Numerical Solution of Nonlinear Systems of Algebraic Equations

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang

doi:doi:10.11648/j.ijdsa.20180401.14

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Numerical Solution of Nonlinear Systems of Algebraic Equations

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang

Published in International Journal of Data Science and Analysis (Volume 4, Issue 1)

Received: 29 January 2018 Accepted: 27 February 2018 Published: 23 March 2018

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Abstract

Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.

Published in	International Journal of Data Science and Analysis (Volume 4, Issue 1)
DOI	10.11648/j.ijdsa.20180401.14
Page(s)	20-23
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

Convergent, Jacobian; Matrix, Approximation, Starting Value, Iteration, Nonlinear System

References

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[2]	Chein-Shan Liu & Satya N. Atluri (2008) A Novel Time Integration Method for Solving A Large System of Non-Linear Algebraic Equations. Technical Science Press CMES, 31(2):71-83
[3]	De Cezaro, A (2008): On Steepest-Descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations
[4]	Gomes, F. A. M & Martinez, J. M. (1992) Parallel implementations of Broyden's method, Springer link 47(3-4):361-366
[5]	Huang, Tsung-Ming (2011): Numerical Solutions of Nonlinear Systems of Equations Department of Mathematics, National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw
[6]	Gilberto E. Urroz (2004) Solution of non-linear equations
[7]	Paul’s online math notes: Algebra-Nonlinear system-Lamar University
[8]	Goh, B. S. and McDonald, D. B. (2015) Newton Methods to Solve a System of Nonlinear Algebraic Equations. Journal of Optimization Theory and Application
[9]	Vincent, T., Grantham, W. (1997): Nonlinear and Optimal Control Systems. Wiley, New York
[10]	Barbashin, E. A. and Krasovskii, N. N. (1952): On the stability of a motion in the large. Dokl. Akad. Nauk. SSR 86, 453–456
[11]	Powell, M. J. D. (1986): How Bad are the BFGS Methods when the Objective Function is Quadratic. Mathematics Programme 34, 34–47 8
[12]	Goh, B. S. (1994) Global attractivity and stability of a scalar nonlinear difference equation. Computer and Mathematics Application 28, 101–107
[13]	Fan, J. Y. and Yuan, Y. X. (2005): On the quadratic convergence of the Levenberg–Marquardt method without non-singularity assumption. Computing 74, 23–39
[14]	Powell, M. J. D (1970) A Hybrid Method for Nonlinear Equations, in Numerical Methods for Nonlinear Algebraic Equations, pp. 87–114. Gordon and Breach, London
[15]	Byrd, R. H., Marazzi, M. and Nocedal, J. (2004): On the convergence of Newton iterations to non-stationary points. Mathematics Programme 99, 127–148
[16]	LaSalle, J. P. (1976): The Stability of Dynamical Systems. SIAM, Philadelphia
[17]	Goh, B. S. (2010): Convergence of numerical methods in unconstrained optimization and the solution of nonlinear equations. Journal of Optimization and Theory Application 144, 43–55

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Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang. (2018). Numerical Solution of Nonlinear Systems of Algebraic Equations. International Journal of Data Science and Analysis, 4(1), 20-23. https://doi.org/10.11648/j.ijdsa.20180401.14

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ACS Style

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang. Numerical Solution of Nonlinear Systems of Algebraic Equations. Int. J. Data Sci. Anal. 2018, 4(1), 20-23. doi: 10.11648/j.ijdsa.20180401.14

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AMA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang. Numerical Solution of Nonlinear Systems of Algebraic Equations. Int J Data Sci Anal. 2018;4(1):20-23. doi: 10.11648/j.ijdsa.20180401.14

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@article{10.11648/j.ijdsa.20180401.14,
  author = {Kamoh Nathaniel Mahwash and Gyemang Dauda Gyang},
  title = {Numerical Solution of Nonlinear Systems of Algebraic Equations},
  journal = {International Journal of Data Science and Analysis},
  volume = {4},
  number = {1},
  pages = {20-23},
  doi = {10.11648/j.ijdsa.20180401.14},
  url = {https://doi.org/10.11648/j.ijdsa.20180401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20180401.14},
  abstract = {Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.},
 year = {2018}
}

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TY  - JOUR
T1  - Numerical Solution of Nonlinear Systems of Algebraic Equations
AU  - Kamoh Nathaniel Mahwash
AU  - Gyemang Dauda Gyang
Y1  - 2018/03/23
PY  - 2018
N1  - https://doi.org/10.11648/j.ijdsa.20180401.14
DO  - 10.11648/j.ijdsa.20180401.14
T2  - International Journal of Data Science and Analysis
JF  - International Journal of Data Science and Analysis
JO  - International Journal of Data Science and Analysis
SP  - 20
EP  - 23
PB  - Science Publishing Group
SN  - 2575-1891
UR  - https://doi.org/10.11648/j.ijdsa.20180401.14
AB  - Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.
VL  - 4
IS  - 1
ER  -

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Author Information

Kamoh Nathaniel Mahwash

Department of Mathematics and Statistics, Bingham University, Karu, Nigeria
Gyemang Dauda Gyang

Department of Mathematics and Statistics, Plateau State Polytechnic, Barkin Ladi, Nigeria

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APA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang. (2018). Numerical Solution of Nonlinear Systems of Algebraic Equations. International Journal of Data Science and Analysis, 4(1), 20-23. https://doi.org/10.11648/j.ijdsa.20180401.14

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ACS Style

Kamoh Nathaniel Mahwash; Gyemang Dauda Gyang. Numerical Solution of Nonlinear Systems of Algebraic Equations. Int. J. Data Sci. Anal. 2018, 4(1), 20-23. doi: 10.11648/j.ijdsa.20180401.14

Copy | Download

AMA Style

Kamoh Nathaniel Mahwash, Gyemang Dauda Gyang. Numerical Solution of Nonlinear Systems of Algebraic Equations. Int J Data Sci Anal. 2018;4(1):20-23. doi: 10.11648/j.ijdsa.20180401.14

Copy | Download

@article{10.11648/j.ijdsa.20180401.14,
  author = {Kamoh Nathaniel Mahwash and Gyemang Dauda Gyang},
  title = {Numerical Solution of Nonlinear Systems of Algebraic Equations},
  journal = {International Journal of Data Science and Analysis},
  volume = {4},
  number = {1},
  pages = {20-23},
  doi = {10.11648/j.ijdsa.20180401.14},
  url = {https://doi.org/10.11648/j.ijdsa.20180401.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20180401.14},
  abstract = {Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Numerical Solution of Nonlinear Systems of Algebraic Equations
AU  - Kamoh Nathaniel Mahwash
AU  - Gyemang Dauda Gyang
Y1  - 2018/03/23
PY  - 2018
N1  - https://doi.org/10.11648/j.ijdsa.20180401.14
DO  - 10.11648/j.ijdsa.20180401.14
T2  - International Journal of Data Science and Analysis
JF  - International Journal of Data Science and Analysis
JO  - International Journal of Data Science and Analysis
SP  - 20
EP  - 23
PB  - Science Publishing Group
SN  - 2575-1891
UR  - https://doi.org/10.11648/j.ijdsa.20180401.14
AB  - Considered in this paper are two basic methods of approximating the solutions of nonlinear systems of algebraic equations. The Steepest Descent method was presented as a way of obtaining good and sufficient initial guess (starting value) which is in turn used for the Broyden’s method. Broyden’s method on the other hand replaces the Newton’s method which requires the use of the inverse of the Jacobian matrix at every new step of iteration with a matrix whose inverse is directly determined at each step by up-dating the previous inverse. The result obtained by this method revealed that the setbacks encountered in computing the inverse of the Jacobian matrix at every step number is eliminated hence saving human effort and computer time. The obtained results also showed that the number of steps that is reduced when compared to Newton’s method used on the same problem.
VL  - 4
IS  - 1
ER  -

Copy | Download