Majorizing sequences for Newton's method under centred conditions for the derivative

References

• S. Amat, C. Bermúdez, S. Busquier, and J.O. Gretay, Convergence by nondiscrete mathematical induction of a two step secant's method, Rocky Mountain J. Math. 37(2) (2007), pp. 359–369. doi: 10.1216/rmjm/1181068756
   Web of Science ®Google Scholar
• S. Amat, S. Busquier, and J.M. Gutiérrez, On the local convergence of secant-type methods, Int. J. Comput. Math. 81(8) (2004), pp. 1153–1161. doi: 10.1080/00207160412331284123
   Web of Science ®Google Scholar
• S. Amat, S. Busquier, and M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25(5) (2004), pp. 397–405. doi: 10.1081/NFA-200042628
   Web of Science ®Google Scholar
• I.K. Argyros, Polynomial Operator Equations in Abstract Spaces and Applications, CRC Press, Boca Raton, FL, 1998.
   Google Scholar
• I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004), pp. 374–397. doi: 10.1016/j.jmaa.2004.04.008
   Web of Science ®Google Scholar
• I.K. Argyros, On the Newton–Kantorovich hypothesis for solving equations, J. Comput. Appl. Math. 169 (2004), pp. 315–332. doi: 10.1016/j.cam.2004.01.029
   Web of Science ®Google Scholar
• I.K. Argyros, Approximating solutions of equations using Newton's method with a modified Newton's method iterate as a starting point, Rev. Anal. Numér. Théor. Approx. 36 (2007), pp. 123–138.
   Google Scholar
• I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publications, New York, 2008.
   Google Scholar
• I.K. Argyros, A semilocal convergence analysis for directional Newton methods, Math. Comp. 80(273) (2010), pp. 327–343. doi: 10.1090/S0025-5718-2010-02398-1
   Google Scholar
• I.K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28(3) (2012), pp. 346–387. doi: 10.1016/j.jco.2011.12.003
   Web of Science ®Google Scholar
• I.K. Argyros and S. Hilout, Estimating upper bounds on the limit points of majorizing sequences for Newton's method, Numer. Algorithms 62 (2013), pp. 115–132. doi: 10.1007/s11075-012-9570-1
   Web of Science ®Google Scholar
• I.K. Argyros, Y.J. Cho, and S. Hilout, Numerical Methods for Equations and its Applications, CRC Press/Taylor & Francis, New York, 2012.
   Google Scholar
• I.K. Argyros, S. Hilout, and M.A. Tabatabai, Mathematical Modelling with Applications in Biosciences and Engineering, Nova Publishers, New York, 2011.
   Google Scholar
• W.E. Bosarge and P.L. Falb, A multipoint method of third order, J. Optim. Theory Appl. 4 (1969), pp. 156–166. doi: 10.1007/BF00930576
   Google Scholar
• J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in Functional Analysis and Applications, L.B. Rall, ed., Academic Press, New York, 1971.
   Google Scholar
• J.A. Ezquerro and M.J. Rubio, A uniparametric family of iterative processes for solving nondifferentiable equations, J. Math. Anal. Appl. 275 (2002), pp. 821–834. doi: 10.1016/S0022-247X(02)00432-8
   Web of Science ®Google Scholar
• J.A. Ezquerro, D. González, and M.A. Hernández, Majorizing sequences for Newton's method from initial value problems, J. Comput. Appl. Math. 236 (2012), pp. 2246–2258. doi: 10.1016/j.cam.2011.11.012
   Web of Science ®Google Scholar
• J.A. Ezquerro, D. González, and M.A. Hernández, A general semilocal convergence result for Newton's method under centered conditions for the second derivative, ESAIM: Math. Model. Numer. Anal. 47 (2013), pp. 149–167. doi: 10.1051/m2an/2012026
   Web of Science ®Google Scholar
• J.A. Ezquerro, M. Grau-Sánchez, M.A. Hernández, and M. Noguera, Semilocal convergence of secant-like methods for differentiable and nondifferentiable operators equations, J. Math. Anal. Appl. 398(1) (2013), pp. 100–112. doi: 10.1016/j.jmaa.2012.08.040
   Web of Science ®Google Scholar
• J.A. Ezquerro, J.M. Gutiérrez, M.A. Hernández, N. Romero, and M.J. Rubio, The Newton method: From Newton to Kantorovich. (Spanish), La Gaceta de la Real Sociedad Matemática Española 13 (2010), pp. 53–76.
   Google Scholar
• J.M. Gutiérrez, A new semilocal convergence theorem for Newton's method, J. Comput. Appl. Math. 79 (1997), pp. 131–145. doi: 10.1016/S0377-0427(97)81611-1
   Web of Science ®Google Scholar
• L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
   Google Scholar
• L.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
   Google Scholar
• A.M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, New York, 1973.
   Google Scholar
• F.A. Potra and V. Pták, Nondiscrete Induction and Iterative Processes. Research Notes in Mathematic, 103, Pitman (Advanced Publishing Program), Boston, MA, 1984.
   Google Scholar
• P.D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton's method, J. Complexity 25 (2009), pp. 38–62. doi: 10.1016/j.jco.2008.05.006
   Web of Science ®Google Scholar
• P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems, J. Complexity 26 (2010), pp. 3–42. doi: 10.1016/j.jco.2009.05.001
   Web of Science ®Google Scholar
• J.F. Traub, 0Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964.
   Google Scholar
• T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987), pp. 545–557. doi: 10.1007/BF01400355
   Web of Science ®Google Scholar