Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations

Azure Isaac; Twum Boakye Stephen; Anas Musah; Aloliga Golbert

doi:doi:10.11648/j.ijssam.20230801.13

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Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations

Azure Isaac, Twum Boakye Stephen, Anas Musah, Aloliga Golbert

Published in International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 1)

Received: 25 February 2023 Accepted: 3 April 2023 Published: 18 April 2023

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Abstract

Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.ijssam.20230801.13
Page(s)	12-16
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, Error Analysis

References

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[2]	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.
[3]	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.
[4]	Dhamacharoen, A. (2014). An efficient hybrid method for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 263, 59-68.
[5]	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.
[6]	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.
[7]	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 Vol. 6, No. 3, 2021 pp. 77-94. doi.org/10.11648/j.ijssam.20210603.11.
[8]	Jain, M. K. (2003). Numerical methods for scientific and engineering computation. New Age International.
[9]	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.
[10]	Li, Y., Wei, Y., & Chu, Y. (2015). Research on solving systems of nonlinear equations based on improved PSO. Mathematical Problems in Engineering, 2015.
[11]	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]	Mahwash, K. N., & Gyang, G. D. (2018). Numerical Solution of Nonlinear Systems of Algebriac Equations.
[13]	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.
[14]	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.
[15]	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.
[16]	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.
[17]	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.
[18]	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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APA Style

Azure Isaac, Twum Boakye Stephen, Anas Musah, Aloliga Golbert. (2023). Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. International Journal of Systems Science and Applied Mathematics, 8(1), 12-16. https://doi.org/10.11648/j.ijssam.20230801.13

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ACS Style

Azure Isaac; Twum Boakye Stephen; Anas Musah; Aloliga Golbert. Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. Int. J. Syst. Sci. Appl. Math. 2023, 8(1), 12-16. doi: 10.11648/j.ijssam.20230801.13

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AMA Style

Azure Isaac, Twum Boakye Stephen, Anas Musah, Aloliga Golbert. Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. Int J Syst Sci Appl Math. 2023;8(1):12-16. doi: 10.11648/j.ijssam.20230801.13

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@article{10.11648/j.ijssam.20230801.13,
  author = {Azure Isaac and Twum Boakye Stephen and Anas Musah and Aloliga Golbert},
  title = {Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {12-16},
  doi = {10.11648/j.ijssam.20230801.13},
  url = {https://doi.org/10.11648/j.ijssam.20230801.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230801.13},
  abstract = {Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.},
 year = {2023}
}

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TY  - JOUR
T1  - Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Anas Musah
AU  - Aloliga Golbert
Y1  - 2023/04/18
PY  - 2023
N1  - https://doi.org/10.11648/j.ijssam.20230801.13
DO  - 10.11648/j.ijssam.20230801.13
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  - 12
EP  - 16
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20230801.13
AB  - Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.
VL  - 8
IS  - 1
ER  -

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Author Information

Azure Isaac

Computer Science Department, Regentropfen College of Applied Sciences, Kansoe, Ghana
Twum Boakye Stephen

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

Computer Science Department, Regentropfen College of Applied Sciences, Kansoe, Ghana
Aloliga Golbert

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

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Plain Text BibTeX RIS

APA Style

Azure Isaac, Twum Boakye Stephen, Anas Musah, Aloliga Golbert. (2023). Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. International Journal of Systems Science and Applied Mathematics, 8(1), 12-16. https://doi.org/10.11648/j.ijssam.20230801.13

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ACS Style

Azure Isaac; Twum Boakye Stephen; Anas Musah; Aloliga Golbert. Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. Int. J. Syst. Sci. Appl. Math. 2023, 8(1), 12-16. doi: 10.11648/j.ijssam.20230801.13

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AMA Style

Azure Isaac, Twum Boakye Stephen, Anas Musah, Aloliga Golbert. Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations. Int J Syst Sci Appl Math. 2023;8(1):12-16. doi: 10.11648/j.ijssam.20230801.13

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@article{10.11648/j.ijssam.20230801.13,
  author = {Azure Isaac and Twum Boakye Stephen and Anas Musah and Aloliga Golbert},
  title = {Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {12-16},
  doi = {10.11648/j.ijssam.20230801.13},
  url = {https://doi.org/10.11648/j.ijssam.20230801.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20230801.13},
  abstract = {Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.},
 year = {2023}
}

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TY  - JOUR
T1  - Error Analysis of Newly Developed Numerical Methods for Solving System of Nonlinear Equations
AU  - Azure Isaac
AU  - Twum Boakye Stephen
AU  - Anas Musah
AU  - Aloliga Golbert
Y1  - 2023/04/18
PY  - 2023
N1  - https://doi.org/10.11648/j.ijssam.20230801.13
DO  - 10.11648/j.ijssam.20230801.13
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  - 12
EP  - 16
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20230801.13
AB  - Solution methods are the major tools in research especially in the area of applied mathematics. This is because, most real-life problems result into system of nonlinear equations, and the right solution method with less computational error is required to obtain an approximated solution to these system of nonlinear equations. The introduction of the Broyden method set the groundwork for the development of several other methods, many of which are referred to as Broyden-like approaches by various researchers. In most cases, these methods have proven to be superior to the original classical Broyden method in terms of the number of iterations and CPU time needed to acquire a solution. Using the solutions of the traditional Broyden method as a point of comparison, this study aimed to examine the error associated with two newly developed numerical methods, the Trapezoidal-Simpson-3/8 (TS-3/8) and Midpoint-Simpson-3/8 (MS-3/8) methods. Results gathered after applying the classical Broyden, MS-3/8 and TS-3/8 methods to solve some bench-mark problems involving system of nonlinear equations and estimating the errors associated with each of the methods considered in the study, using the formula of the approximate error, showed that the error associated with the MS-3/8 method was minimal compared to that of the Broyden and the TS-3/8 methods. At the end of the study, the results gathered suggested the MS-3/8 technique as the most highly advised numerical approach among the other methods. This means that, MS-3/8 method is a more accurate solution method for solving system of nonlinear equations considering the results in this paper.
VL  - 8
IS  - 1
ER  -

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