Abstract
We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.
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References
Argyros, I.K.: The theory and application of abstract polynomial equations. Mathematics Series. St. Lucie/CRC/Lewis, Boca Raton (1998)
Argyros, I.K.: On the Newton–Kantorovich hypothesis for solving equations. J. Comput. Appl. Math. 169, 315–332 (2004)
Argyros, I.K.: A unifying local–semilocal convergence analysis and applications for two-point Newton–like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004)
Argyros, I.K.: Convergence and applications of Newton–type iterations. Springer, New York (2008)
Argyros, I.K.: On a class of Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 228, 115–122 (2009)
Chandrasekhar, S.: Radiative transfer. Dover, New York (1960)
Chen, X.: On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms. Ann. Inst. Stat. Math. 42, 387–401 (1990)
Chen, X., Yamamoto, T.: Convergence domains of certain iterative methods for solving nonlinear equations. Numer. Funct. Anal. Optim. 10, 37–48 (1989)
Cianciaruso, F., De Pascale, E.: Newton–Kantorovich aproximations when the derivative is Hölderian: old and new results. Numer. Funct. Anal. Optim. 24, 713–723 (2003)
Cianciaruso, F.: A further journey in the “terra incognita” of the Newton–Kantorovich method. Nonlinear Funct. Anal. Appl. (2009, in press)
Dennis, J.E.: Toward a unified convergence theory for Newton–like methods. In: Rall, L.B. (ed.) Nonlinear Functional Analysis and Applications, pp. 425–472. Academic, New York (1971)
Deuflhard, P.: Newton methods for nonlinear problems. In: Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, vol. 35. Springer, Berlin (2004)
Deuflhard, P., Heindl, G.: Affine invariant convergence theorems for Newton’s method and extensions to related methods. SIAM J. Numer. Anal. 16, 1–10 (1979)
Gutiérrez, J.M.: A new semilocal convergence theorem for Newton’s method. J. Comput. Appl. Math. 79, 131–145 (1997)
Hernández, M.A.: The Newton method for operators with Hölder continuous first derivatives. J. Optim. Theory Appl. 109(3), 631–648 (2001)
Hernández, M.A., Rubio, M.J., Ezquerro, J.A.: Secant-like methods for slving nonlinear integral equations of the Hammerstein type. J. Comput. Appl. Math. 115, 245–254 (2000)
Huang, Z.: A note of Kantorovich theorem for Newton iteration. J. Comput. Appl. Math. 47, 211–217 (1993)
Kantorovich, L.V., Akilov, G.P.: Functional analysis. Pergamon, Oxford (1982)
Miel, G.J.: Unified error analysis for Newton–type methods. Numer. Math. 33, 391–396 (1979)
Miel, G.J.: Majorizing sequences and error bounds for iterative methods. Math. Comput. 34, 185–202 (1980)
Moret, I.: A note on Newton type iterative methods. Computing 33, 65–73 (1984)
Potra, F.A.: Sharp error bounds for a class of Newton–like methods. Libertas Mathematica 5, 71–84 (1985)
Rheinboldt, W.C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal. 5, 42–63 (1968)
Yamamoto, T.: A convergence theorem for Newton–like methods in Banach spaces. Numer. Math. 51, 545–557 (1987)
Zabrejko, P.P., Nguen, D.F.: The majorant method in the theory of Newton–Kantorovich approximations and the Pták error estimates. Numer. Funct. Anal. Optim. 9, 671–684 (1987)
Zinc̆enko, A.I.: Some approximate methods of solving equations with non-differentiable operators. (Ukrainian), Dopovidi Akad, pp. 156–161. Nauk Ukraïn. RSR (1963)
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Argyros, I.K., Hilout, S. On the convergence of Newton-type methods under mild differentiability conditions. Numer Algor 52, 701–726 (2009). https://doi.org/10.1007/s11075-009-9308-x
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DOI: https://doi.org/10.1007/s11075-009-9308-x