American Journal of Mathematical and Computer Modelling

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Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink® and MAPLE®

Received: 25 November 2017    Accepted: 07 December 2017    Published: 03 January 2018
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Abstract

Interest in Science, Technology, Engineering and Mathematics (STEM)-based courses at tertiary institution is on a steady decline. To curd this trend, among others, teaching and learning of STEM subjects must be made less mental tasking. This can be achieved by the aid of technical computing software. In this study, a novel approach to explaining and implementing Newton’s method as a numerical approach for solving Nonlinear System of Algebraic Equations (NLSAEs) was presented using MATLAB® and MAPLE® in a complementary manner. Firstly, the analytical based computational software MAPLE® was used to substitute the initial condition values into the NLSAEs and then evaluate them to get a constant value column vector. Secondly, MAPLE® was used to obtain partial derivative of the NLSAEs hence, a Jacobean matrix. Substituting initial condition into the Jacobean matrix and evaluating resulted in a constant value square matrix. Both vector and matrix represent a Linear System of Algebraic Equations (LSAEs) for the related initial condition. This LSAEs was then solved using Gaussian Elimination method in the numerical-based computational software of MATLAB/Simulink®. This process was repeated until the solution to the NLSAEs converged. To explain the concept of quadratic convergence of the Newton’s method, power function of degree 2 (quad) relates the errors and successive errors in each iteration. This was achieved with the aid of Curve Fitting Toolbox of MATLAB®. Finally, a script file and a function file in MATLAB® were written that implements the complete solution process to the NLSAEs.

DOI 10.11648/j.ajmcm.20170204.14
Published in American Journal of Mathematical and Computer Modelling (Volume 2, Issue 4, November 2017)
Page(s) 117-131
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

Newton’s Method, MAPLE®, MATLAB®, Non-Linear System of Algebraic Equations

References
[1] Lyn Haynes (2008). Studying Stem? What are the Barriers. A Literature Review of the Choices Students Make. A Fact-file Provided By The Institution of Engineering and Technology http://www.theiet.org/factfiles
[2] Frank Y. Wang (2015). Physics with MAPLE: The Computer Algebra Resource for Mathematical Methods in Physics. ISBN: 3-527-40640-9
[3] Ralph E. White and Venkat R. Subramanian (2010). Computational Methods in Chemical Engineering with MAPLE. ISBN 978-3-642-04310-9
[4] Robin C. and Murat T. (1994). Engineering Explorations with MAPLE. ISBN: 0-15-502338-7
[5] George Z. V and Peter I. K (2005). Mechanics of Composite Materials with MATLAB, ISBN-10 3-540-24353-4
[6] Wndy L. M and Angel R. M (2002). Computational Statistics Handbook with MATLAB. ISBN 1-58488-229-8
[7] Bassem R. M (2000). Radar Systems Analysis and Design Using MATLAB. ISBN 1-58488-182-8
[8] Glyn James et al (2015). Modern Engineering Mathematics, ISBN: 978-1-292-08073-4
[9] Linge S., Langtangen H. P. (2016) Solving Nonlinear Algebraic Equations. In: Programming for Computations - MATLAB/Octave. Texts in Computational Science and Engineering, vol 14. Springer, Cham. ISBN 978-3-319-32452-4.
[10] Burden, Richard L. and J. Douglas Faires, (2010). Numerical Analysis, 9th Ed., Brooks Cole, ISBN 0538733519
[11] Robin Carr and Murat Tanyel (1994). Engineering Exploration with MAPLE. ISBN: 0-15-502338-7
[12] MAPLE User Manual Copyright © Maplesoft, a division of Waterloo Maple Inc.1996-2009. ISBN 978-1-897310-69-4
[13] Harold Klee and Randal Allen (2011). Simulation of Dynamic Systems with MATLAB and Simulink, ISBN -13: 978-1-4398-3674-3
[14] O. B euchre and M. Week (2006). Introduction to MATLAB & Simulink: A project Approach, Third Edition. ISBN: 978-1-934015-04-9
[15] Steven T. Karris (2006). Introduction to Simulink with Engineering Applications, ISBN 978-0-9744239-8-2
[16] Cheney, Ward and David Kincaid, (2007) Numerical Mathematics and Computing, 6th Ed., Brooks Cole, 2007, ISBN 0495114758
[17] Kincaid, David and Ward Cheney, (2002). Numerical Analysis: Mathematics of Scientific Computing, Vol. 2, 2002, ISBN 0821847880
[18] Curve Fitting Toolbox™ User’s Guide (2014). The Math Works, Inc. www.mathworks.com
Author Information
  • Department of Dynamics & Control System, Centre for Space Transport & Propulsion (CSTP), Lagos, Nigeria

  • Department of Marine Engineering, Fleet Support Unit BEECROFT, Lagos, Nigeria

  • Department of Chemical Propulsion System, Centre for Space Transport & Propulsion (CSTP), Lagos, Nigeria

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    Aliyu Bhar Kisabo, Nwojiji Cornelius Uchenna, Funmilayo Aliyu Adebimpe. (2018). Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink® and MAPLE®. American Journal of Mathematical and Computer Modelling, 2(4), 117-131. https://doi.org/10.11648/j.ajmcm.20170204.14

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    Aliyu Bhar Kisabo; Nwojiji Cornelius Uchenna; Funmilayo Aliyu Adebimpe. Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink® and MAPLE®. Am. J. Math. Comput. Model. 2018, 2(4), 117-131. doi: 10.11648/j.ajmcm.20170204.14

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    AMA Style

    Aliyu Bhar Kisabo, Nwojiji Cornelius Uchenna, Funmilayo Aliyu Adebimpe. Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink® and MAPLE®. Am J Math Comput Model. 2018;2(4):117-131. doi: 10.11648/j.ajmcm.20170204.14

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  • @article{10.11648/j.ajmcm.20170204.14,
      author = {Aliyu Bhar Kisabo and Nwojiji Cornelius Uchenna and Funmilayo Aliyu Adebimpe},
      title = {Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink® and MAPLE®},
      journal = {American Journal of Mathematical and Computer Modelling},
      volume = {2},
      number = {4},
      pages = {117-131},
      doi = {10.11648/j.ajmcm.20170204.14},
      url = {https://doi.org/10.11648/j.ajmcm.20170204.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajmcm.20170204.14},
      abstract = {Interest in Science, Technology, Engineering and Mathematics (STEM)-based courses at tertiary institution is on a steady decline. To curd this trend, among others, teaching and learning of STEM subjects must be made less mental tasking. This can be achieved by the aid of technical computing software. In this study, a novel approach to explaining and implementing Newton’s method as a numerical approach for solving Nonlinear System of Algebraic Equations (NLSAEs) was presented using MATLAB® and MAPLE® in a complementary manner. Firstly, the analytical based computational software MAPLE® was used to substitute the initial condition values into the NLSAEs and then evaluate them to get a constant value column vector. Secondly, MAPLE® was used to obtain partial derivative of the NLSAEs hence, a Jacobean matrix. Substituting initial condition into the Jacobean matrix and evaluating resulted in a constant value square matrix. Both vector and matrix represent a Linear System of Algebraic Equations (LSAEs) for the related initial condition. This LSAEs was then solved using Gaussian Elimination method in the numerical-based computational software of MATLAB/Simulink®. This process was repeated until the solution to the NLSAEs converged. To explain the concept of quadratic convergence of the Newton’s method, power function of degree 2 (quad) relates the errors and successive errors in each iteration. This was achieved with the aid of Curve Fitting Toolbox of MATLAB®. Finally, a script file and a function file in MATLAB® were written that implements the complete solution process to the NLSAEs.},
     year = {2018}
    }
    

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    AU  - Aliyu Bhar Kisabo
    AU  - Nwojiji Cornelius Uchenna
    AU  - Funmilayo Aliyu Adebimpe
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    DO  - 10.11648/j.ajmcm.20170204.14
    T2  - American Journal of Mathematical and Computer Modelling
    JF  - American Journal of Mathematical and Computer Modelling
    JO  - American Journal of Mathematical and Computer Modelling
    SP  - 117
    EP  - 131
    PB  - Science Publishing Group
    SN  - 2578-8280
    UR  - https://doi.org/10.11648/j.ajmcm.20170204.14
    AB  - Interest in Science, Technology, Engineering and Mathematics (STEM)-based courses at tertiary institution is on a steady decline. To curd this trend, among others, teaching and learning of STEM subjects must be made less mental tasking. This can be achieved by the aid of technical computing software. In this study, a novel approach to explaining and implementing Newton’s method as a numerical approach for solving Nonlinear System of Algebraic Equations (NLSAEs) was presented using MATLAB® and MAPLE® in a complementary manner. Firstly, the analytical based computational software MAPLE® was used to substitute the initial condition values into the NLSAEs and then evaluate them to get a constant value column vector. Secondly, MAPLE® was used to obtain partial derivative of the NLSAEs hence, a Jacobean matrix. Substituting initial condition into the Jacobean matrix and evaluating resulted in a constant value square matrix. Both vector and matrix represent a Linear System of Algebraic Equations (LSAEs) for the related initial condition. This LSAEs was then solved using Gaussian Elimination method in the numerical-based computational software of MATLAB/Simulink®. This process was repeated until the solution to the NLSAEs converged. To explain the concept of quadratic convergence of the Newton’s method, power function of degree 2 (quad) relates the errors and successive errors in each iteration. This was achieved with the aid of Curve Fitting Toolbox of MATLAB®. Finally, a script file and a function file in MATLAB® were written that implements the complete solution process to the NLSAEs.
    VL  - 2
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    ER  - 

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