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

Determination of multiple roots of nonlinear equations and applications

The final publication is available at Springer via https://dx.doi.org/10.1007/s10910-014-0460-8[EN] In this work we focus on the problem of approximating multiple roots of nonlinear equations. Multiple roots appear in some applications such as the compression of band-limited signals and the multipactor effect in electronic devices. We present a new family of iterative methods for multiple roots whose multiplicity is known. The methods are optimal in Kung-Traub's sense (Kung and Traub in J Assoc Comput Mach 21:643-651, [1]), because only three functional values per iteration are computed. By adding just one more function evaluation we make this family derivative free while preserving the convergence order. To check the theoretical results, we codify the new algorithms and apply them to different numerical examples.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and by Vicerrectorado de Investigacion, Universitat Politecnica de Valencia PAID-SP-2012-0474.Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel Ferragud, C. (2015). Determination of multiple roots of nonlinear equations and applications. Journal of Mathematical Chemistry. 53(3):880-892. https://doi.org/10.1007/s10910-014-0460-8S880892533H.T. Kung, J.F. Traub, Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 255, 105–112 (2009)W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–245 (2009)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, New modifications of Potra-Pták’s method with optimal fourth and eighth order of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870)C. Chun, B. Neta, A third-order modification of Newtons method for multiple roots. Appl. Math. Comput. 211, 474–479 (2009)Y.I. Kim, S.D. Lee, A third-order variant of NewtonSecant method finding a multiple zero. J. Chungcheong Math. Soc. 23(4), 845–852 (2010)B. Neta, Extension of Murakamis high-order nonlinear solver to multiple roots. Int. J. Comput. Math. 8, 1023–1031 (2010)H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence. Appl. Math. Comput. 209, 206–210 (2009)Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants. Appl. Math. Comput. 214, 10–16 (2009)S. Amat, S. Busquier, On a Steffensen’s type method and its behavior for semismooth equations. Appl. Math. Comput. 177, 819–823 (2006)X. Feng, Y. He, High order iterative methods without derivatives for solving nonlinear equations. Appl. Math. Comput. 186, 1617–1623 (2007)A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. doi: 10.1016/j.amc.2011.02.067F. Marvasti, A. Jain, Zero crossings, bandwidth compression, and restoration of nonlinearly distorted band-limited signals. J. Opt. Soc. Am. A 3, 651–654 (1986)S. Anza, C. Vicente, B. Gimeno, V.E. Boria, J. Armendáriz, Long-term multipactor discharge in multicarrier systems. Physics of Plasmas 14(8), 082–112 (2007)J.L. Hueso, E. Martínez, C. Teruel, New families of iterative methods with fourth and sixth order of convergence and their dynamics, in Proceedings of the 13th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2013, 24–27 June 2013A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. doi: 10.10016/j.cam.2014.01.024 (2014)J.R. Sharma, R. Sharma, Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010

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

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

Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces

In this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in Banach spaces that allows us to improve the convergence domains. Specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the Frechet derivative satisfies the Lipschitz continuity condition. For some values of the parameter, these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives 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 have considered a number of numerical examples including a nonlinear Hammerstein equation and computed the radii of the convergence balls. It is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods. The global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials.Martínez Molada, E.; Singh, S.; Hueso Pagoaga, JL.; Gupta, D. (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.2016.01.036S25226528

Dynamical study while searching equilibrium solutions in N-body problem

The theory of complex dynamics is usually applied to compare the global convergence properties of different iterative methods, by obtaining the attraction basins for simple polynomial equations in the complex domain. However, in this work, we use it in quite another context: the study of a nontrivial nonlinear system that describes the motion of interacting bodies in celestial mechanics, namely, Newtonian planar circular restricted four-body problem and its relative equilibrium solutions. These have been investigated from a dynamical point of view. New properties of the solutions of this system have been obtained. Practical guidelines for efficient search of relative equilibrium solutions of Nbody problem have been given.This research has been supported by Ministerio de Ciencia y Tecnologia MTM2014-52016-C2-02.Budzko, D.; Hueso Pagoaga, JL.; Martínez Molada, E.; Teruel-Ferragud, C. (2016). Dynamical study while searching equilibrium solutions in N-body problem. Journal of Computational and Applied Mathematics. 297:26-40. https://doi.org/10.1016/j.cam.2015.11.010S264029

Efficient high-order methods based on golden ratio for nonlinear system

We derive new iterative methods with order of convergence four or higher, for solving nonlinear systems, by composing iteratively golden ratio methods with a modified Newton's method. We use different efficiency indices in order to compare the new methods with other ones and present several numerical tests which confirm the theoretical results. © 2010 Elsevier Inc. All rights reserved.This research was supported by Ministerio de Ciencia y Tecnologia MTM2010-18539.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2011). Efficient high-order methods based on golden ratio for nonlinear system. Applied Mathematics and Computation. 217(9):4548-4556. https://doi.org/10.1016/j.amc.2010.11.006S45484556217

Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants

[EN] We present new high-order optimal iterativemethods for solving a nonlinear equation, f(x) = 0, by using Pade-like approximants. We compose optimal methods of order 4 with Newton's step and substitute the derivative by using an appropriate rational approximant, getting optimal methods of order 8. In the same way, increasing the degree of the approximant, we obtain optimal methods of order 16. We also perform different numerical tests that confirm the theoretical results.This work has been supported by Ministerio de Ciencia e Innovacion de Espana MTM2014-52016-C2-02-P and Generalitat Valenciana PROMETEO/2016/089.Cordero Barbero, A.; Hueso Pagoaga, JL.; Martínez Molada, E.; Torregrosa Sánchez, JR. (2017). Multistep High-Order Methods for Nonlinear Equations Using Pade-Like Approximants. Discrete Dynamics in Nature and Society. 1-6. https://doi.org/10.1155/2017/3204652S16Kung, H. T., & Traub, J. F. (1974). Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21(4), 643-651. doi:10.1145/321850.321860Petković, M. S., Neta, B., Petković, L. D., & Džunić, J. (2013). Basic concepts. Multipoint Methods, 1-26. doi:10.1016/b978-0-12-397013-8.00001-7Bi, W., Ren, H., & Wu, Q. (2009). Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics, 225(1), 105-112. doi:10.1016/j.cam.2008.07.004Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2011). Three-step iterative methods with optimal eighth-order convergence. Journal of Computational and Applied Mathematics, 235(10), 3189-3194. doi:10.1016/j.cam.2011.01.004Liu, L., & Wang, X. (2010). Eighth-order methods with high efficiency index for solving nonlinear equations. Applied Mathematics and Computation, 215(9), 3449-3454. doi:10.1016/j.amc.2009.10.040Sharma, J. R., & Sharma, R. (2009). A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms, 54(4), 445-458. doi:10.1007/s11075-009-9345-5Cordero, A., Hueso, J. L., Martínez, E., & Torregrosa, J. R. (2010). New modifications of Potra–Pták’s method with optimal fourth and eighth orders of convergence. Journal of Computational and Applied Mathematics, 234(10), 2969-2976. doi:10.1016/j.cam.2010.04.009Wang, X., & Liu, L. (2010). New eighth-order iterative methods for solving nonlinear equations. Journal of Computational and Applied Mathematics, 234(5), 1611-1620. doi:10.1016/j.cam.2010.03.002Neta, B., & Petković, M. S. (2010). Construction of optimal order nonlinear solvers using inverse interpolation. Applied Mathematics and Computation, 217(6), 2448-2455. doi:10.1016/j.amc.2010.07.045Fidkowski, K. J., Oliver, T. A., Lu, J., & Darmofal, D. L. (2005). p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations. Journal of Computational Physics, 207(1), 92-113. doi:10.1016/j.jcp.2005.01.005Amat, S., & Busquier, S. (Eds.). (2016). Advances in Iterative Methods for Nonlinear Equations. SEMA SIMAI Springer Series. doi:10.1007/978-3-319-39228-8Bruns, D. D., & Bailey, J. E. (1977). Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state. Chemical Engineering Science, 32(3), 257-264. doi:10.1016/0009-2509(77)80203-0He, Y., & Ding, C. H. Q. (2001). The Journal of Supercomputing, 18(3), 259-277. doi:10.1023/a:1008153532043Revol, N., & Rouillier, F. (2005). Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library. Reliable Computing, 11(4), 275-290. doi:10.1007/s11155-005-6891-yKing, R. F. (1973). A Family of Fourth Order Methods for Nonlinear Equations. SIAM Journal on Numerical Analysis, 10(5), 876-879. doi:10.1137/0710072Maheshwari, A. K. (2009). A fourth order iterative method for solving nonlinear equations. Applied Mathematics and Computation, 211(2), 383-391. doi:10.1016/j.amc.2009.01.047Weerakoon, S., & Fernando, T. G. I. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87-93. doi:10.1016/s0893-9659(00)00100-2Cordero, A., & Torregrosa, J. R. (2007). Variants of Newton’s Method using fifth-order quadrature formulas. Applied Mathematics and Computation, 190(1), 686-698. doi:10.1016/j.amc.2007.01.06

Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces

[EN] The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered.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, DK.; Badoni, RP.; Martínez Molada, E.; Hueso Pagoaga, JL. (2017). Local convergence of a parameter based iteration with Holder continuous derivative in Banach spaces. CALCOLO. 54(2):527-539. doi:10.1007/s10092-016-0197-9S527539542Argyros, 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)Singh, S., Gupta, D.K., Martínez, E., Hueso, J.L.: Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition. Appl. Math. Comput. 276, 266–277 (2016)Traub, J.F.: Iterative methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)Rall, L.B.: Computational solution of nonlinear operator equations, reprint edn. R. E. Krieger, New York (2007)Cordero, A., Ezquerro, J.A., Hernández-Verón, M.A., Torregrosa, J.R.: On the local convergence of a fifth-order iterative method in Banach spaces. Appl. Math. Comput. 251, 396–403 (2015)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)Argyros, I.K., Behl, R., Motsa, S.S.: Local convergence of an efficient high convergence order method using hypothesis only on the first derivative. Algorithms 8, 1076–1087 (2015)Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon Press, Oxford (1982)Argyros, I.K., Magreñán, A.A.: A study on the local convergence and dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Algorithms 71, 1–23 (2016)Li, D., Liu, P., Kou, J.: An improvement ofthe Chebyshev-Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)Argyros, I.K., George, S.: Local convergence of deformed Halley method in Banach space under Holder continuity conditions. J. Nonlinear Sci. Appl. 8, 246–254 (2015)Argyros, I.K., Khattri, S.K.: Local convergence for a family of third order methods in Banach spaces. J. Math. 46, 53–62 (2014)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., George, S.: Local convergence of modified Halley-like methods with less computation of inversion. Novi. Sad. J. Math. 45, 47–58 (2015)Xiao, X.Y., Yin, H.W.: Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo (2015). doi: 10.1007/s10092-015-0149-9Martínez, E., Singh, S., Hueso, J.L., Gupta, D.K.: Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces. Appl. Math. Comput. 281, 252–265 (2016

Looking at Rattleback

[EN] Celtic or Rattleback Stone is a solid that if it is rotated as top exhibits a surprising behaviour. Using a stereoscopic camera and digital monitoring techniques, we capture its movement. According the laws of classical mechanics, justify its evolution “rattling” and emphasize the high didactic potential that have these amazing systems for engineering students.[ES] La Piedra Celta es un sólido que cuando se le hace girar como peonza exhibe un comportamiento sorprendente. Utilizando una cámara estereoscópica y técnicas de seguimiento digital, capturamos su movimiento. De acuerdo con las leyes clásicas de la mecánica, justificamos su evolución "traqueteante", y destacamos el alto potencial didáctico que para los alumnos de Ingeniería poseen estos sorprendentes sistemas.El presente trabajo se ha realizado con la financiaci´on del Proyecto PAID-SP-2012-0498 del Vicerrectorado de Investigaci´on, Universitat Polit`ecnica de Val`encia. Jaime Riera es miembro del Equipo de Innovaci´on y Calidad Educativa de la UPV, E-MACAFI.Riera Guasp, J.; Ballester Sarrias, E.; Ginestar Peiro, D.; Hueso Pagoaga, JL. (2014). Mirando la Piedra Celta. Modelling in Science Education and Learning. 7:103-110. https://doi.org/10.4995/msel.2014.2124SWORD103110