Previous |  Up |  Next

Article

Keywords:
Newton-Jarratt's method; radius of convergence; local convergence; decomposition techniques; restricted convergence domain
Summary:
A. Cordero et. al (2010) considered a modified Newton-Jarratt's composition to solve nonlinear equations. In this study, using decomposition technique under weaker assumptions we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.
References:
[1] Amat S., Busquier S., Negra M.: Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim. 25 (2004), no. 5–6, 397–405. DOI 10.1081/NFA-200042628 | MR 2106266 | Zbl 1071.65077
[2] Argyros I. K.: Computational Theory of Iterative Methods. Studies in Computational Mathematics, 15, Elsevier, Amsterdam, 2007. MR 2356038 | Zbl 1147.65313
[3] Argyros I. K., Cho Y. J., George S.: Local convergence for some third-order iterative methods under weak conditions. J. Korean Math. Soc. 53 (2016), no. 4, 781–793. DOI 10.4134/JKMS.j150244 | MR 3521238
[4] Argyros I. K., George S.: Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions. Calcolo 53 (2016), no. 4, 585–595. DOI 10.1007/s10092-015-0163-y | MR 3574604
[5] Argyros I. K., Magreñán Á. A.: Local convergence analysis of proximal Gauss-Newton method for penalized nonlinear least squares problems. Appl. Math. Comput. 241 (2014), 401–408. MR 3223438
[6] Argyros I. K., Szidarovszky F.: The Theory and Applications of Iteration Methods. Systems Engineering Series, CRC Press, Boca Raton, 1993. MR 1272012
[7] Cordero A., Hueso J., Martinez E., Torregrosa J. R.: A modified Newton-Jarratt's composition. Numer. Algorithms 55 (2010), no. 1, 87–99. DOI 10.1007/s11075-009-9359-z | MR 2679752
[8] Cordero A., Torregrosa J. R.: Variants of Newton's method for functions of several variables. Appl. Math. Comput. 183 (2006), no. 1, 199–208. MR 2282802
[9] Cordero A., Torregrosa J. R.: Variants of Newton's method using fifth-order quadrature formulas. Appl. Math. Comput. 190 (2007), no. 1, 686–698. MR 2338747
[10] Ezquerro J. A., Hernández M. A., Romero A. N.: Approximacion de soluciones de algunas equacuaciones integrals de Hammerstein mediante metodos iterativos tipo. Newton, XXI Congresode ecuaciones diferenciales y aplicaciones Universidad de Castilla-La Mancha, Ciudad Real, 2009, 8 pages.
[11] Grau-Sánchez M., Grau À., Noguera M.: On the computational efficiency index and some iterative methods for solving systems of non-linear equations. J. Comput. Appl. Math. 236 (2011), no. 6, 1259–1266. DOI 10.1016/j.cam.2011.08.008 | MR 2854048
[12] Gutiérrez J. M., Hernández M. A.: Newton's method under weak Kantorovich conditions. IMA J. Numer. Anal. 20 (2000), no. 4, 521–532. DOI 10.1093/imanum/20.4.521 | MR 1795296
[13] Homeier H. H. H.: A modified Newton method with cubic convergence, the multivariate case. J. Comput. Appl. Math. 169 (2004), no. 1, 161–169. DOI 10.1016/j.cam.2003.12.041 | MR 2071267
[14] Homeier H. H. H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176 (2005), no. 2, 425–432. DOI 10.1016/j.cam.2004.07.027 | MR 2116403
[15] Kou J., Li Y., Wang X.: Some modification of Newton's method with fifth-order convergence. J. Comput. Appl. Math. 209 (2007), no. 2, 146–152. DOI 10.1016/j.cam.2006.10.072 | MR 2387121
[16] Noor M. A., Waseem M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57 (2009), no. 1, 101–106. DOI 10.1016/j.camwa.2008.10.067 | MR 2484261
[17] Ren H., Argyros I. K.: Improved local analysis for a certain class of iterative methods with cubic convergence. Numer. Algorithms 59 (2012), no. 4, 505–521. DOI 10.1007/s11075-011-9501-6 | MR 2892562
[18] Rheinboldt W. C.: An adaptive continuation process for solving systems of nonlinear equations. Mathematical models and numerical methods, Banach Center Publ., 3, PWN, Warszawa, 1978, pages 129–142. MR 0514377 | Zbl 0378.65029
[19] Shah F. A., Noor M. A.: Some numerical methods for solving nonlinear equations by using decomposition technique. Appl. Math. Comput. 251 (2015), 378–386. MR 3294725
[20] Sharma J. R., Gupta P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67 (2014), no. 3, 591–601. DOI 10.1016/j.camwa.2013.12.004 | MR 3149734
[21] Traub J. F.: Iterative Methods for the Solution of Equations. AMS Chelsea Publishing, New York, 1982. Zbl 0672.65025
Partner of
EuDML logo