×

zbMATH — the first resource for mathematics

On the local convergence of a family of Euler-halley type iterations with a parameter. (English) Zbl 1181.65081
The authors are going to study the local convergence of a family of Euler–Halley type iterations depending on a parameter, for solving a nonlinear operator equation in a real or complex Banach space. The idea of introducing one real parameter \(\alpha\) in the algorithm for finding the solution of equation is used also, by other authors, for example by W. Werner [“Some improvement of classical methods for the solution of nonlinear equations”, Lect. Notes Math. 878, 426–440 (1981; Zbl 0494.65033)]. For particular values of the parameter \(\alpha\), are obtained some famous methods: \(\alpha=\frac12\), the Halley method; \(\alpha=0\), the Chebyshev-Euler method; \(\alpha=1\), the super-Halley method.
Under the so called second-order generalized Lipshitz assumption of the nonlinear operator from the left-hand side of the equation, the local convergence of the family of Euler-Halley type iterations, is discussed and the radius of the optimal convergence ball is estimated, for each real value of the parameter. The convergence analysis is done separately for positive values and for negative values of the parameter. It is verified that for the local convergence there is no universal constant for the iterations, which is quite different from the semi-local behaviour of iterations.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
47J25 Iterative procedures involving nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amat, S., Busquier, S., Gutierrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003) · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5
[2] Argyros, I.K.: On an improved unified convergence analysis for a certain class of Euler-Halley type methods. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 13, 207–215 (2006) · Zbl 1143.65346
[3] Argyros, I.K., Chen, D.: Results on the Chebyshev method in Banach spaces. Proyecciones 12, 119–128 (1993) · Zbl 1082.65536
[4] Argyros, I.K., Chen, D., Qian, Q.: A convergence analysis for rational methods with a parameter in Banach space. Pure Math. Appl. 5, 59–73 (1994) · Zbl 0817.65043
[5] Candela, V., Marquina, A.: Recurrence relations for rational cubic methods I. The Halley method. Computing 44, 169–184 (1990) · Zbl 0701.65043 · doi:10.1007/BF02241866
[6] Candela, V., Marquina, A.: Recurrence relations for rational cubic methods II. The Chebyshev method. Computing 45, 355–367 (1990) · Zbl 0714.65061 · doi:10.1007/BF02238803
[7] Chen, D., Argyros, I.K., Qian, Q.: A local convergence theorem for the Supper-Halley method in a Banach space. Appl. Math. Lett. 7, 49–52 (1994) · Zbl 0811.65043 · doi:10.1016/0893-9659(94)90071-X
[8] Ezquerro, J.A., Hernandez, M.A.: New Kantorovich-type conditions for Halley’s method. Appl. Numer. Anal. Comput. Math. 2, 70–77 (2005) · Zbl 1076.65052 · doi:10.1002/anac.200410024
[9] Ezquerro, J.A., Hernandez, M.A.: On the R-order of the Halley method. J. Math. Anal. Appl. 303, 591–601 (2005) · Zbl 1079.65064 · doi:10.1016/j.jmaa.2004.08.057
[10] Ezquerro, J.A., Hernandez, M.A.: Halley’s method for operators with unbounded second derivative. Appl. Numer. Math. 57, 354–360 (2007) · Zbl 1252.65098 · doi:10.1016/j.apnum.2006.05.001
[11] Gutierrez, J.M., Hernandez, M.A.: A family of Chebyshev-Halley type methods in Banach space. Bull. Aust. Math. Soc. 55, 113–130 (1997) · Zbl 0893.47043 · doi:10.1017/S0004972700030586
[12] Gutierrez, J.M., Hernandez, M.A.: An acceleration of Newton’s method: Super-Halley method. Appl. Math. Comput. 117, 223–239 (2001) · Zbl 1023.65051 · doi:10.1016/S0096-3003(99)00175-7
[13] Han, D.: The convergence on a family of iterations with cubic order. J. Comput. Math. 19, 467–474 (2001) · Zbl 1008.65035
[14] Hernandez, M.A., Salanova, M.A.: A family of Chebyshev-Halley type methods. Int. J. Comput. Math. 47, 59–63 (1993) · Zbl 0812.65038 · doi:10.1080/00207169308804162
[15] Huang, Z.: On a family of Chebyshev-Halley type methods in Banach space under weaker Smale condition. Numer. Math. JCU 9, 37–44 (2000) · Zbl 0960.65066
[16] Wang, H., Li, C., Wang, X.: On relationship between convergence ball of Euler iteration in Banach spaces and its dynamical behavior on Riemann spheres. Sci. China Ser. A 46, 376–382 (2003) · Zbl 1217.37045
[17] Wang, X.: Convergence on the iteration of Halley family in weak conditions. Chin. Sci. Bull. 42, 552–555 (1997) · Zbl 0884.30004 · doi:10.1007/BF03182614
[18] Wang, X.: Convergence of the iteration of Halley’s family and Smale operator class in Banach space. Sci. China Ser. A 41, 700–709 (1998) · Zbl 0910.65036 · doi:10.1007/BF02901952
[19] Wang, X.: Convergence of iterations of Euler family under weak condition. Sci. China Ser. A 43, 958–962 (2000) · Zbl 0999.65047 · doi:10.1007/BF02879801
[20] Wang, X., Li, C.: Local and global behavior for algorithms of solving equations. Chin. Sci. Bull. 46, 441–448 (2001) · Zbl 1044.65049 · doi:10.1007/BF03187252
[21] Wang, X., Li, C.: On the united theory of the family of Euler-Halley type methods with cubical convergence in Banach spaces. J. Comput. Math. 21, 195–200 (2003) · Zbl 1057.65033
[22] Werner, W.: Some improvement of classical methods for the solution of nonlinear equations. In: Allgower, E.L., Glashoff, K., Peitgen, H.-O. (eds.) Numerical Solution of Nonlinear Equations. Lecture Notes in Mathematics, vol. 878, pp. 426–440. Springer, Berlin (1981)
[23] Ye, X., Li, C.: Convergence of the family of the deformed Euler-Halley iterations under the Holder condition of the second derivative. J. Comput. Appl. Math. 194, 294–308 (2006) · Zbl 1100.47057 · doi:10.1016/j.cam.2005.07.019
[24] Ye, X., Li, C., Shen, W.: Convergence of the variants of the Chebyshev-Halley iteration family under the Holder condition of the first derivative. J. Comput. Appl. Math. 203, 279–288 (2007) · Zbl 1118.65059 · doi:10.1016/j.cam.2006.04.003
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.