## 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 $$(5 + \sqrt {17})/2 \approx 4.562$$ 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.

### MSC:

 65H05 Numerical computation of solutions to single equations 65Y20 Complexity and performance of numerical algorithms

gmp
Full Text:

### References:

 [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 [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 [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 [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 [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 [7] Jarratt, P., Some fourth order multipoint methods for solving equations, Math. comput., 20, 434-437, (1966) · Zbl 0229.65049 [8] Jarratt, P., Some efficient fourth-order multipoint methods for solving equations, Bit, 9, 119-124, (1969) · Zbl 0188.22101 [9] King, R.F., A fifth order family of modified Newton methods, Bit, 11, 409-412, (1971) · Zbl 0231.65052 [10] King, R., A family of fourth order methods for nonlinear equations, SIAM J. numer. anal., 10, 876-879, (1973) · Zbl 0266.65040 [11] Kung, H.T.; Traub, J.F., Optimal order of one-point and multipoint iteration, J. ACM, 21, 643-651, (1974) · Zbl 0289.65023 [12] Liu, L.; Wang, X., Eighth-order methods with high efficiency index for solving nonlinear equations, Appl. math. comput., 215, 3449-3454, (2010) · Zbl 1183.65051 [13] Maheshwari, A.K., A fourth-order iterative method for solving nonlinear equations, Appl. math. comput., 211, 383-391, (2009) · Zbl 1162.65346 [14] Neta, B., A sixth order family of methods for nonlinear equations, Int. J. comput. math., 7, 157-161, (1979) · Zbl 0397.65032 [15] Neta, B., On a family of multipoint methods for nonlinear equations, Int. J. comput. math., 9, 353-361, (1981) · Zbl 0466.65027 [16] Neta, B., A new family of higher order methods for solving equations, Int. J. comput. math., 14, 191-195, (1983) · Zbl 0514.65029 [17] Neta, B., Several new methods for solving equations, Int. J. comput., 23, 265-282, (1988) · Zbl 0661.65048 [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 [20] Ostrowski, A.M., Solution of equations and systems of equations, (1960), Academic Press New York · 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 [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 [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) · Zbl 1299.65094 [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 [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 [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 [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 [29] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice-Hall Englewood Cliffs, New Jersey · 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 [31] Yun, B.I., A non-iterative method for solving non-linear equations, Appl. math. comput., 198, 691-699, (2008) · Zbl 1138.65035 [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
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.