M. K., S. R. K. Iyengar., and R. K. Jain, Numerical Methods for Scientific and Engineering Computation, Sixth Edition, New Age International Publishers, (Formerly Wiley Eastern Limited), New Delhi, 2012. Pp 128-177.
K. E. Atkinson, An Introduction to Numerical Analysis, John Wiley and Sons, New York, NY, USA, Second Edition, 1989.
Rao, G. S. (2006), Numerical Analysis, Revised Third Edition (New Age International (P) Limited Publishers).
T. Ramachandran, R. Parimala (2015), Open Newton - Cotes Quadrature With Midpoint Derivative For Integration Of Algebraic Functions, International Journal of Research in Engineering and Technology, Volume: 04 Issue: 10.