Extended Kung-Traub-type method for solving equations. (English) Zbl 1491.65036
Summary: We are motivated by a Kung-Traub-type method for solving equations on the real line. In particular, we extend this method for Banach space valued operators. The radius of convergence is also obtained as well as error bounds on the distances involved and a uniqueness result. Our convergence analysis avoids Taylor expansions and the computation of higher order than one derivatives.
MSC:
| 65H05 | Numerical computation of solutions to single equations |
| 65J15 | Numerical solutions to equations with nonlinear operators |
References:
| [1] | Amat, S., Hern´andez, M.A., Romero, N., (2012), Semilocal convergence of a sixth order iterative method for quadratic equations, Appll. Numer.Math., 62, pp.833-841. · Zbl 1387.65049 |
| [2] | Argyros, I.K., (2007), Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, 15, Editors: C.K. Chui and L. Wuytack, Elsevier Publ. Co. New York, USA. · Zbl 1147.65313 |
| [3] | Argyros, I.K., (2003), On the convergence and application of Newton’s method under H¨older continuity assumptions, Int. J. Comput. Math., 80, pp.767-780. · Zbl 1050.65056 |
| [4] | Argyros, I.K., George, S., (2018), Local convergence of a Hansen-Patrick-like family of of optimal fourth order methods, TWMS J. Pure Appl. Math., 9(1), pp.32-39. · Zbl 1433.65096 |
| [5] | Bi, W., Ren, H.M., Wu, Q., (2011), Convergence of the modified Halley’s method for multiple zeros under H¨older continuous derivatives, Numer. Algor., 58, pp.497-512. · Zbl 1242.65098 |
| [6] | Chun, C., Neta, B., (2009), A third order modification of Newton’s method for multiple roots, Appl. Math. Comput., 211, pp.474-479. · Zbl 1162.65342 |
| [7] | Hansen, E., Patrick, M., (1977), A family of root finding methods, Numer. Math., 27, pp.257-269. · Zbl 0361.65041 |
| [8] | Magre˜n´an, A.A., (2014), Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput., 233, pp.29-38. · Zbl 1334.65083 |
| [9] | Magre˜n´an, A. A., (2014), A new tool to study real dynamics: The convergence plane, Appl. Math. Comput., 248, pp.29-38. |
| [10] | Neta, B., (2008), New third order nonlinear solvers for multiple roots, Appl. Math. Comput., 202, pp.162-170. · Zbl 1151.65041 |
| [11] | Obreshkov, N., (1963), On the numerical solution of equations (Bulgarian), Annuaire Univ. Sofia. fac. Sci. Phy. Math., 56, pp.73-83. · Zbl 0158.33802 |
| [12] | Osada, N., (1994), An optimal mutiple root-finding method of order three, J. Comput. Appl. Math., 52, pp.131-133. · Zbl 0814.65045 |
| [13] | Petkovic, M.S., Neta, B., Petkovic, L., Dˇzuniˇc, J., (2013), Multipoint Methods for Solving Nonlinear Equations, Elsevier, 344p. · Zbl 1286.65060 |
| [14] | Ren, H.M., Argyros, I.K., (2010), Convergence radius of the modified Newton method for multiple zeros under H¨older continuity derivative, Appl. Math. Comput., 217, pp.612-621. · Zbl 1205.65174 |
| [15] | Schr¨oder, E., (1870), Uber unendlich viele Algorithmen zur Auflosung der Gieichunger, Math. Ann., 2, pp.317-365. · JFM 02.0042.02 |
| [16] | Traub, J.F., (1982), Iterative Methods for the Solution of Equations, AMS Chelsea Publishing, 310p. Ioannis K. Argyros, for a photograph and biography, see TWMS J. Pure Appl. Math., V.9, N.1, 2018, p.39 Santhosh George, for a photograph and biography, see TWMS J. Pure Appl. Math., V.9, N.1, 2018, p |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
