## Fifth-order iterative method for finding multiple roots of nonlinear equations.(English)Zbl 1222.65047

The authors propose a modification of Newton’s method based on some existing modified Newton’s methods to find multiple roots of nonlinear equations with unknown multiplicity. The fifth-order convergence of the method is established. Some numerical results are presented to show that the proposed method is more efficient than the classical Newton’s method and some other existing methods.

### MSC:

 65H05 Numerical computation of solutions to single equations
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### References:

 [1] Schröder, E.: Über unendlich viele Algorithmen zur Auflösung der Gleichungen. Math. Ann. 2, 317–365 (1870) · doi:10.1007/BF01444024 [2] Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977) · Zbl 0361.65041 · doi:10.1007/BF01396176 [3] Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for Multiple roots. Comput. Math. Appl. 59, 126–135 (2010) · Zbl 1189.65093 · doi:10.1016/j.camwa.2009.08.066 [4] 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 · doi:10.1080/00207168208803346 [5] 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 · doi:10.1080/00207168708803576 [6] Osada, N.: An optimal multiple root-finding method of order three. J. Comput. Appl. Math. 51, 131–133 (1994) · Zbl 0814.65045 · doi:10.1016/0377-0427(94)00044-1 [7] 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 · doi:10.1016/j.amc.2009.01.087 [8] Neta, B.: New third order nonlinear solvers for multiple roots. Appl. Math. Comput. 202, 162–170 (2008) · Zbl 1151.65041 · doi:10.1016/j.amc.2008.01.031 [9] Chun, C., Bae, H.J., Neta, B.: New families of nonlinear third-order solvers for finding multiple roots. Comput. Math. Appl. 57, 1574–1582 (2009) · Zbl 1186.65060 · doi:10.1016/j.camwa.2008.10.070 [10] Neta, B., Johnson, A.N.: High-order nonlinear solver for multiple roots. Comput. Math. Appl. 55, 2012–2017 (2008) · Zbl 1142.65044 · doi:10.1016/j.camwa.2007.09.001 [11] Neta, B.: Extension of Murakami’s high order nonlinear solver to multiple roots. Int. J. Comput. Math. 87, 1023–1031 (2010) · Zbl 1192.65052 · doi:10.1080/00207160802272263 [12] Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood (1964) · Zbl 0121.11204 [13] King, R.F.: A secant method for multiple roots. BIT 17, 321–328 (1977) · Zbl 0367.65026 · doi:10.1007/BF01932152 [14] 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 · doi:10.1016/S0898-1221(00)00290-X [15] Wu, X.Y., Xia, J.L., Shao, R.: Quadratically convergent multiple roots finding method without derivatives. Comput. Math. Appl. 42, 115–119 (2001) · Zbl 0985.65048 · doi:10.1016/S0898-1221(01)00136-5 [16] Steffensen, I.F.: Remark on Iteration, vol. 16, pp. 64–72. Skand, Aktuarietidskr (1933) [17] Wu, X.Y.: A new continuation Newton-like method and its deformation. Appl. Math. Comput. 112, 75–78 (2000) · Zbl 1023.65043 · doi:10.1016/S0096-3003(99)00049-1 [18] Parida, P.K., Gupta, D.K.: An improved method for finding multiple roots and it’s multiplicity of nonlinear equations in R. Appl. Math. Comput. 202, 498–503 (2008) · Zbl 1151.65042 · doi:10.1016/j.amc.2008.02.030 [19] Yun, B.I.: A derivative free iterative method for finding multiple roots of nonlinear equations. Appl. Math. Lett. 22, 1859–1863 (2009) · Zbl 1205.65176 · doi:10.1016/j.aml.2009.07.013 [20] Kioustelidis, J.B.: A derivative-free transformation preserving the order of convergence of iteration methods in case of multiple zeros. Numer. Math. 33, 385–389 (1979) · Zbl 0422.65033 · doi:10.1007/BF01399321 [21] Bi, W., Ren, H., Wu, Q.: New family of seventh-order methods for nonlinear equations. Appl. Math. Comput. 203, 408–412 (2008) · Zbl 1154.65323 · doi:10.1016/j.amc.2008.04.048 [22] 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 · doi:10.1016/j.cam.2008.07.004 [23] Yun, B.I.: Transformation methods for finding multiple roots of nonlinear equations. Appl. Math. Comput. 217, 599–606 (2010) · Zbl 1205.65177 · doi:10.1016/j.amc.2010.05.094 [24] Halley, E.: A new, exact and easy method of finding the roots of equations generally and that without any previous reduction. Phil. Trans. Roy. Soc. Lond. 18, 136–148 (1694) [25] Laguerre, E.N.: Sur une méthode pour obtener par approximation les racines d’une équation algébrique qui a toutes ses racines réelles. Nouvelles Ann. de Math. 2e séries 19, 88–103 (1880) [26] Dong, C.: A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation. Math. Numer. Sinica 11, 445–450 (1982) · Zbl 0511.65030 [27] Werner, W.: Iterationsverfahren höherer Ordnung zur Lösung nicht linearer Gleichungen. Z. Angew. Math. Mech. 61, T322–T324 (1981) [28] Gautschi, W.: Numerical Analysis: an Introduction. Birkhäuser (1997) · Zbl 0877.65001 [29] Neta, B.: On a family of multipoint methods for non-linear equations. Int. J. Comput. Math. 9, 353–361 (1981) · Zbl 0466.65027 · doi:10.1080/00207168108803257
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