American Journal of Mathematical and Computer Modelling

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

Received: 18 December 2020    Accepted: 04 January 2021    Published: 15 January 2021
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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.

DOI 10.11648/j.ajmcm.20210601.11
Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 1, March 2021)
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), 2024. 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
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[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.
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[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.
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Author Information
  • Mathematics Department, University for Development Studies, Tamale, Ghana

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

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

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  • 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

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    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

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    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

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  • @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://download.sciencepg.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}
    }
    

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    AU  - Azure Isaac
    AU  - Twum Boakye Stephen
    AU  - Baba Seidu
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    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
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    PB  - Science Publishing Group
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    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.
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