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

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

Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations

[EN] In this work, we performed an study about the domain of existence and uniqueness for an efficient fifth order iterative method for solving nonlinear problems treated in their infinite dimensional form. The hypotheses for the operator and starting guess are weaker than in the previous studies. We assume omega continuity condition on second order Frechet derivative. This fact it is motivated by showing different problems where the nonlinear operators that define the equation do not verify Lipschitz and Holder condition; however, these operators verify the omega condition established. Then, the semilocal convergence balls are obtained and the R-order of convergence and error bounds can be obtained by following thee main theorem. Finally, we perform a numerical experience by solving a nonlinear Hammerstein integral equations in order to show the applicability of the theoretical results by obtaining the existence and uniqueness balls.This research was partially supported by Ministerio de Economia y Competitividad under grant PGC2018-095896-B-C22.Singh, S.; Martínez Molada, E.; Kumar, A.; Gupta, DK. (2020). Domain of Existence and Uniqueness for Nonlinear Hammerstein Integral Equations. Mathematics. 8(3):1-11. https://doi.org/10.3390/math8030384S11183Hernández, M. A. (2001). Chebyshev’s approximation algorithms and applications. Computers & Mathematics with Applications, 41(3-4), 433-445. doi:10.1016/s0898-1221(00)00286-8Amat, S., Hernández, M. A., & Romero, N. (2008). A modified Chebyshev’s iterative method with at least sixth order of convergence. Applied Mathematics and Computation, 206(1), 164-174. doi:10.1016/j.amc.2008.08.050Argyros, I. K., Ezquerro, J. A., Gutiérrez, J. M., Hernández, M. A., & Hilout, S. (2011). On the semilocal convergence of efficient Chebyshev–Secant-type methods. Journal of Computational and Applied Mathematics, 235(10), 3195-3206. doi:10.1016/j.cam.2011.01.005Hueso, J. L., & Martínez, E. (2013). Semilocal convergence of a family of iterative methods in Banach spaces. Numerical Algorithms, 67(2), 365-384. doi:10.1007/s11075-013-9795-7Zhao, Y., & Wu, Q. (2008). Newton–Kantorovich theorem for a family of modified Halley’s method under Hölder continuity conditions in Banach space. Applied Mathematics and Computation, 202(1), 243-251. doi:10.1016/j.amc.2008.02.004Parida, P. K., & Gupta, D. K. (2007). Recurrence relations for a Newton-like method in Banach spaces. Journal of Computational and Applied Mathematics, 206(2), 873-887. doi:10.1016/j.cam.2006.08.027Parida, P. K., & Gupta, D. K. (2008). Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces. Journal of Mathematical Analysis and Applications, 345(1), 350-361. doi:10.1016/j.jmaa.2008.03.064Cordero, A., Ezquerro, J. A., Hernández-Verón, M. A., & Torregrosa, J. R. (2015). On the local convergence of a fifth-order iterative method in Banach spaces. Applied Mathematics and Computation, 251, 396-403. doi:10.1016/j.amc.2014.11.084Argyros, I. K., & Hilout, S. (2013). On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 245, 1-9. doi:10.1016/j.cam.2012.12.002Argyros, I. K., George, S., & Magreñán, Á. A. (2015). Local convergence for multi-point-parametric Chebyshev–Halley-type methods of high convergence order. Journal of Computational and Applied Mathematics, 282, 215-224. doi:10.1016/j.cam.2014.12.023Wang, X., Kou, J., & Gu, C. (2012). Semilocal Convergence of a Class of Modified Super-Halley Methods in Banach Spaces. Journal of Optimization Theory and Applications, 153(3), 779-793. doi:10.1007/s10957-012-9985-9Argyros, I. K., & Magreñán, Á. A. (2015). A study on the local convergence and the dynamics of Chebyshev–Halley–type methods free from second derivative. Numerical Algorithms, 71(1), 1-23. doi:10.1007/s11075-015-9981-xWu, Q., & Zhao, Y. (2007). Newton–Kantorovich type convergence theorem for a family of new deformed Chebyshev method. Applied Mathematics and Computation, 192(2), 405-412. doi:10.1016/j.amc.2007.03.018Martínez, E., Singh, S., Hueso, J. L., & Gupta, D. K. (2016). Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Applied Mathematics and Computation, 281, 252-265. doi:10.1016/j.amc.2016.01.036Kumar, A., Gupta, D. K., Martínez, E., & Singh, S. (2018). Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces. Journal of Computational and Applied Mathematics, 330, 732-741. doi:10.1016/j.cam.2017.02.042Singh, S., Gupta, D. K., Martínez, E., & Hueso, J. L. (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-

Extending the Applicability of an Efficient Fifth Order Method Under Weak Conditions in Banach Space

We extend the applicability of an efficient fifth order method for solving Banach space valued equations. To achieve this we use weaker Lipschitz-type conditions in combination with our idea of the restricted convergence region. Numerical examples are used to compare our results favorably to the ones in earlier works

Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition

The semilocal and local convergence in Banach spaces is described for a fifth order iteration for the solutions of nonlinear equations when the Frechet derivative satisfies the Holder condition. The Holder condition generalizes the Lipschtiz condition. The importance of our work lies in the fact that many examples are available which fail to satisfy the Lipschtiz condition but satisfy the Holder condition. The existence and uniqueness theorem is established with error bounds for the solution. The convergence analysis is finally worked out on different examples and convergence balls for each of them are obtained. These examples include nonlinear Hammerstein and Fredholm integral equations and a boundary value problem. It is found that the larger radius of convergence balls is obtained for all the examples in comparison to existing methods using stronger conditions. (C) 2015 Elsevier Inc. All rights reserved.Singh, S.; Gupta, D.; Martínez Molada, E.; Hueso Pagoaga, JL. (2016). Semilocal and local convergence of a fifth order iteration with Frechet derivative satisfying Holder condition. Applied Mathematics and Computation. 276:266-277. doi:10.1016/j.amc.2015.11.062S26627727

Local convergence of a family of iterative methods for Hammerstein equations

[EN] In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Fr,chet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.This research was supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.This research was supported by Ministerio de Ciencia y Tecnología MTM2014-52016-C2-02.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (2016). Local convergence of a family of iterative methods for Hammerstein equations. Journal of Mathematical Chemistry. 54(7):1370-1386. https://doi.org/10.1007/s10910-016-0602-2S13701386547I.K. Argyros, S. Hilout, M.A. Tabatabai, Mathematical Modelling with Applications in Biosciences and Engineering (Nova Publishers, New York, 2011)J.F. Traub, Iterative Methods for the Solution of Equations (Prentice-Hall, Englewood Cliffs, New Jersey, 1964)A.M. Ostrowski, Solutions of Equations in Euclidean and Banach Spaces (Academic Press, New York, 1973)I.K. Argyros, J.A. Ezquerro, J.M. Gutiárrez, M.A. Hernández, S. Hilout, On the semilocal convergence of efficient ChebyshevSecant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)José L. Hueso, E. Martínez, Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algorithms 67, 365–384 (2014)X. Wang, C. Gu, J. Kou, Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algorithms 54, 497–516 (2011)J. Kou, Y. Li, X. Wang, A variant of super Halley method with accelerated fourth-order convergence. Appl. Math. Comput. 186, 535–539 (2007)L. Zheng, C. Gu, Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces. Numer. Algorithms 59, 623–638 (2012)S. Amat, M.A. Hernández, N. Romero, A modified Chebyshevs iterative method with at least sixth order of convergence. Appl. Math. Comput. 206, 164–174 (2008)X. Wang, J. Kou, C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algorithms 57, 441–456 (2011)A. Cordero, J.A. Ezquerro, M.A. Hernández-Verón, J.R. Torregrosa, On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)I.K. Argyros, S. Hilout, On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)X. Feng, Y. He, High order oterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)X. Wang, J. Kou, Y. Li, Modified Jarratt method with sixth-order convergence. Appl. Math. Lett. 22, 1798–1802 (2009)A.D. Polyanin, A.V. Manzhirov, Handbook of Integral Equations (CRC Press, Boca Raton, 1998)S. Plaza, N. Romero, Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)A. Cordero, J.R. Torregrosa, P. Vindel, Study of the dynamics of third-order iterative methods on quadratic polynomials. Int. J. Comput. Math. 89(13–14), 1826–1836 (2012)Gerardo Honorato, Sergio Plaza, Natalia Romero, Dynamics of a higher-order family of iterative methods. J. Complex. 27(2), 221–229 (2011)J.M. Gutirrez, M.A. Hernández, N. Romero, Dynamics of a new family of iterative processes for quadratic polynomials. J. Comput. Appl. Math. 233(10), 2688–2695 (2010)I.K. Argyros, A.A. Magreñan, A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms. doi: 10.1007/s11075-015-9981-xI.K. Argyros, S. George, Local convergence of modified Halley-like methods with less computation of inversion (Novi Sad J. Math, Draft version, 2015

On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions

[EN] In this paper the semilocal convergence for an alternative to the three steps Newton's method with frozen derivative is presented. We analyze the generalization of convergence conditions given by w-conditioned non-decreasing functions instead of the first derivative Lipschitz or Holder continuous given by other authors. A nonlinear integral equation of mixed Hammerstein type is considered for illustrating the new theoretical results obtained in this paper, where previous results can not be satisfied.This work was supported in part by the project MTM2011-28636-C02-01-{01,02} of the Spanish Ministry of Science and Innovation.Hernandez Verón, MA.; Martínez Molada, E. (2015). On the semilocal convergence of a three steps Newton-type iterative process under mild convergence conditions. Numerical Algorithms. 70(2):377-392. doi:10.1007/s11075-014-9952-7S377392702Amat, S., Busquier, S., Bermúdez, C., Plaza, S.: On two families of high order Newton type methods. Appl. Math. Comput. 25, 2209–2217 (2012)Argyros, I.K.: The Newton-Kantorovich method under mild differentiability conditions and the Pták error estimates. Monatsh. Math. 101, 175–193 (1990)Argyros, I.K.: Remarks on the convergence of Newton’s method under Hölder continuity conditions. Tamkang J. Math. 23(4), 269–277 (1992)Bruns, D.D., Bailey, J.E.: Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chem. Eng. Sci. 32, 257–264 (1977)Ezquerro, J.A., Hernández, M.A.: Generalized differentiability conditions for Newton’s method. IMA J. Numer. Anal. 22(2), 187–205 (2002)Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)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)Hueso, J.L., Martínez, E.: Semilocal convergence of a family of iterative methods in Banach spaces. Numer. Algor. In Press. doi: 10.1007/s11075-013-9795-7Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)Ostrowski, A.M.: Solutions of equations and system of equations. Academic Press, New York (1960)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. Math. 49, 203–220 (1986