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Modified Jarratt method with sixth-order convergence. (English) Zbl 1184.65054
Summary: We present a variant of Jarratt method with order of convergence six for solving non-linear equations [cf. J. Kou and Y. Li, Appl. Math. Comput. 189, No. 2, 1816–1821 (2007; Zbl 1122.65338)]. Per iteration the method requires two evaluations of the function and two of its first derivatives. The new multistep iteration scheme, based on the new method, is developed and numerical tests verifying the theory are also given.

MSC:
65H05Single nonlinear equations (numerical methods)
References:
[1]Ostrowski, A. M.: Solutions of equations and system of equations, (1960) · Zbl 0115.11201
[2]Argyros, I. K.; Chen, D.; Qian, Q.: The jarratt method in Banach space setting, J. comput. Appl. math. 51, 103-106 (1994) · Zbl 0809.65054 · doi:10.1016/0377-0427(94)90093-0
[3]Kou, J.; Li, Y.; Wang, X.: An improvement of the jarratt method, Appl. math. Comput. 189, 1816-1821 (2007)
[4]Chun, C.: Some improvements of jarratt’s method with sixth-order convergence, Appl. math. Comput. 190, 1432-1437 (2007) · Zbl 1122.65329 · doi:10.1016/j.amc.2007.02.023
[5]Grau, M.; Díaz-Barrero, J. L.: An improvement to Ostrowski root-finding method, Appl. math. Comput. 173, 450-456 (2006) · Zbl 1090.65053 · doi:10.1016/j.amc.2005.04.043
[6]Grau, M.; Noguera, M.: A variant of Cauchy’s method with accelerated fifth-order convergence, Appl. math. Lett. 17, 509-517 (2004) · Zbl 1070.65034 · doi:10.1016/S0893-9659(04)90119-X
[7]Grau, M.: An improvement to the computing of nonlinear equation solutions, Numer. algorithms 34, 1-12 (2003) · Zbl 1043.65071 · doi:10.1023/A:1026100500306
[8]Grau, M.; Díaz-Barrero, J. L.: An improvement of the Euler–Chebyshev iterative method, J. math. Anal. appl. 315, 1-7 (2006) · Zbl 1113.65048 · doi:10.1016/j.jmaa.2005.09.086
[9]Kou, J.; Li, Y.; Wang, X.: A family of fifth-order iterations composed of Newton and third-order methods, Appl. math. Comput. 186, 1258-1262 (2007) · Zbl 1119.65037 · doi:10.1016/j.amc.2006.07.150