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On a reduced cost derivative-free higher-order numerical algorithm for nonlinear systems. (English) Zbl 07241627
Summary: A derivative-free iterative method of convergence order five for solving systems of nonlinear equations is presented. The computational efficiency is examined and the comparison between efficiencies of the proposed technique with existing most efficient techniques is performed. It is shown that the new method has less computational cost than the existing counterparts, which implies that the method is computationally more efficient. Numerical problems, including those resulting from discretization of boundary value problem and integral equation, are given to compare the performance of the proposed method with existing methods and to confirm the theoretical results concerning the order of convergence and efficiency. The numerical results, including the elapsed CPU time, confirm the accurate and efficient character of the proposed technique.
MSC:
65H10 Numerical computation of solutions to systems of equations
41A25 Rate of convergence, degree of approximation
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
Software:
Mathematica; MPFR
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References:
[1] Argyros, IK; Hilout, S., Numerical Methods in Nonlinear Analysis (2013), New Jersey: World Scientific Publishing Company, New Jersey
[2] Chandrasekhar, S., Radiative Transfer (1960), New York: Dover, New York
[3] Constantinides, A.; Mostoufi, N., Numerical Methods for Chemical Engineers with MATLAB Applications (1999), New Jersey: Prentice Hall PTR, New Jersey
[4] Fousse L, Hanrot G, Lefèvre V, Pélissier P, Zimmermann P (2007) MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2):13 · Zbl 1365.65302
[5] Grau-Sánchez, M.; Noguera, M.; Amat, S., On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237, 363-372 (2013) · Zbl 1308.65074
[6] Liu, Z.; Zheng, Q.; Zhao, P., A variant of Steffensen’s method of fourth-order convergence and its applications, Appl. Math. Comput., 216, 1978-1983 (2012) · Zbl 1208.65064
[7] Moré, JJ; Allgower, EL; Georg, K., A collection of nonlinear model problems, Computational Solution of Nonlinear Systems of Equations Lectures in Applied Mathematics, 723-762 (1990), RI: American Mathematical Society, Providence, RI
[8] Ortega, JM; Rheinboldt, WC, Iterative Solution of Nonlinear Equations in Several Variables (1970), New York: Academic Press, New York
[9] Ren, H.; Wu, Q.; Bi, W., A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput., 209, 206-210 (2009) · Zbl 1166.65338
[10] Sharma, JR; Arora, H., Efficient derivative-free numerical methods for solving systems of nonlinear equations, Comp. Appl. Math., 35, 269-284 (2016) · Zbl 1342.65131
[11] Sharma, JR; Arora, H., An efficient derivative-free iterative method for solving systems of nonlinear equations, Appl. Anal. Discrete Math., 7, 390-403 (2013) · Zbl 1299.65102
[12] Steffensen, JF, Remarks on iteration., Skand. Aktuar Tidskr., 16, 64-72 (1933) · Zbl 0007.02601
[13] Traub, JF, Iterative Methods for the Solution of Equations (1982), New York: Chelsea Publishing Company, New York
[14] Wang, X.; Zhang, T., A family of Steffensen type methods with seventh-order convergence, Numer. Algor., 62, 429-444 (2013) · Zbl 1276.65028
[15] Wang, X.; Zhang, T.; Qian, W.; Teng, M., Seventh-order derivative-free iterative method for solving nonlinear syatems, Numer. Algor., 70, 545-558 (2015) · Zbl 1331.65077
[16] Weerkoon, S.; Fernando, TGI, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett., 13, 87-93 (2000) · Zbl 0973.65037
[17] Wolfram S (2003) The mathematica book, 5th ed. Wolfram Media, Champaign, IL · Zbl 0878.65001
[18] Xiao, XY; Yin, HW, Increasing the order of convergence for iterative methods to solve nonlinear systems, Calcolo, 53, 285-300 (2016) · Zbl 1356.65140
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