Abstract
We present a local convergence analysis of a two-point four parameter Jarratt-like method of high convergence order in order to approximate a locally unique solution of a nonlinear equation. In contrast to earlier studies such us (Amat et al. Aequat. Math. 69(3), 212–223 2015; Amat et al. J. Math. Anal. Appl. 366(3), 24–32 2010; Behl, R. 2013; Bruns and Bailey Chem. Eng. Sci. 32, 257–264 1977; Candela and Marquina. Computing 44, 169–184 1990; Candela and Marquina. Computing 45(4), 355–367 1990; Chun. Appl. Math. Comput. 190(2), 1432–1437 2007; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2007; Deghan. Comput. Appl Math. 29(1), 19–30 2010; Deghan. Comput. Math. Math. Phys. 51(4), 513–519 2011; Deghan and Masoud. Eng. Comput. 29(4), 356–365 15; Cordero and Torregrosa. Appl. Math. Comput. 190, 686–698 2012; Deghan and Masoud. Eng. Comput. 29(4), 356–365 2012; Ezquerro and Hernández. Appl. Math. Optim. 41(2), 227–236 2000; Ezquerro and Hernández. BIT Numer. Math. 49, 325–342 2009; Ezquerro and Hernández. J. Math. Anal. Appl. 303, 591–601 2005; Gutiérrez and Hernández. Comput. Math. Appl. 36(7), 1–8 1998; Ganesh and Joshi. IMA J. Numer. Anal. 11, 21–31 1991; González-Crespo et al. Expert Syst. Appl. 40(18), 7381–7390 2013; Hernández. Comput. Math. Appl. 41(3-4), 433–455 2001; Hernández and Salanova. Southwest J. Pure Appl. Math. 1, 29–40 1999; Jarratt. Math. Comput. 20(95), 434–437 1966; Kou and Li. Appl. Math. Comput. 189, 1816–1821 2007; Kou and Wang. Numer. Algor. 60, 369–390 2012; Lorenzo et al. Int. J. Interact. Multimed. Artif. Intell. 1(3), 60–66 2010; Magreñán. Appl. Math. Comput. 233, 29–38 2014; Magreñán. Appl. Math. Comput. 248, 215–224 2014; Parhi and Gupta. J. Comput. Appl. Math. 206(2), 873–887 2007; Rall 1979; Ren et al. Numer. Algor. 52(4), 585–603 2009; Rheinboldt Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 1978; Sicilia et al. J. Comput. Appl. Math. 291, 468–477 2016; Traub 1964; Wang et al. Numer. Algor. 57, 441–456 2011) using hypotheses up to the fifth derivative, our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. The dynamics of the family for choices of the parameters such that it is optimal is also shown. Numerical examples are also provided in this study
Similar content being viewed by others
References
Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequat. Math. 69(3), 212–223 (2005)
Amat, S., Busquier, S., Plaza, S.: Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl. 366(1), 24–32 (2010)
Amat, S., Hernández, M.A., Romero, N.: A modified Chebyshev’s iterative method with at least sixth order of convergence. Appl. Math. Comput. 206(1), 164–174 (2008)
Argyros, I.K.: Convergence and Application of Newton-type Iterations. Springer (2008)
Argyros, I.K., Hilout, S.: Numerical methods in Nonlinear Analysis. World Scientific Publication Computing, New Jersey (2013)
Argyros, I.K., Magreñán, Á.A.: On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput. 252, 336–346 (2015)
Behl, R.: Development and analysis of some new iterative methods for numerical solutions of nonlinear equations. (PhD Thesis) Punjab University (2013)
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)
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(4), 355–367 (1990)
Chun, C.: Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190(2), 1432–1437 (2007)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
Deghan, M.: Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations. Comput. Appl Math. 29(1), 19–30 (2010)
Deghan, M.: On derivative free cubic convergence iterative methods for solving nonlinear equations. Comput. Math. Math. Phys. 51(4), 513–519 (2011)
Deghan, M., Masoud, H.: Fourth-order variants of Newton’s method without second derivatives for solving non-linear equations. Eng. Comput. 29(4), 356–365 (2012)
Ezquerro, J.A., Hernández, M.A.: Recurrence relations for Chebyshev-type methods. Appl. Math. Optim. 41(2), 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)
Ezquerro, J.A., Hernández, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005)
Gutiérrez, J.M., Hernández, M.A.: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36(7), 1–8 (1998)
Ganesh, M., Joshi, M.C.: Numerical solvability of Hammerstein integral equations of mixed type. IMA J. Numer. Anal. 11, 21–31 (1991)
González-Crespo, R., Ferro, R., Joyanes, L., Velazco, S., Castillo, A.G.: Use of ARIMA mathematical analysis to model the implementation of expert system courses by means of free software OpenSim and Sloodle platforms in virtual university campuses. Expert Syst. Appl. 40(18), 7381–7390 (2013)
Hernández, M.A.: Chebyshev’s approximation algorithms and applications. Comput. Math. Appl. 41(3-4), 433–455 (2001)
Hernández, M.A., Salanova, M.A.: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29–40 (1999)
Jarratt, P.: Some fourth order multipoint methods for solving equations. Math. Comput. 20(95), 434–437 (1966)
Kou, J., Li, Y.: An improvement of the Jarratt method. Appl. Math. Comput. 189, 1816–1821 (2007)
Kou, J., Wang, X.: Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity conditions. Numer. Algor. 60, 369–390 (2012)
Lorenzo, W., Crespo, R.G., Castillo, A.: A prototype for linear features generalization. Int. J. Interact. Multimed. Artif. Intell. 1(3), 60–66 (2010)
Magreñán, Á.A.: Different anoMalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)
Magreñán, Á.A., Some fourth order multipoint methods for solving equations: Appl. Math. Comput. 248, 215–224 (2014)
Parhi, S.K., Gupta, D.K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206(2), 873–887 (2007)
Rall, L.B.: Computational solution of nonlinear operator equations. Robert E. Krieger, New York (1979)
Ren, H., Wu, Q., Bi, W.: New variants of Jarratt method with sixth-order convergence. Numer. Algor. 52(4), 585–603 (2009)
Rheinboldt, W.C.: An adaptive continuation process for solving systems of nonlinear equations. Pol. Acad. Sci. Banach Ctr. Publ. 3, 129–142 (1978)
Sicilia, J.A., Quemada, C., Royo, B., Escuín, D.: An optimization algorithm for solving the rich vehicle routing problem based on Variable Neighborhood Search and Tabu Search metaheuristics. J. Comput. Appl. Math. 291, 468–477 (2016)
Traub, J.F.: Iterative methods for the solution of equations. Prentice- Hall Series in Automatic Computation, Englewood Cliffs, NJ (1964)
Wang, X., Kou, J., Gu, C.: Semilocal convergence of a sixth-order Jarratt method in Banach spaces. Numer. Algor. 57, 441–456 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amat, S., Argyros, I.K., Busquier, S. et al. Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions. Numer Algor 74, 371–391 (2017). https://doi.org/10.1007/s11075-016-0152-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0152-5