# zbMATH — the first resource for mathematics

Iterative methods for solving nonlinear equations with finitely many roots in an interval. (English) Zbl 1247.65064
A nonlinear equation $$f(x) = 0$$ having finitely many roots in general, in a given interval, is considered. Based on the so-called numerical integration method (NIM) the author develops a new iterative method to find all of the roots. The paper is organized as follows. The first Section is preliminaries. In Section 2 a basic algorithm combining NIM and existing iterative methods for a unique simple root in an interval is studied. In the next section the method to the case of finitely many roots via partitioning the given interval is extended. In Section 4 the method for finding multiple roots by using a transformation is generalized and, additionally it is shown that it is also available to find the extrema of $$f(x)$$. The usefulness of the proposed method by performing several numerical examples is demonstrated. In the last section the method with a concluding remark is summarized.

##### MSC:
 65H05 Numerical computation of solutions to single equations
Full Text:
##### References:
 [1] Basto, M.; Semiao, V.; Calheiros, F.L., A new iterative method to compute nonlinear equations, Appl. math. comput., 173, 468-483, (2006) · Zbl 1091.65043 [2] Bi, W.; Ren, H.; Wu, Q., Three-step iterative methods with eighth-order convergence for solving nonlinear equations, J. comput. appl. math., 225, 105-112, (2009) · Zbl 1161.65039 [3] Chen, J., New modified regula falsi method for nonlinear equations, Appl. math. comput., 184, 965-971, (2007) · Zbl 1114.65042 [4] Chun, C., A family of composite fourth-order iterative methods for solving nonlinear equations, Appl. math. comput., 187, 951-956, (2007) · Zbl 1116.65054 [5] Chun, C.; Ham, Y., Some fourth-order modifications of newton’s method, Appl. math. comput., 197, 654-658, (2008) · Zbl 1137.65028 [6] Cordero, A.; Torregrosa, J.R., Variants of newton’s method using fifth-order quadrature formulas, Appl. math. comput., 190, 686-698, (2007) · Zbl 1122.65350 [7] Cordero, A.; Torregrosa, J.R.; Vassileva, M.P., Three-step iterative methods with optimal eighth-order convergence, J. comput. appl. math., 235, 3189-3194, (2011) · Zbl 1215.65091 [8] Dzunić, 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 [9] 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 [10] Geum, Y.H.; Kim, Y.I., A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots, Appl. math. lett., 24, 929-935, (2011) · Zbl 1215.65092 [11] Homeier, H.H.H., A modified Newton method with third-order convergence, J. comput. appl. math., 157, 227-230, (2003) · Zbl 1070.65541 [12] Kou, J.; Li, Y.; Wang, A., A modification of Newton method with third-order convergence, Appl. math. comput., 181, 1106-1111, (2006) · Zbl 1172.65021 [13] Kou, J.; Wang, X.; Li, Y., Some eighth-order root-finding three-step methods, Commun. nonlinear sci. numer. simul., 15, 536-544, (2010) · Zbl 1221.65115 [14] Noor, M.A., New class of iterative methods for nonlinear equations, Appl. math. comput., 191, 128-131, (2007) · Zbl 1193.65069 [15] Peng, Y.; Feng, H.; Li, Q.; Zhang, X., A fourth-order derivative-free algorithm for nonlinear equations, J. comput. appl. math., 235, 2551-2559, (2011) · Zbl 1229.65083 [16] Petković, L.D.; Petković, M.S., A note on some recent methods for solving nonlinear equations, Appl. math. comput., 185, 368-374, (2007) · Zbl 1121.65321 [17] 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 [18] 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 [19] Petković, M.S.; Rancić, L.; Milosević, M.R., On the new fourth-order methods for the simultaneous approximation of polynomial zeros, J. comput. appl. math., 235, 4059-4075, (2011) · Zbl 1222.65045 [20] 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 [21] Sharma, J.R.; Guha, R.K., Second-derivative free methods of third and fourth order for solving nonlinear equations, Int. J. comput. math., 88, 163-170, (2011) · Zbl 1215.65096 [22] Wang, X.; Liu, L., Modified ostrowski’s method with eighth-order convergence and high efficiency index, Appl. math. lett., 23, 549-554, (2010) · Zbl 1191.65050 [23] Wu, X.Y.; Fu, D.S., New higher-order convergence iteration methods without employing derivatives for solving nonlinear equations, Comput. math. appl., 41, 489-495, (2001) · Zbl 0985.65047 [24] Yun, B.I., A non-iterative method for solving non-linear equations, Appl. math. comput., 198, 691-699, (2008) · Zbl 1138.65035 [25] Yun, B.I.; Petković, M.S., Iterative methods based on the signum function approach for solving nonlinear equations, Numer. algorithms, 52, 649-662, (2009) · Zbl 1178.65046 [26] Chun, C.; Neta, B., A third-order modification of newton’s method for multiple roots, Appl. math. comput., 211, 474-479, (2009) · Zbl 1162.65342 [27] Dong, C., A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation, Math. numer. sin., 11, 445-450, (1982) · Zbl 0511.65030 [28] Dong, C., A family of multipoint iterative functions for finding multiple roots of equations, Int. J. comput. math., 21, 363-367, (1987) · Zbl 0656.65050 [29] Li, S.; Cheng, L.; Neta, B., Some fourth-order nonlinear solvers with closed formulae for multiple roots, Comput. math. appl., 59, 126-135, (2010) · Zbl 1189.65093 [30] Neta, B., New third order nonlinear solvers for multiple roots, Appl. math. comput., 202, 162-170, (2008) · Zbl 1151.65041 [31] Neta, B.; Johnson, A.N., High order nonlinear solvers for multiple roots, Comput. math. appl., 55, 2012-2017, (2008) · Zbl 1142.65044 [32] Neta, B., Extension of murakami’s high order nonlinear solver to multiple roots, Int. J. comput. math., 87, 1023-1031, (2010) · Zbl 1192.65052 [33] Osada, N., An optimal multiple root finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045 [34] Parida, P.K.; Gupta, D.K., An improved method for finding multiple roots and it’s multiplicity of nonlinear equations in $$\mathbb{R}$$, Appl. math. comput., 202, 498-503, (2008) · Zbl 1151.65042 [35] Petković, M.S.; Petković, L.D.; Dzunić, J., Accelerating generators of iterative methods for finding multiple roots of nonlinear equations, Comput. math. appl., 59, 2784-2793, (2010) · Zbl 1193.65071 [36] Sharma, J.R.; Sharma, R., Modified jarratt method for computing multiple roots, Appl. math. comput., 217, 878-881, (2010) · Zbl 1203.65084 [37] Victory, H.D.; Neta, B., A higher order method for multiple zeros of nonlinear functions, Int. J. comput. math., 12, 329-335, (1983) · Zbl 0499.65026 [38] Yun, B.I., Transformation methods for finding multiple roots of nonlinear equations, Appl. math. comput., 217, 599-606, (2010) · Zbl 1205.65177 [39] Yun, B.I., New higher order methods for solving nonlinear equations with multiple roots, J. comput. appl. math., 235, 1533-1555, (2011) [40] Jarratt, P., Some efficient fourth order multipoint methods for solving equations, Bit, 9, 119-124, (1969) · Zbl 0188.22101 [41] King, R.F., A family of fourth order methods for nonlinear equations, SIAM J. numer. anal., 10, 876-879, (1973) · Zbl 0266.65040 [42] Neta, B., Several new methods for solving equations, Int. J. comput. math, 23, 265-282, (1988) · Zbl 0661.65048 [43] Ostrowski, A.M., On approximation of equations by algebraic equations, SIAM J. numer. anal., 1, 104-130, (1964) · Zbl 0138.09602 [44] Ostrowski, A.M., Solution of equations and system of equations, (1973), Academic Press New York · Zbl 0304.65002 [45] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice Hall New York · Zbl 0121.11204
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.