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Interpolatory multipoint methods with memory for solving nonlinear equations. (English) Zbl 1243.65054
Summary: A general way to construct multipoint methods for solving nonlinear equations by using inverse interpolation is presented. The proposed methods belong to the class of multipoint methods with memory. In particular, a new two-point method with memory with the order (5+17)/24·562 is derived. Computational efficiency of the presented methods is analyzed and their comparison with existing methods with and without memory is performed on numerical examples. It is shown that a special choice of initial approximations provides a considerably great accuracy of root approximations obtained by the proposed interpolatory iterative methods.
65H05Single nonlinear equations (numerical methods)
65Y20Complexity and performance of numerical algorithms
[1]Bi, W.; Wu, Q.; Ren, H.: A new family of eight-order iterative methods for solving nonlinear equations, Appl. math. Comput. 214, 236-245 (2009) · Zbl 1173.65030 · doi:10.1016/j.amc.2009.03.077
[2]Bi, W.; Ren, H.; Wu, Q.: Three-step iterative methods with eight-order convergence for solving nonlinear equations, J. comput. Appl. math. 225, 105-112 (2009) · Zbl 1161.65039 · doi:10.1016/j.cam.2008.07.004
[3]Džunić, J.; Petković, M. S.; Petković, L. D.: A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. math. Comput. 217, 7612-7619 (2011) · Zbl 1216.65056 · doi:10.1016/j.amc.2011.02.055
[4]Geum, Y. H.; Kim, Y. I.: A multi-parameter family of three-step eighth-order iterative methods locating a simple root, Appl. math. Comput. 215, 3375-3382 (2010) · Zbl 1183.65049 · doi:10.1016/j.amc.2009.10.030
[5]T. Granlund, GNU MP; The GNU Multiple Precision Arithmetic Library, edition 5.0.1, 2010.
[6]Herzberger, J.: Über matrixdarstellungen für iterationverfahren bei nichtlinearen gleichungen, Computing 12, 215-222 (1974) · Zbl 0278.65054 · doi:10.1007/BF02293107
[7]Jarratt, P.: Some fourth order multipoint methods for solving equations, Math. comput. 20, 434-437 (1966) · Zbl 0229.65049 · doi:10.2307/2003602
[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]King, R. F.: A fifth order family of modified Newton methods, Bit 11, 409-412 (1971) · Zbl 0231.65052 · doi:10.1007/BF01939409
[10]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
[11]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
[12]Liu, L.; Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations, Appl. math. Comput. 215, 3449-3454 (2010)
[13]Maheshwari, A. K.: A fourth-order iterative method for solving nonlinear equations, Appl. math. Comput. 211, 383-391 (2009)
[14]Neta, B.: A sixth order family of methods for nonlinear equations, Int. J. Comput. math. 7, 157-161 (1979)
[15]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
[16]Neta, B.: A new family of higher order methods for solving equations, Int. J. Comput. math. 14, 191-195 (1983) · Zbl 0514.65029 · doi:10.1080/00207168308803384
[17]Neta, B.: Several new methods for solving equations, Int. J. Comput. 23, 265-282 (1988) · Zbl 0661.65048 · doi:10.1080/00207168808803622
[18]Neta, B.; Johnson, A. N.: High order nonlinear solver, J. comput. Methods sci. Eng. 8, 245-250 (2008) · Zbl 1168.65345
[19]Neta, B.; Petković, M. S.: Construction of optimal order nonlinear solvers using inverse interpolation, Appl. math. Comput. 217, 2448-2455 (2010) · Zbl 1202.65062 · doi:10.1016/j.amc.2010.07.045
[20]Ostrowski, A. M.: Solution of equations and systems of equations, (1960) · Zbl 0115.11201
[21]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
[22]Petković, M. S.; Ilić, S.; Džunić, J.: Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. math. Comput. 217, 1887-1895 (2010) · Zbl 1200.65034 · doi:10.1016/j.amc.2010.06.043
[23]M.S. Petković, B. Neta, L.D. Petković, On the Kung-Traub family of multipoint methods with memory, private communication.
[24]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)
[25]Petković, M. S.; Petković, L. D.; Džunić, J.: A class of three-point root-solvers of optimal order of convergence, Appl. math. Comput. 216, 671-676 (2010) · Zbl 1188.65068 · doi:10.1016/j.amc.2010.01.123
[26]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
[27]Sharma, J. R.; Sharma, R.: A new family of modified ostrowskis methods with accelerated eighth order convergence, Numer. algor. 54, 445-458 (2010) · Zbl 1195.65067 · doi:10.1007/s11075-009-9345-5
[28]Thukral, R.; Petković, M. S.: 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
[29]Traub, J. F.: Iterative methods for the solution of equations, (1964) · Zbl 0121.11204
[30]Wang, X.; Liu, L.: New eighth-order iterative methods for solving nonlinear equations, J. comput. Appl. math. 234, 1611-1620 (2010) · Zbl 1190.65081 · doi:10.1016/j.cam.2010.03.002
[31]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
[32]Yun, B. I.; Petković, M. S.: Iterative methods based on the signum function approach for solving nonlinear equations, Numer. algor. 52, 649-662 (2009) · Zbl 1178.65046 · doi:10.1007/s11075-009-9305-0