Abstract
We study the local convergence of a method presented by Cordero et al. of convergence order at least five to approximate a locally unique solution of a nonlinear equation. These studies show the convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analysis of this method is also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.
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References
Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequ. Math. 69 (2005) 3, 212223
Amat, S., Busquier, S., Plaza, S.: Chaotic dynamics of a third-order Newton-type method. J. Math. Anal. Appl. 366(1), 2432 (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, Berlin (2008)
Argyros, I.K., Hilout, S.: Numerical Methods in Nonlinear Analysis. World Scientific Publ. Comp, New Jersey (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)
Budzko, D., Cordero, A., Torregrosa, J.R.: A new family of iterative methods widening areas of convergence. Appl. Math. Comput. 252, 405–417 (2015)
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)
Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, Article ID 780153
Chun, C.: Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190(2), 1432–1437 (1990)
Cordero, A., García-Maimó, J., Torregrosa, J.R., Vassileva, M.P., Vindel, P.: Chaos in King’s iterative family. Appl. Math. Lett. 26, 842–848 (2013)
Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev–Halley type methods. Appl. Math. Comput. 219, 8568–8583 (2013)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
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)
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. Algorithms 60, 369–390 (2012)
Li, D., Liu, P., Kou, J.: An improvement of the Chebyshev–Halley methods free from second derivative. Appl. Math. Comput. 235, 221–225 (2014)
Magreñán, Á.A.: Different anomalies in a Jarratt family of iterative root-finding methods. Appl. Math. Comput. 233, 29–38 (2014)
Magreñán, Á.A.: A new tool to study real dynamics: the convergence plane. Appl. Math. Comput. 248, 215–224 (2014)
Petković, M., Neta, B., Petković, L., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)
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. Algorithms 52(4), 585–603 (2009)
Rheinboldt, W.C.: An Adaptive Continuation Process for Solving Systems of Nonlinear Equations, vol. 3. Polish Academy of Science. Banach Ctr. Publ., Polish, pp. 129–142 (1978)
Traub, J.F.: Iterative methods for the solution of equations. In: 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. Algorithms 57, 441–456 (2011)
Acknowledgments
This work has been partially supported by the project MTM2014-52016-C2-1-P of Spanish Ministry of Economy and Competitiveness and by the Universidad Internacional de La Rioja (UNIR, http://www.unir.net), under the Plan Propio de Investigación, Desarrollo e Innovación [2013–2015]. Research group: Matemática aplicada al mundo real (MAMUR).
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Alberto Magreñán, Á., Argyros, I.K. Ball convergence theorems and the convergence planes of an iterative method for nonlinear equations. SeMA 71, 39–55 (2015). https://doi.org/10.1007/s40324-015-0047-8
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DOI: https://doi.org/10.1007/s40324-015-0047-8