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Derivative free two-point methods with and without memory for solving nonlinear equations. (English) Zbl 1200.65034
Summary: Two families of derivative free two-point iterative methods for solving nonlinear equations are constructed. These methods use a suitable parametric function and an arbitrary real parameter. It is proved that the first family has the convergence order four requiring only three function evaluations per iteration. In this way it is demonstrated that the proposed family without memory supports the Kung-Traub hypothesis [H. T. Kung and J. F. Traub, J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] on the upper bound 2 n of the order of multipoint methods based on n+1 function evaluations. Further acceleration of the convergence rate is attained by varying a free parameter from step to step using information available from the previous step. This approach leads to a family of two-step self-accelerating methods with memory whose order of convergence is at least 2+54·236 and even 2+64·449 in special cases. The increase of convergence order is attained without any additional calculations so that the family of methods with memory possesses a very high computational efficiency. Numerical examples are included to demonstrate exceptional convergence speed of the proposed methods using only few function evaluations.
MSC:
65H05Single nonlinear equations (numerical methods)
References:
[1]Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[2]Ostrowski, A. M.: Solution of equations and systems of equations, (1960) · Zbl 0115.11201
[3]Jarratt, P.: Some fourth order multipoint methods for solving equations, Math. comput. 20, 434-437 (1966) · Zbl 0229.65049 · doi:10.2307/2003602
[4]Jarratt, P.: Some efficient fourth-order multipoint methods for solving equations, Bit 9, 119-124 (1969) · Zbl 0188.22101 · doi:10.1007/BF01933248
[5]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
[6]Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, J. ACM 21, 643-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
[7]Neta, B.: On a family of multipoint methods for nonlinear equations, Int. J. Comput. math. 9, 353-361 (1981) · Zbl 0466.65027 · doi:10.1080/00207168108803257
[8]Petković, M. S.; Petković, L. D.: Families of optimal multipoint methods for solving nonlinear equations: a survey, Appl. anal. Discrete math. 4, 1-22 (2010)
[9]Steffensen, J. F.: Remark on iteration, Skand. aktuar tidskr. 16, 64-72 (1933) · Zbl 0007.02601
[10]Ren, H.; Wu, Q.; Bi, W.: A class of two-step Steffensen type methods with fourth-order convergence, Appl. math. Comput. 209, 206-210 (2009) · Zbl 1166.65338 · doi:10.1016/j.amc.2008.12.039
[11]Thukral, R.; Petković, M. S.: A family of three-point methods of optimal order for solving nonlinear equations, J. comput. Appl. math. 233, 2278-2284 (2010) · Zbl 1180.65058 · doi:10.1016/j.cam.2009.10.012
[12]Ortega, J. M.; Rheiboldt, W. C.: Iterative solution of nonlinear equations in several variables, (1970) · Zbl 0241.65046
[13]Alefeld, G.; Herzberger, J.: Introduction to interval computation, (1983) · Zbl 0552.65041
[14]Petković, M. S.: On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. anal. 47, 4402-4414 (2010) · Zbl 1209.65053 · doi:10.1137/090758763
[15]Maheshwari, A. K.: A fourth-order iterative method for solving nonlinear equations, Appl. math. Comput. 211, 383-391 (2009) · Zbl 1162.65346 · doi:10.1016/j.amc.2009.01.047
[16]Basu, D.: From third to fourth order variant of Newton’s method for simple roots, Appl. math. Comput. 202, 886-892 (2008) · Zbl 1147.65037 · doi:10.1016/j.amc.2008.02.021
[17]Chun, C.: A family of composite fourth-order iterative methods for solving nonlinear equations, Appl. math. Comput. 187, 951-956 (2007) · Zbl 1116.65054 · doi:10.1016/j.amc.2006.09.009
[18]Chun, C.: Some variants of King’s fourth-order family of methods for nonlinear equations, Appl. math. Comput. 190, 57-62 (2007) · Zbl 1122.65328 · doi:10.1016/j.amc.2007.01.006
[19]Chun, C.: Some fourth-order iterative methods for solving nonlinear equations, Appl. math. Comput. 195, 454-459 (2008) · Zbl 1173.65031 · doi:10.1016/j.amc.2007.04.105
[20]Chun, C.; Ham, Y.: A one-parameter fourth-order family of iterative methods for nonlinear equations, Appl. math. Comput. 189, 610-614 (2007) · Zbl 1122.65330 · doi:10.1016/j.amc.2006.11.113
[21]Chun, C.; Neta, B.: Certain improvements of Newton’s method with fourth-order convergence, Appl. math. Comput. 215, 821-828 (2009) · Zbl 1192.65049 · doi:10.1016/j.amc.2009.06.007
[22]Petković, M. S.; Petković, L. D.: A one parameter square root family of two-step root-finders, Appl. math. Comput. 188, 339-344 (2007) · Zbl 1118.65042 · doi:10.1016/j.amc.2006.09.122
[23]Yun, B. I.: A non-iterative method for solving non-linear equations, Appl. math. Comput. 198, 691-699 (2008) · Zbl 1138.65035 · doi:10.1016/j.amc.2007.09.006