# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\left(5+\sqrt{17}\right)/2\approx 4·562$ 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 Single nonlinear equations (numerical methods) 65Y20 Complexity and performance of numerical algorithms
##### 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 · 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