Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems

[EN] In this paper, we analyze the semilocal convergence of k-steps Newton's method with frozen first derivative in Banach spaces. The method reaches order of convergence k + 1. By imposing only the assumption that the Fr,chet derivative satisfies the Lipschitz continuity, we define appropriate recurrence relations for obtaining the domains of convergence and uniqueness. We also define the accessibility regions for this iterative process in order to guarantee the semilocal convergence and perform a complete study of their efficiency. Our final aim is to apply these theoretical results to solve a special kind of conservative systems.Hernández-Verón, MA.; Martínez Molada, E.; Teruel-Ferragud, C. (2017). Semilocal convergence of a k-step iterative process and its application for solving a special kind of conservative problems. Numerical Algorithms. 76(2):309-331. https://doi.org/10.1007/s11075-016-0255-zS309331762Amat, 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., Hilout, S., Tabatabai, M.A.: Mathematical Modelling with Applications in Biosciences and Engineering. Nova Publishers, New York (2011)Argyros, I.K., George, S.: A unified local convergence for Jarratt-type methods in Banach space under weak conditions. Thai. J. Math. 13, 165–176 (2015)Argyros, I.K., Hilout, S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)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, 3195–2206 (2011)Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation. Math. Comput. Mod. 57, 1950–1956 (2013)Ezquerro, J.A., Grau-Sánchez, M., Hernández, M. A., Noguera, M.: Semilocal convergence of secant-like methods for differentiable and nondifferentiable operators equations. J. Math. Anal. Appl. 398(1), 100–112 (2013)Honorato, G., Plaza, S., Romero, N.: Dynamics of a higher-order family of iterative methods. J. Complexity 27(2), 221–229 (2011)Jerome, J.W., Varga, R.S.: Generalizations of Spline Functions and Applications to Nonlinear Boundary Value and Eigenvalue Problems, Theory and Applications of Spline Functions. Academic Press, New York (1969)Kantorovich, L.V., Akilov, G.P.: Functional analysis Pergamon Press. Oxford (1982)Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Dover Publications, New York (1992)Na, T.Y.: Computational Methods in Engineering Boundary Value Problems. Academic Press, New York (1979)Ortega, J.M.: The Newton-Kantorovich theorem. Amer. Math. Monthly 75, 658–660 (1968)Ostrowski, A.M.: Solutions of Equations in Euclidean and Banach Spaces. Academic Press, New York (1973)Plaza, S., Romero, N.: Attracting cycles for the relaxed Newton’s method. J. Comput. Appl. Math. 235(10), 3238–3244 (2011)Porter, D., Stirling, D.: Integral Equations: A Practical Treatment, From Spectral Theory to Applications. Cambridge University Press, Cambridge (1990)Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall. Englewood Cliffs, New Jersey (1964)Argyros, I.K., George, S.: Extending the applicability of Gauss-Newton method for convex composite optimization on Riemannian manifolds using restricted convergence domains. Journal of Nonlinear Functional Analysis 2016 (2016). Article ID 27Xiao, J.Z., Sun, J., Huang, X.: Approximating common fixed points of asymptotically quasi-nonexpansive mappings by a k+1-step iterative scheme with error terms. J. Comput. Appl. Math 233, 2062–2070 (2010)Qin, X., Dehaish, B.A.B., Cho, S.Y.: Viscosity splitting methods for variational inclusion and fixed point problems in Hilbert spaces. J. Nonlinear Sci. Appl. 9, 2789–2797 (2016

Multipoint secant and interpolation methods with nonmonotone line search for solving systems of nonlinear equations

Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way

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-

Composing Scalable Nonlinear Algebraic Solvers

Most efficient linear solvers use composable algorithmic components, with the most common model being the combination of a Krylov accelerator and one or more preconditioners. A similar set of concepts may be used for nonlinear algebraic systems, where nonlinear composition of different nonlinear solvers may significantly improve the time to solution. We describe the basic concepts of nonlinear composition and preconditioning and present a number of solvers applicable to nonlinear partial differential equations. We have developed a software framework in order to easily explore the possible combinations of solvers. We show that the performance gains from using composed solvers can be substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table