A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure Isaac; Twum Boakye Stephen; Baba Seidu

doi:doi:10.11648/j.ajmcm.20210601.11

| Peer-Reviewed

A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations

Azure Isaac, Twum Boakye Stephen, Baba Seidu

Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1)

Received: 18 December 2020 Accepted: 4 January 2021 Published: 15 January 2021

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS – 1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS – 3/8 with the other methods revealed that TS – 3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS – 1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.

Published in	American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1)
DOI	10.11648/j.ajmcm.20210601.11
Page(s)	1-8
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), 2021. Published by Science Publishing Group

Keywords

Broyden Method, Newton-Raphson Method, Quadrature Rules, Trapizoidal-Simpson- 3/8 Rule, Nonlinear Systems, Convergence, Numerical Examples, Trapizoidal-Simpson- 1/3 Rule

References

[1]	Al-Towaiq, M. H., & Abu Hour, Y. S. (2017). Two improved classes of Broyden's methods for solving nonlinear systems of equations. JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE-JMCS, 17 (1), 22-31.
[2]	Autar, K., Egwu, E. K., Duc, N. (2017). Numerical Methods with Applications. http://numericalmethodng.usf.edu.
[3]	Azure, I., Aloliga, G., & Doabil, L. (2020). Comparative Study of Numerical Methods for Solving Non-linear Equations Using Manual Computation. Mathematics Letters, 5 (4), 41.
[4]	Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2012). Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Applied Mathematics and Computation, 218 (23), 11496-11504.
[5]	Cordero, A., & Torregrosa, J. R. (2006). Variants of Newton’s method for functions of several variables. Applied Mathematics and Computation, 183 (1), 199-208.
[6]	Darvishi, M. T., & Shin, B. C. (2011). High-order Newton-Krylov methods to solve systems of nonlinear equations. Journal of the Korean Society for Industrial and Applied Mathematics, 15 (1), 19-30.
[7]	Dhamacharoen, A. (2014). An efficient hybrid method for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 263, 59-68.
[8]	Frontini, M. A. R. C. O., & Sormani, E. (2003). Some variant of Newton’s method with third-order convergence. Applied Mathematics and Computation, 140 (2-3), 419-426.
[9]	Hafiz, M. A., & Bahgat, M. S. (2012). An efficient two-step iterative method for solving system of nonlinear equations. Journal of Mathematics Research, 4 (4), 28.
[10]	Jain, M. K, Iyengar, S. R. K & Jain, R. K. (2012). Numerical Methods for Scientific and Engineering Computation (Sixth Edition). New Age International (P) Limited, Publishers, New Delhi.
[11]	Kelley, C. T. (1995). Iterative methods for linear and nonlinear equations. Society for Industrial and Applied Mathematics.
[12]	Kou, J., Li, Y., & Wang, X. (2007). A composite fourth-order iterative method for solving non-linear equations. Applied Mathematics and Computation, 184 (2), 471-475 (11).
[13]	Li, Y., Wei, Y., & Chu, Y. (2015). Research on solving systems of nonlinear equations based on improved PSO. Mathematical Problems in Engineering, 2015 (1).
[14]	Luo, Y. Z., Tang, G. J., & Zhou, L. N. (2008). Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing, 8 (2), 1068-1073 (12).
[15]	Mo, Y., Liu, H., & Wang, Q. (2009). Conjugate direction particle swarm optimization solving systems of nonlinear equations. Computers & Mathematics with Applications, 57 (11-12), 1877-1882 (13).
[16]	Mohammad, H., & Waziri, M. Y. (2015). On Broyden-like update via some quadratures for solving nonlinear systems of equations. Turkish Journal of Mathematics, 39 (3), 335-345.
[17]	Muhammad, K., Mamat, M., & Waziri, M. Y. (2013). A Broyden’s-like Method for solving systems of Nonlinear Equations. World Appl Sc J, 21, 168-173.
[18]	Osinuga, I. A., & Yusuff, S. O. (2018). Quadrature based Broyden-like method for systems of nonlinear equations. Statistics, Optimization & Information Computing, 6 (1), 130-138.
[19]	Osinuga I. A. & Yusuff, S. O. (2018). A robust Broyden-like method for systems of nonlinear equations. Transactions of the Nigerian Association of Mathematical Physics, 6 (Jan., 2018), 81–89.
[20]	Osinuga, I. A., & Yusuff, S. O. (2017). Construction of a Broyden-like method for Nonlinear systems of equations. Annals. Computer Science Series, 15 (2), 128-135.
[21]	Van de Rotten, B., & Lunel, S. V. (2005). A limited memory Broyden method to solve high-dimensional systems of nonlinear equations. In EQUADIFF 2003 (pp. 196-201).
[22]	Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton's method with accelerated third-order convergence. Applied Mathematics Letters, 13 (8), 87-93.
[23]	Yusuff, S. O., Osinuga, I. A., Adeniran, O. J., & Onashoga, S. A. (2019). ROBUST THREE-STEP BROYDEN–LIKE ALGORITHMS FOR FUNCTIONS OF SEVERAL VARIABLES. Annals. Computer Science Series, 17 (2).

Cite This Article

Plain Text BibTeX RIS

APA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. (2021). A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. American Journal of Mathematical and Computer Modelling, 6(1), 1-8. https://doi.org/10.11648/j.ajmcm.20210601.11

Copy | Download

ACS Style

Azure Isaac; Twum Boakye Stephen; Baba Seidu. A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. Am. J. Math. Comput. Model. 2021, 6(1), 1-8. doi: 10.11648/j.ajmcm.20210601.11

Copy | Download

AMA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. Am J Math Comput Model. 2021;6(1):1-8. doi: 10.11648/j.ajmcm.20210601.11

Copy | Download

@article{10.11648/j.ajmcm.20210601.11,
  author = {Azure Isaac and Twum Boakye Stephen and Baba Seidu},
  title = {A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ajmcm.20210601.11},
  url = {https://doi.org/10.11648/j.ajmcm.20210601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210601.11},
  abstract = {Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS – 1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS – 3/8 with the other methods revealed that TS – 3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS – 1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Baba Seidu
Y1  - 2021/01/15
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210601.11
DO  - 10.11648/j.ajmcm.20210601.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210601.11
AB  - Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS – 1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS – 3/8 with the other methods revealed that TS – 3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS – 1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.
VL  - 6
IS  - 1
ER  -

Copy | Download

Author Information

Azure Isaac

Mathematics Department, University for Development Studies, Tamale, Ghana
Twum Boakye Stephen

Mathematics Department, C. K. Tedam University of Technology and Applied Sciences, Navrongo, Ghana
Baba Seidu

Mathematics Department, C. K. Tedam University of Technology and Applied Sciences, Navrongo, Ghana

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. (2021). A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. American Journal of Mathematical and Computer Modelling, 6(1), 1-8. https://doi.org/10.11648/j.ajmcm.20210601.11

Copy | Download

ACS Style

Azure Isaac; Twum Boakye Stephen; Baba Seidu. A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. Am. J. Math. Comput. Model. 2021, 6(1), 1-8. doi: 10.11648/j.ajmcm.20210601.11

Copy | Download

AMA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations. Am J Math Comput Model. 2021;6(1):1-8. doi: 10.11648/j.ajmcm.20210601.11

Copy | Download

@article{10.11648/j.ajmcm.20210601.11,
  author = {Azure Isaac and Twum Boakye Stephen and Baba Seidu},
  title = {A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {1},
  pages = {1-8},
  doi = {10.11648/j.ajmcm.20210601.11},
  url = {https://doi.org/10.11648/j.ajmcm.20210601.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210601.11},
  abstract = {Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS – 1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS – 3/8 with the other methods revealed that TS – 3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS – 1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - A New Trapezoidal-Simpson 3/8 Method for Solving Systems of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Baba Seidu
Y1  - 2021/01/15
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210601.11
DO  - 10.11648/j.ajmcm.20210601.11
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 1
EP  - 8
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210601.11
AB  - Since its introduction, the Broyden method has been used as the foundation to develop several other Broyden-like methods (or hybrid Broyden methods) which in many cases have turned out to be improved forms of the original method. The modified classical Broyden methods developed by many authors to solve system of nonlinear equations have been effective in overcoming the deficiency of the classical Newton Raphson method, however there are new trends of methods proposed by authors, which have proven to be more efficient than some already existing ones. This work introduces two Broyden-like method developed from a weighted combination of quadrature rules, namely the Trapizoidal, Simpson 3/8 and Simpson 1/3 quadrature rules. Hence the new Broyden-like methods named by the authors as TS-3/8 and TS – 1/3 methods have been developed from these rules. After subjecting the proposed methods together with some other existing Broyden-like methods to solve four bench-mark problems, the results of numerical test confirm that the TS-3/8 method is promising (in terms of speed and in most cases accuracy) when compared with other proposed Broyden-like methods. Results gathered after the comparison of TS – 3/8 with the other methods revealed that TS – 3/8 method performed better than all the methods in terms of speed and the number of iterations needed to reach a solution. On the other hand, TS – 1/3 method yielded results for all the benchmark problems but with a relatively higher number of iterations compared with the other methods selected for comparison.
VL  - 6
IS  - 1
ER  -

Copy | Download