A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations

Azure Isaac; Twum Boakye Stephen; Baba Seidu

doi:doi:10.11648/j.ijssam.20210603.11

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A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations

Azure Isaac, Twum Boakye Stephen, Baba Seidu

Published in International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 3)

Received: 10 July 2021 Accepted: 20 July 2021 Published: 24 July 2021

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Abstract

The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 6, Issue 3)
DOI	10.11648/j.ijssam.20210603.11
Page(s)	77-94
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), 2024. Published by Science Publishing Group

Keywords

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

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://numericalmethods.eng.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]	Isaac, A., Stephen, T. B., Seidu, B. (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.
[11]	Jain, M. K. (2003). Numerical methods for scientific and engineering computation. New Age International.
[12]	Kelley, C. T. (1995). Iterative methods for linear and nonlinear equations. Society for Industrial and Applied Mathematics.
[13]	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)
[14]	Li, Y., Wei, Y., & Chu, Y. (2015). Research on solving systems of nonlinear equations based on improved PSO. Mathematical Problems in Engineering, 2015. (1).
[15]	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).
[16]	Mahwash, K. N., & Gyang, G. D. (2018). Numerical Solution of Nonlinear Systems of Algebriac Equations.
[17]	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).
[18]	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.
[19]	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.
[20]	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.
[21]	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.
[22]	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).
[23]	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.

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Azure Isaac, Twum Boakye Stephen, Baba Seidu. (2021). A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. International Journal of Systems Science and Applied Mathematics, 6(3), 77-94. https://doi.org/10.11648/j.ijssam.20210603.11

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ACS Style

Azure Isaac; Twum Boakye Stephen; Baba Seidu. A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. Int. J. Syst. Sci. Appl. Math. 2021, 6(3), 77-94. doi: 10.11648/j.ijssam.20210603.11

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AMA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. Int J Syst Sci Appl Math. 2021;6(3):77-94. doi: 10.11648/j.ijssam.20210603.11

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@article{10.11648/j.ijssam.20210603.11,
  author = {Azure Isaac and Twum Boakye Stephen and Baba Seidu},
  title = {A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {3},
  pages = {77-94},
  doi = {10.11648/j.ijssam.20210603.11},
  url = {https://doi.org/10.11648/j.ijssam.20210603.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210603.11},
  abstract = {The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.},
 year = {2021}
}

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TY  - JOUR
T1  - A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Baba Seidu
Y1  - 2021/07/24
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210603.11
DO  - 10.11648/j.ijssam.20210603.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 77
EP  - 94
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210603.11
AB  - The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.
VL  - 6
IS  - 3
ER  -

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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, Novrongo, Ghana
Baba Seidu

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

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APA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. (2021). A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. International Journal of Systems Science and Applied Mathematics, 6(3), 77-94. https://doi.org/10.11648/j.ijssam.20210603.11

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ACS Style

Azure Isaac; Twum Boakye Stephen; Baba Seidu. A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. Int. J. Syst. Sci. Appl. Math. 2021, 6(3), 77-94. doi: 10.11648/j.ijssam.20210603.11

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AMA Style

Azure Isaac, Twum Boakye Stephen, Baba Seidu. A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations. Int J Syst Sci Appl Math. 2021;6(3):77-94. doi: 10.11648/j.ijssam.20210603.11

Copy | Download

@article{10.11648/j.ijssam.20210603.11,
  author = {Azure Isaac and Twum Boakye Stephen and Baba Seidu},
  title = {A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {6},
  number = {3},
  pages = {77-94},
  doi = {10.11648/j.ijssam.20210603.11},
  url = {https://doi.org/10.11648/j.ijssam.20210603.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20210603.11},
  abstract = {The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - A Comparison of Newly Developed Broyden – Like Methods for Solving System of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Baba Seidu
Y1  - 2021/07/24
PY  - 2021
N1  - https://doi.org/10.11648/j.ijssam.20210603.11
DO  - 10.11648/j.ijssam.20210603.11
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 77
EP  - 94
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20210603.11
AB  - The search for a more efficient and robust numerical method for solving problems have become an interesting area for many researchers as most problems resulting into nonlinear system of equations would require a very good numerical method for its computation. The introduction of the Broyden method has served as the foundation to developing several others, which are referred to as Broyden – like methods by some authors. These methods, in most cases, have proven to be superior to the original classical Broyden method in terms of the number of iterations needed and the CPU time required to reach a solution. This research sought to develop new Broyden – like methods using weighted combinations of quadrature rules (i.e., Simpson -1/3 and Simpson -3/8 rules against Midpoint, Trapezoidal, and Simpson quadrature rules). The weighted combination of the quadrature rules in the development of the new methods led to the discovery of several new methods. Some of which have proven to be more efficient and robust when compared with some existing methods. A comparison of these newly developed methods with the classical Broyden method together with some existing improved Broyden method revealed that, one of the newly developed methods namely, Midpoint–Simpson-3/8 (MS–3/8) method, outperformed all the others, with the MS–3/8 method giving the best of numerical results in all the benchmark problems considered in the study.
VL  - 6
IS  - 3
ER  -

Copy | Download