zbMATH — the first resource for mathematics

A faster King-Werner-type iteration and its convergence analysis. (English) Zbl 07250981
Summary: We introduce a new faster two-step King-Werner-type iterative method for solving nonlinear equations. The methodology is based on rational Hermite interpolation. The local as well as semi-local convergence analyses are presented under weak center Lipschitz and Lipschitz conditions. The convergence order is increased from \(1 + \sqrt{2}\) to 3 without any additional function calculations. Another advantage is the convenient fact that this method does not use derivatives. Numerical examples further validate the theoretical results.
65H10 Numerical computation of solutions to systems of equations
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
41A25 Rate of convergence, degree of approximation
Full Text: DOI
[1] Traub, JF., Iterative methods for the solution of equations (1984), Englewood Cliffs: Prentice Hall, Englewood Cliffs
[2] Argyros, IK.In: Chui CK, Wuytack L, editors, Computational theory of iterative methods: series, studies in computational mathematics. Vol. 15. New York: Elsevier; 2007. · Zbl 1147.65313
[3] Argyros, IK., Covergence and applications of newton-type iterations (2008), New York: Springer-Verlag, New York
[4] Argyros, IK; Cho, YJ; Hilout, S., Numerical methods for equations and its applications (2012), New York: CRC Press, New York
[5] Argyros, IK; Magreñán, ÁA., Iterative methods and their dynamics with applications (2017), New York: CRC Press, New York
[6] Dehghan, M.; Hajarian, M., Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations, Comput Appl Math, 29, 19-30 (2010) · Zbl 1189.65091
[7] Dehghan, M.; Hajarian, M., New iterative method for solving non-linear equations with fourth-order convergence, Int J Comput Math, 87, 834-839 (2010) · Zbl 1193.65056
[8] Dehghan, M.; Hajarian, M., On some cubic convergence iterative formulae without derivatives for solving nonlinear equations, Int J Numer Meth Biomed Eng, 27, 722-731 (2011) · Zbl 1227.65042
[9] Dehghan, M.; Hajarian, M., On derivative free cubic convergence iterative methods for solving nonlinear equations, Comput Math Math Phys, 51, 513-519 (2011) · Zbl 1249.65107
[10] King, RF., Tangent methods for nonlinear equations, Numer Math, 18, 298-304 (1971) · Zbl 0215.27403
[11] Gutiérrez, JM; Hernández, MA., A family of Chebyshev-Halley type methods in Banach spaces, Bull Aust Math Soc, 55, 113-130 (1997) · Zbl 0893.47043
[12] Hernández, MA., Chebyshev’s approximation algorithms and applications, Comput Math Appl, 41, 433-445 (2001) · Zbl 0985.65058
[13] Parida, PK; Gupta, DK., Recurrence relations for a Newton-like method in Banach spaces, J Comput Appl Math, 206, 873-887 (2007) · Zbl 1119.47063
[14] Ezquerro, JA; Grau-Sánchez, M.; Hernández, MA., Solving non-differentiable equations by a new one-point iterative method with memory, J Complex, 28, 48-58 (2012) · Zbl 1417.65132
[15] Magreñán, ÁA; Argyros, IK., New improved convergence analysis for the secant method, Math Comput Simul, 119, 161-170 (2016)
[16] Argyros, IK; Cordero, A.; Magreñán, ÁA, On the convergence of a damped Newton-like method with modified right hand side vector, Appl Math Comput, 266, 927-936 (2015) · Zbl 1410.65209
[17] Ezquerro, JA; Hernández, MA., Enlarging the domain of starting points for Newton’s method under center conditions on the first Fréchet-derivative, J Complex, 33, 89-106 (2016) · Zbl 1333.65056
[18] Behl, R.; Cordero, A.; Motsa, SS, Stable high-order iterative methods for solving nonlinear models, Appl Math Comput, 303, 70-88 (2017) · Zbl 1411.65074
[19] Cordero, A.; Ezquerro, JA; Hernández-Veron, MA, On the local convergence of fifth-order iterative method in Banach spaces, Appl Math Comput, 251, 396-403 (2015) · Zbl 1328.65130
[20] Sharma, JR; Gupta, P., An efficient family of Traub-Steffensen-type methods for solving systems of nonlinear equations, Adv Numer Anal (2003)
[21] Sharma, JR; Guha, RK; Sharma, R., An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer Algor, 62, 307-323 (2013) · Zbl 1283.65051
[22] Sharma, JR; Sharma, R.; Bahl, A., An improved Newton-Traub composition for solving systems of nonlinear equations, Appl Math Comput, 290, 98-110 (2016) · Zbl 1410.65198
[23] Argyros, IK; Hilout, S., Weaker conditions for the convergence of Newton’s method, J Complex, 28, 364-387 (2012) · Zbl 1245.65058
[24] Argyros, IK; Ren, H., On the convergence of efficient King-Werner-type methods of order \(####\), J Comp Appl Math, 285, 169-180 (2015) · Zbl 1314.65073
[25] Werner, W., Some supplementary results on the \(####\) order method for the solution of nonlinear equations, Numer Math, 38, 383-392 (1982) · Zbl 0478.65029
[26] Ren, H.; Argyors, IK., On the convergence of King-Werner-type methods of order \(####\) free of derivative, Appl Math Comput, 256, 148-159 (2015) · Zbl 1338.65148
[27] Werner, W., Uber ein Verfahren der Ordung \(####\) zur Nullstellenbestimmung, Numer Math, 32, 333-342 (1979) · Zbl 0431.65040
[28] McDougall, TJ; Wotherspoon, SJ., A simple modification of Newton’s method to achieve convergence of order \(####\), Appl Math Lett, 29, 20-25 (2014) · Zbl 1311.65049
[29] Ortega, JM; Rheinboldt, WC., Iterative solution of nonlinear equations in several variables (1970), New York: Academic Press, New York
[30] Ostrowski, AM., Solutions of equations and system of equations (1960), New York: Academic Press, New York
[31] Cordero, A.; Torregrosa, JR., Variants of Newton’s method using fifth-order quadrature formulas, Appl Math Comput, 190, 686-698 (2007) · Zbl 1122.65350
[32] Hansen, E.; Patrick, M., A family of root finding methods, Numer Math, 27, 257-269 (1976) · Zbl 0361.65041
[33] Noor, MA; Waseem, M., Some iterative methods for solving a system of nonlinear equations, Comput Math Appl, 57, 101-106 (2009) · Zbl 1165.65349
[34] Weerakoon, S.; Fernando, TGI., A variant of Newton’s method with accelerated third-order convergence, Appl Math Lett, 13, 87-93 (2000) · Zbl 0973.65037
[35] Grau-Sánchez, M.; Grau, Á.; Noguera, M., Ostrowski type methods for solving systems of nonlinear equations, Appl Math Comput, 218, 2377-2385 (2011) · Zbl 1243.65056
[36] Fang, X.; Ni, Q.; Zeng, M., A modified quasi-Newton method for nonlinear equations, J Comput Appl Math, 328, 44-58 (2018) · Zbl 1372.65146
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.