Two improved classes of Broyden's methods for solving nonlinear systems of equations
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Authors
Mohammad H. Al-Towaiq
- Deparment of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid, Jordan.
Yousef S. Abu hour
- Deparment of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid, Jordan.
Abstract
In this paper, we propose two efficient algorithms based on Broyden’s methods by using the central finite difference and
modification of Newton’s method for solving systems of nonlinear equations. The most significant features of these algorithms
are their simplicity and excellent accuracy. Some numerical examples are given to test the validity of the proposed algorithms
and for comparison reasons. Superior results show the efficiency and accuracy of the proposed algorithms and a tremendous
improvements in Broyden’s methods.
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ISRP Style
Mohammad H. Al-Towaiq, Yousef S. Abu hour, Two improved classes of Broyden's methods for solving nonlinear systems of equations, Journal of Mathematics and Computer Science, 17 (2017), no. 1, 22-31
AMA Style
Al-Towaiq Mohammad H., hour Yousef S. Abu, Two improved classes of Broyden's methods for solving nonlinear systems of equations. J Math Comput SCI-JM. (2017); 17(1):22-31
Chicago/Turabian Style
Al-Towaiq, Mohammad H., hour, Yousef S. Abu. "Two improved classes of Broyden's methods for solving nonlinear systems of equations." Journal of Mathematics and Computer Science, 17, no. 1 (2017): 22-31
Keywords
- Nonlinear systems of equations
- Newton’s method
- Broyden’s methods
- quasi Newton method
- finite difference
- secant equation.
MSC
References
-
[1]
Y. Abu-Hour, Improved Classes of Broyden methods for solving a nonlinear systems of equations, MSc. thesis, Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan (2016)
-
[2]
M. H. Al-Towaiq, Y. S. Abu Hour, Two Improved Methods Based on Broyden’s Newton Methods for the solution of nonlinear systems of equations, J. Teknologi, (accepted)
-
[3]
K. Amini, F. Rostami, Three-steps modified Levenberg-Marquardt method with a new line search for systems of nonlinear equations, J. Comput. Appl. Math., 300 (2016), 30–42.
-
[4]
L. Berenguer, D. Tromeur-Dervout, Developments on the Broyden procedure to solve nonlinear problems arising in CFD, Comput. & Fluids, 88 (2013), 891–896.
-
[5]
C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp., 19 (1965), 577–593.
-
[6]
C. G. Broyden, J. E. Dennis, J. J. Moré, On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl., 12 (1973), 223–245.
-
[7]
V. Candela, R. Peris, The rate of multiplicity of the roots of nonlinear equations and its application to iterative methods, Appl. Math. Comput., 264 (2015), 417–430.
-
[8]
S. Chandrasekhar, Radiative transfer, Dover Publications, Inc., New York (1960)
-
[9]
C. Y. Deng, A generalization of the Sherman-Morrison-Woodbury formula, Appl. Math. Lett., 24 (2011), 1561–1564.
-
[10]
J. E. Dennis Jr., On some methods based on Broyden’s secant approximation to the Hessian , Numerical methods for non-linear optimization, Conf., Dundee, (1971), 19–34, Academic Press, London (1972)
-
[11]
C. T. Kelley, Solving nonlinear equations with Newton’s method. Fundamentals of Algorithms, , Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003)
-
[12]
J. J. Moré, A collection of nonlinear model problems, Computational solution of nonlinear systems of equations, Fort Collins, CO, (1988), 723–762, Lectures in Appl. Math., Amer. Math. Soc., Providence, RI, 26, (1990)
-
[13]
J. J. Moré, B. S. Garbow, K. E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software, 7 (1981), 17–41.
-
[14]
J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London (1970)
-
[15]
K. Sayevanda, H. Jafari, On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth- and fifth-order convergence, Appl. Math. Model., 40 (2016), 1467–1476.
-
[16]
S. Sharifi, M. Salimi, S. Siegmund, T. Lotfi, A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations, Math. Comput. Simulation, 119 (2016), 69–90.
-
[17]
J. R. Sharma, R. Sharma, N. Kalra, A novel family of composite Newton-Traub methods for solving systems of nonlinear equations, Appl. Math. Comput., 269 (2015), 520–535.
-
[18]
X.-Y. Xiao, H.-W. Yin, A simple and efficient method with high order convergence for solving systems of nonlinear equations, Comput. Math. Appl., 69 (2015), 1220–1231.