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), 2021. 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

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[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.
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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

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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