×

zbMATH — the first resource for mathematics

On the semilocal convergence behavior for Halley’s method. (English) Zbl 06348494
Summary: The present paper is concerned with the semilocal convergence problems of Halley’s method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley’s method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.

MSC:
47J05 Equations involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65H10 Numerical computation of solutions to systems of equations
Software:
NewtonLib
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Argyros, IK, On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, 315-332, (2004) · Zbl 1055.65066
[2] Argyros, IK, Ball convergence theorems for hally’s method in Banach space, J. Appl. Math. Comput., 38, 453-465, (2012) · Zbl 1295.65064
[3] Candela, V; Marquina, A, Recurrence relations for rational cubic methods I: the Halley method, Computing, 44, 169-184, (1990) · Zbl 0701.65043
[4] Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin Heidelberg (2004) · Zbl 1056.65051
[5] Deuflhard, P; Heindl, G, Affine invariant convergence theorems for newton’s method and extensions to related methods, SIAM J. Numer. Anal., 16, 1-10, (1979) · Zbl 0395.65028
[6] Ezquerro, JA; Hernández, MA, On the R-order of the Halley method, J. Math. Anal. Appl., 303, 591-601, (2005) · Zbl 1079.65064
[7] Ferreira, OP, Local convergence of newton’s method in Banach space from the viewpoint of the majorant principle, IMA J. Numer. Anal., 29, 746-759, (2009) · Zbl 1175.65067
[8] Ferreira, OP; Svaiter, BF, Kantorovich’s majorants principle for newton’s method, Comput. Optim. Appl., 42, 213-229, (2009) · Zbl 1191.90095
[9] Gragg, WB; Tapia, RA, Optimal error bounds for the Newton-Kantorovich theorem, SIAM J. Numer. Anal., 11, 10-13, (1974) · Zbl 0284.65042
[10] Gutiérrez, JM; Hernández, MA, Newton’s method under weak Kantorovich conditions, IMA J. Numer. Anal., 20, 521-532, (2000) · Zbl 0965.65081
[11] Han, D, The convergence on a family of iterations with cubic order, J. Comput. Math., 19, 467-474, (2001) · Zbl 1008.65035
[12] Han, D; Wang, X, The error estimates of halley’s method, Numer. Math. JCU Engl. Ser., 6, 231-240, (1997) · Zbl 0905.65067
[13] Hernández, MA; Romero, N, On a characterization of some Newton-like methods of R-order at least three, J. Comput. Appl. Math., 183, 53-66, (2005) · Zbl 1087.65057
[14] Hernández, MA; Romero, N, Application of iterative processes of R-order at least three to operators with unbounded second derivative, Appl. Math. Comput., 185, 737-747, (2007) · Zbl 1115.65063
[15] Hernández, MA; Romero, N, Toward a unified theory for third R-order iterative methods for operators with unbounded second derivative, Appl. Math. Comput., 215, 2248-2261, (2009) · Zbl 1181.65080
[16] Jay, LO, A note on Q-order of convergence, BIT Numer. Math., 41, 422-429, (2001) · Zbl 0973.40001
[17] Kantorvich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
[18] Potra, FA, On Q-order and R-order of convergence, J. Optim. Theory Appl., 63, 415-431, (1989) · Zbl 0663.65049
[19] Potra, FA; Pták, V, Sharp error bounds for newton’s process, Numer. Math., 34, 63-72, (1980) · Zbl 0434.65034
[20] Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Processes, Number 103 in Research Notes in Mathematics. Wiley, Boston (1984)
[21] Proinov, PD, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, 3-42, (2010) · Zbl 1185.65095
[22] Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185-196. Springer, New York (1986) · Zbl 1115.65063
[23] Wang, X, Convergence on the iteration of Halley family in weak conditions, Chin. Sci. Bull., 42, 552-555, (1997) · Zbl 0884.30004
[24] Wang, X, Convergence of newton’s method and inverse functions theorem in Banach space, Math. Comput., 68, 169-186, (1999) · Zbl 0923.65028
[25] Wang, X, Convergence of newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal., 20, 123-134, (2000) · Zbl 0942.65057
[26] Wang, X; Han, D, On the dominating sequence method in the point estimates and smale’s theorem, Scientia Sinica Ser. A., 33, 135-144, (1990) · Zbl 0699.65046
[27] Wang, X; Han, D, Criterion \(α \) and newton’s method in the weak conditions (in Chinese), Math. Numer. Sinica, 19, 103-112, (1997) · Zbl 0879.65031
[28] Xu, X; Li, C, Convergence of newton’s method for systems of equations with constant rank derivatives, J. Comput. Math., 25, 705-718, (2007) · Zbl 1150.49011
[29] Xu, X; Li, C, Convergence criterion of newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl., 345, 689-701, (2008) · Zbl 1154.65332
[30] Ye, X; Li, C, Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative, J. Comput. Appl. Math., 194, 294-308, (2006) · Zbl 1100.47057
[31] Yamamoto, T, Error bounds for newton’s iterates derived from the Kantorovich assumptions, Numer. Math., 49, 91-98, (1986) · Zbl 0567.65027
[32] Yamamoto, T, A method for finding sharp error bounds for newton’s method under the Kantorovich assumptions, Numer. Math., 49, 203-220, (1986) · Zbl 0607.65033
[33] Ypma, TJ, Affine invariant convergence results for newton’s method, BIT Numer. Math., 22, 108-118, (1982) · Zbl 0481.65027
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.