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A new family of eighth-order iterative methods for solving nonlinear equations. (English) Zbl 1173.65030
A new family of iterative methods for the numerical solution of nonlinear equations is constructed an analyzed. All methods in the family show eigth-order convergence and use four evaluations. The family agrees with the Kung-Traub conjecture. Numerical comparisons are presented.

MSC:
65H05Single nonlinear equations (numerical methods)
References:
[1]Ortega, J. M.; Rheinbolt, W. C.: Iterative solution of nonlinear equations in several variables, (1970) · Zbl 0241.65046
[2]I.K. Argyros, in: C.K. Chui, L. Wuytack (Eds.), Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, 2007.
[3]King, R.: A family of fourth order methods for nonlinear equations, SIAM J. Numer. anal. 10, 876-879 (1973) · Zbl 0266.65040 · doi:10.1137/0710072
[4]Ostrowski, A. M.: Solution of equations in Euclidean and Banach spaces, (1960)
[5]Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[6]Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, J. assoc. Comput. math. 21, 634-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
[7]Steffensen, I. F.: Remarks on iteration, Skand. aktuarietidskr. 16, 64-72 (1933) · Zbl 0007.02601
[8]Jarratt, P.: Some efficient fourth order multipoint methods for solving equations, Bit 9, 119-124 (1969) · Zbl 0188.22101 · doi:10.1007/BF01933248
[9]Ezquerro, J. A.; Hernández, M. A.; Salanova, M. A.: Construction of iterative processes with high order of convergence, Int. J. Comp. math. 69, 191-201 (1998) · Zbl 0908.65031 · doi:10.1080/00207169808804717
[10]Gutiérrez, J. M.; Hernández, 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
[11]Argyros, I. K.: The jarratt method in a Banach space setting, J. comp. Appl. math. 51, 103-106 (1994) · Zbl 0809.65054 · doi:10.1016/0377-0427(94)90093-0
[12]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
[13]Sharma, J. R.; Guha, R. K.: A family of modified Ostrowski methods with accelerated sixth-order convergence, Appl. math. Comput. 190, 111-115 (2007) · Zbl 1126.65046 · doi:10.1016/j.amc.2007.01.009
[14]Chun, C.; Ham, Y.: Some sixth-order variants of Ostrowski root-finding methods, Appl. math. Comput. 193, 389-394 (2007) · Zbl 1193.65055 · doi:10.1016/j.amc.2007.03.074
[15]Kou, J.: The improvements of modified Newton’s method, Appl. math. Comput. 189, 602-609 (2007) · Zbl 1122.65332 · doi:10.1016/j.amc.2006.11.115
[16]Kou, J.; Li, Y.; Wang, X.: An improvement of the jarrat method, Appl. math. Comput. 189, 1816-1821 (2007)
[17]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
[18]Noor, K. I.; Noor, M. A.; Momani, S.: Modified householder iterative method for nonlinear equations, Appl. math. Comput. 190, 1534-1539 (2007) · Zbl 1122.65341 · doi:10.1016/j.amc.2007.02.036
[19]Parhi, S. K.; Gupta, D. K.: A sixth order method for nonlinear equations, Appl. math. Comput. 203, 50-55 (2008) · Zbl 1154.65327 · doi:10.1016/j.amc.2008.03.037
[20]Kou, J.; Li, Y.; Wang, X.: Some variants of Ostrowski’s method with seventh-order convergence, J. comput. Appl. math. 209, 153-159 (2007) · Zbl 1130.41006 · doi:10.1016/j.cam.2006.10.073
[21]Mir, N. A.; Zaman, T.: Some quadrature based three-step iterative methods for non-linear equations, Appl. math. Comput. 193, 366-373 (2007) · Zbl 1193.65068 · doi:10.1016/j.amc.2007.03.071
[22]Bi, W.; Ren, H.; Wu, Q.: New family of seventh-order methods for nonlinear equations, Appl. math. Comput. 203, 408-412 (2008) · Zbl 1154.65323 · doi:10.1016/j.amc.2008.04.048
[23]Bi, W.; Ren, H.; Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations, J. comput. Appl. math. 255, 105-112 (2009) · Zbl 1161.65039 · doi:10.1016/j.cam.2008.07.004
[24]Quarteroni, A.; Sacco, R.; Saleri, F.: Numerical mathematics, (2000)
[25]Xu, L.; Wang, X.: Topics on methods and examples of mathematical analysis (in chinese), (1983)
[26]Gautschi, W.: Numerical analysis: an introduction, (1997)
[27]Levin, Y.; Ben-Israel, A.: Directional Newton methods in n variables, Math. comput. 71, 251-262 (2002) · Zbl 0985.65049 · doi:10.1090/S0025-5718-01-01332-1
[28]An, H.; Bai, Z.: Broyden method for nonlinear equation in several variables, Math. numer. Sinica. 26, 385-400 (2004)
[29]An, H.; Bai, Z.: Directional secant method for nonlinear equations, J. comput. Appl. math. 175, 291-304 (2005) · Zbl 1076.65046 · doi:10.1016/j.cam.2004.05.013
[30]Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third-order convergence, Appl. math. Lett. 13, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2