Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces

[EN] Semilocal convergence for an iteration of order five for solving nonlinear equations in Banach spaces is established under second-order Fr,chet derivative satisfying the Lipschitz condition. It is done by deriving a number of recurrence relations. A theorem for the existence-uniqueness along with the estimation of error bounds of the solution is established. Its R-order is shown to be equal to five. Both efficiency and computational efficiency indices are given. A variety of examples are worked out to show its applicability. In comparison to existing methods having similar R-orders, improved results in terms of computational efficiency index and error bounds are found using our methodology.The authors thank the referees for their valuable comments which have improved the presentation of the paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal Convergence Analysis of an Iteration of Order Five Using Recurrence Relations in Banach Spaces. Mediterranean Journal of Mathematics. 13(6):4219-4235. doi:10.1007/s00009-016-0741-5S42194235136Cordero A., Hueso J.L., Martinez E., Torregrosa J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)Chen, L., Gu, C., Ma Y.: Semilocal convergence for a fifth order Newton’s method using Recurrence relations in Banach spaces. J. Appl. Math. 2011, 1–15 (2011)Wang X., Kou J., Gu C.: Semilocal convergence of a sixth order Jarrat method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)Zheng L., Gu C.: Semilocal convergence of a sixth order method in Banach spaces. Numer. Algorithms 61, 413–427 (2012)Zheng L., Gu C.: Recurrence relations for semilocal convergence of a fifth order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)Proinov P.D., Ivanov S.I.: On the convergence of Halley’s method for multiple polynomial zeros. Mediterr. J. Math. 12, 555–572 (2015)Ezquerro, J.A., Hernández-Verón M.A.: On the domain of starting points of Newton’s method under center lipschitz conditions. Mediterr. J. Math. (2015). doi: 10.1007/s00009-015-0596-1Cordero A., Hernández-Verón M.A., Romero N., Torregrosa J.R.: Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces. J. Comput. Appl. Math. 273, 205–213 (2015)Parida P.K., Gupta D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206, 873–887 (2007)Hueso J.L., Martínez E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)Argyros, I.K., Hilout S.: Numerical methods in nonlinear analysis. World Scientific Publ. Comp., New Jersey (2013)Argyros, I.K., Hilout, S., Tabatabai, M.A.: Mathematical modelling with applications in biosciences and engineering. Nova Publishers, New York (2011)Argyros I.K., Khattri S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2004)Argyros I.K., Hilout A.S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)Kantorovich, L.V., Akilov G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros I.K., George S., Magreñán A.A.: Local convergence for multi-point-parametric Chebyshev-Halley-type methods of higher convergence order. J. Comput. Appl. Math. 282, 215–224 (2015)Argyros I.K., Magreñán A.A.: A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2015)Amat S., Hernández M.A., Romero N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Chun, C., St $${\breve{a}}$$ a ˘ nic $${\breve{a}}$$ a ˘ , P., Neta, B.: Third-order family of methods in Banach spaces. Comput. Math. Appl. 61, 1665–1675 (2011)Ostrowski, A.M.: Solution of equations in Euclidean and Banach spaces, 3rd edn. Academic Press, New-York (1977)Jaiswal J.P.: Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency. Numer. Algorithms 71, 933–951 (2015)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964

Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

AbstractIn this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1] for the solution x∗ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F″, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α, our analysis reduces to those for the Chebyshev method (α=0) and the convex acceleration of Newton’s method (α=1) respectively with improved results

Local Convergence for an Improved Jarratt-type Method in Banach Space

We present a local convergence analysis for an improved Jarratt-type methods of order at least five to approximate a solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given using hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the third Fréchet derivative. Numerical examples are also provided in this study, where the older hypotheses are not satisfied to solve equations but the new hypotheses are satisfied

Majorizing sequences for Newton’s method from initial value problems

AbstractThe most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems

Directional k-Step Newton Methods in n Variables and its Semilocal Convergence Analysis

[EN] The directional k-step Newton methods (k a positive integer) is developed for solving a single nonlinear equation in n variables. Its semilocal convergence analysis is established by using two different approaches (recurrent relations and recurrent functions) under the assumption that the first derivative satisfies a combination of the Lipschitz and the center-Lipschitz continuity conditions instead of only Lipschitz condition. The convergence theorems for the existence and uniqueness of the solution for each of them are established. Numerical examples including nonlinear Hammerstein-type integral equations are worked out and significantly improved results are obtained. It is shown that the second approach based on recurrent functions solves problems failed to be solved by first one using recurrent relations. This demonstrates the efficacy and applicability of these approaches. This work extends the directional one and two-step Newton methods for solving a single nonlinear equation in n variables. Their semilocal convergence analysis using majorizing sequences are studied in Levin (Math Comput 71(237): 251-262, 2002) and Ioannis (Num Algorithms 55(4): 503-528, 2010) under the assumption that the first derivative satisfies the Lipschitz and the combination of the Lipschitz and the center-Lipschitz continuity conditions, respectively. Finally, the computational order of convergence and the computational efficiency of developed method are studied.The authors thank the referees for their fruitful suggestions which have uncovered several weaknesses leading to the improvement in the paper. A. Kumar wishes to thank UGC-CSIR(Grant no. 2061441001), New Delhi and IIT Kharagpur, India, for their financial assistance during this work.Kumar, A.; Gupta, D.; Martínez Molada, E.; Singh, S. (2018). 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Semilocal convergence of a family of iterative methods in Banach spaces

[EN] In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory.This work has been supported by Ministerio de Ciencia e Innovaci´on MTM2011-28636-C02-02 and by Vicerrectorado de Investigaci´on. Universitat Polit`ecnica de Val`encia PAID-SP-2012-0498Hueso Pagoaga, JL.; Martínez Molada, E. (2014). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms. 67(2):365-384. https://doi.org/10.1007/s11075-013-9795-7S365384672Traub, J.F.: Iterative Methods for the Solution of Nonlinear Equations. Prentice Hall, New York (1964)Kantorovich, L.V.: On the newton method for functional equations. Doklady Akademii Nauk SSSR 59, 1237–1240 (1948)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, I: The Halley method. Computing 44, 169–184 (1990)Candela, V., Marquina, A.: Recurrence relations for rational cubic methods, II: The Chebyshev method. Computing 45, 355–367 (1990)Hernández, M.A.: Reduced recurrence relations for the Chebyshev method. J. Optim. Theory Appl. 98, 385–397 (1998)Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for super-Halley method. J. Comput. Math. Appl. 7, 1–8 (1998)Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-like methods. Appl. Math. Optim. 41, 227–236 (2000)Ezquerro, J.A., Hernández, M.A.: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49, 325–342 (2009)Argyros, I., K., Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Hilout, S.: On the semilocal convergence of efficient Chebyshev Secant-type methods. J. Comput. Appl. Math. 235–10, 3195–3206 (2011)Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newtons method. J. Complex. 28(3), 364–387 (2012)Wang, X., Gu, C., Kou, J.: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algoritm. 54, 497–516 (2011)Kou, J., Li, Y., Wang, X.: A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)Zheng, L., Gu, C.: Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algoritm. 59, 623–638 (2012)Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algoritm. 57, 441–456 (2011)Hernández, M.A.: The newton method for operators with hlder continuous first derivative. J. Optim. Appl. 109, 631–648 (2001)Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Hlder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006)Zhao, Y., Wu, Q.: Newton-Kantorovich theorem for a family of modified Halleys method under Hlder continuity conditions in Banach spaces. Appl. Math. Comput. 202, 243–251 (2008)Argyros, I.K.: Improved generalized differentiability conditions for Newton-like methods. J. Complex. 26, 316–333 (2010)Hueso, J.L., Martínez. E., Torregrosa, J.R.: Third and fourth order iterative methods free from second derivative for nonlinear systems. Appl. Math. Comput. 211, 190–197 (2009)Taylor, A.Y., Lay, D.: Introduction to Functional Analysis, 2nd edn.New York, Wiley (1980)Jarrat, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)Cordero, A., Torregrosa, J.R.: Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007