The N-Point Definite Integral Approximation Formula  (N-POINT DIAF)

Francis Oketch Ochieng’; Nicholas Muthama Mutua; Nicholas Mwilu Mutothya

doi:doi:10.11648/j.acm.20170601.11

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The N-Point Definite Integral Approximation Formula (N-POINT DIAF)

Francis Oketch Ochieng’, Nicholas Muthama Mutua, Nicholas Mwilu Mutothya

Published in Applied and Computational Mathematics (Volume 6, Issue 1)

Received: 20 November 2016 Accepted: 26 December 2016 Published: 21 February 2017

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Abstract

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.

Published in	Applied and Computational Mathematics (Volume 6, Issue 1)
DOI	10.11648/j.acm.20170601.11
Page(s)	1-33
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Numerical Integration, Approximation, Definite Integrals, Error, Analytical Techniques, Stability

References

[1]	Burden, R. L. and Faires, J. D., Numerical Analysis, 9th ed., Brooks/Cole, Pacific Grove, 2011. pp 193-256.
[2]	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.
[3]	Dukkipati, R. V., Numerical Methods, New Age International Publishers (P) Ltd., New Delhi, India, 2010. Pp 237-263.
[4]	Gordon K. Smyth (1998), Numerical Integration, Encyclopedia of Biostatistics (ISBN 0471 975761), John Wiley & Sons, Ltd, Chichester.
[5]	Gupta, C. B and Malik, A. K. (2009), Advanced Mathematics, (New Age International Publishers) Pp 90-101.
[6]	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.
[7]	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.
[8]	K. E. Atkinson, An Introduction to Numerical Analysis, John Wiley and Sons, New York, NY, USA, Second Edition, 1989.
[9]	Rao, G. S. (2006), Numerical Analysis, Revised Third Edition (New Age International (P) Limited Publishers).
[10]	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.

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APA Style

Francis Oketch Ochieng’, Nicholas Muthama Mutua, Nicholas Mwilu Mutothya. (2017). The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Applied and Computational Mathematics, 6(1), 1-33. https://doi.org/10.11648/j.acm.20170601.11

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ACS Style

Francis Oketch Ochieng’; Nicholas Muthama Mutua; Nicholas Mwilu Mutothya. The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Appl. Comput. Math. 2017, 6(1), 1-33. doi: 10.11648/j.acm.20170601.11

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AMA Style

Francis Oketch Ochieng’, Nicholas Muthama Mutua, Nicholas Mwilu Mutothya. The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Appl Comput Math. 2017;6(1):1-33. doi: 10.11648/j.acm.20170601.11

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@article{10.11648/j.acm.20170601.11,
  author = {Francis Oketch Ochieng’ and Nicholas Muthama Mutua and Nicholas Mwilu Mutothya},
  title = {The N-Point Definite Integral Approximation Formula  (N-POINT DIAF)},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {1},
  pages = {1-33},
  doi = {10.11648/j.acm.20170601.11},
  url = {https://doi.org/10.11648/j.acm.20170601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170601.11},
  abstract = {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.},
 year = {2017}
}

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TY  - JOUR
T1  - The N-Point Definite Integral Approximation Formula  (N-POINT DIAF)
AU  - Francis Oketch Ochieng’
AU  - Nicholas Muthama Mutua
AU  - Nicholas Mwilu Mutothya
Y1  - 2017/02/21
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170601.11
DO  - 10.11648/j.acm.20170601.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 1
EP  - 33
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170601.11
AB  - 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.
VL  - 6
IS  - 1
ER  -

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Author Information

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

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Plain Text BibTeX RIS

APA Style

Francis Oketch Ochieng’, Nicholas Muthama Mutua, Nicholas Mwilu Mutothya. (2017). The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Applied and Computational Mathematics, 6(1), 1-33. https://doi.org/10.11648/j.acm.20170601.11

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ACS Style

Francis Oketch Ochieng’; Nicholas Muthama Mutua; Nicholas Mwilu Mutothya. The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Appl. Comput. Math. 2017, 6(1), 1-33. doi: 10.11648/j.acm.20170601.11

Copy | Download

AMA Style

Francis Oketch Ochieng’, Nicholas Muthama Mutua, Nicholas Mwilu Mutothya. The N-Point Definite Integral Approximation Formula (N-POINT DIAF). Appl Comput Math. 2017;6(1):1-33. doi: 10.11648/j.acm.20170601.11

Copy | Download

@article{10.11648/j.acm.20170601.11,
  author = {Francis Oketch Ochieng’ and Nicholas Muthama Mutua and Nicholas Mwilu Mutothya},
  title = {The N-Point Definite Integral Approximation Formula  (N-POINT DIAF)},
  journal = {Applied and Computational Mathematics},
  volume = {6},
  number = {1},
  pages = {1-33},
  doi = {10.11648/j.acm.20170601.11},
  url = {https://doi.org/10.11648/j.acm.20170601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20170601.11},
  abstract = {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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - The N-Point Definite Integral Approximation Formula  (N-POINT DIAF)
AU  - Francis Oketch Ochieng’
AU  - Nicholas Muthama Mutua
AU  - Nicholas Mwilu Mutothya
Y1  - 2017/02/21
PY  - 2017
N1  - https://doi.org/10.11648/j.acm.20170601.11
DO  - 10.11648/j.acm.20170601.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 1
EP  - 33
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20170601.11
AB  - 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.
VL  - 6
IS  - 1
ER  -

Copy | Download