A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method

Tahar Latreche

doi:doi:10.11648/j.acm.20140303.11

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A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method

Tahar Latreche

Received: 3 May 2014 Accepted: 17 May 2014 Published: 30 May 2014

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Abstract

The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.

DOI	10.11648/j.acm.20140303.11
Published in	Applied and Computational Mathematics (Volume 3, Issue 3, June 2014)
Page(s)	68-74
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

Algebraic Equations, Linear and Non-Linear Algebra, Elementary Equations, Polynomial Equations, Runge-Kutta Method

References

[1]	Bird, J., (2010). Higher Engineering Mathematics. Sixth Edition, Elsevier, Ltd.
[2]	Ciarlet, P. G., and Lions, J. L., (2000). Handbook of Numerical Analysis. Vol. 7. First Edition, Elsevier Science.
[3]	Kiusalaas, J., (2005). Numerical Methods in Engineering with Python. First Edition, Cambridge University Press.
[4]	Polyanin, A. D. and Manzhrov, A. V., (2007). Handbook of Mathematics for Engineers and Scientists. First Edition, Taylor and Francis Group, LLC.
[5]	Press, W. H. et al., (2007). Numerical Recipes. Third Edition, Cambridge University Press.
[6]	Press, W. H. et al., (1997). Numerical Recipes in Fortran 77. Second Edition, Cambridge University Press.
[7]	Press, W. H. et al., (1997). Numerical Recipes in Fortran 90. Second Edition, Cambridge University Press.
[8]	Riley, K. F. et al., (2006). Mathematical Methods for Physics and Engineering. Third Edition, Cambridge University Press.
[9]	Soyeur, A. et al., (2011). Cours de Mathématiques. http://www.les-mathematiques.net
[10]	Yang, W. Y., et al., (2005). Applied Numerical Methods using Matlab. First Edition, John-Wiley and Sons, Inc.

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APA Style

Tahar Latreche. (2014). A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Applied and Computational Mathematics, 3(3), 68-74. https://doi.org/10.11648/j.acm.20140303.11

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ACS Style

Tahar Latreche. A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Appl. Comput. Math. 2014, 3(3), 68-74. doi: 10.11648/j.acm.20140303.11

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AMA Style

Tahar Latreche. A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Appl Comput Math. 2014;3(3):68-74. doi: 10.11648/j.acm.20140303.11

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@article{10.11648/j.acm.20140303.11,
  author = {Tahar Latreche},
  title = {A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {3},
  pages = {68-74},
  doi = {10.11648/j.acm.20140303.11},
  url = {https://doi.org/10.11648/j.acm.20140303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140303.11},
  abstract = {The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.},
 year = {2014}
}

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TY  - JOUR
T1  - A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method
AU  - Tahar Latreche
Y1  - 2014/05/30
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140303.11
DO  - 10.11648/j.acm.20140303.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 68
EP  - 74
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140303.11
AB  - The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.
VL  - 3
IS  - 3
ER  -

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Author Information

Tahar Latreche

Doctorate student in Civil Engineering, B.P. 129 Salem Lalmi, 40003 Khenchela, Algeria

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APA Style

Tahar Latreche. (2014). A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Applied and Computational Mathematics, 3(3), 68-74. https://doi.org/10.11648/j.acm.20140303.11

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ACS Style

Tahar Latreche. A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Appl. Comput. Math. 2014, 3(3), 68-74. doi: 10.11648/j.acm.20140303.11

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AMA Style

Tahar Latreche. A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method. Appl Comput Math. 2014;3(3):68-74. doi: 10.11648/j.acm.20140303.11

Copy | Download

@article{10.11648/j.acm.20140303.11,
  author = {Tahar Latreche},
  title = {A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {3},
  pages = {68-74},
  doi = {10.11648/j.acm.20140303.11},
  url = {https://doi.org/10.11648/j.acm.20140303.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140303.11},
  abstract = {The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method
AU  - Tahar Latreche
Y1  - 2014/05/30
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140303.11
DO  - 10.11648/j.acm.20140303.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 68
EP  - 74
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140303.11
AB  - The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.
VL  - 3
IS  - 3
ER  -

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