Abstract
We study a class of at least third order iterative methods for nonlinear equations on Banach spaces. A characterization of the convergence under Gamma-type conditions is presented. Though, in general, these methods are not very extended due to their computational costs, we can find examples in which they are competitive and even cheaper than other simpler methods. Indeed, we propose a new nonlinear mathematical model for the denoising of digital images, where the best method in the family has fourth order of convergence. Moreover, our family includes two-step Newton type methods with good numerical behavior in general. We center our analysis in both, analytic and computational, aspects.
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References
Alefeld, G.E.: On the convergence of Halley’s method. Am. Math. Monthly. 88(7), 530–536 (1981)
Amat, S., Ruiz, J., Trillo, J.C.: Fast multiresolution algorithms and their related variational problems for image denoising. J. Sci. Comp. 43(1), 1–23 (2010)
Argyros, I.K., Chen, D.: Results on the Chebyshev method in Banach spaces. Proyecciones 12(2), 119–128 (1993)
Argyros, I.K.: The convergence of a Halley-Chebyshev-type method under Newton-Kantorovich hypotheses. Appl. Math. Lett. 6(5), 71–74 (1993)
Argyros, I.K.: An inverse free Newton-Jarrat-type iterative method for solving equations under gamma condition. J. Appl. Math. Comput. 28(1–2), 15–28 (2008)
Candela, V.F., Marquina, A.: Recurrence relations for rational cubic methods I: the Halley method. Computing 44, 169–184 (1990)
Candela, V.F., Marquina, A.: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing 45, 355–367 (1990)
Chan, T., Esedoglu, S., Park, F., Yip, A.: Total variation image restoration: overview and recent developments. Handbook of mathematical models in computer vision, pp. 17–31. Springer, New York (2006)
Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)
Chan, T.F., Mulet, P.: Iterative methods for total variation image restoration. Iterative methods in scientific computing (Hong Kong, 1995), pp/ 359–381. Springer, Singapore (1997)
Chan, T.F., Shen, J., Vese, L.: Variational PDE models in image processing. Not. Am. Math. Soc. 50(1), 14–26 (2003)
Chun, C.: Construction of third-order modifications of Newton’s method. Appl. Math. Comput. 189, 662–668 (2007)
Ezquerro, J.A., Gutiérrez, J.M., Hernández, M.A., Salanova, M.A.: Chebyshev-like methods and quadratic equations. Rev. Anal. Numer. Theor. Approx. 28(1), 23–35 (2000)
Ezquerro, J.A., Hernández, M.A., Romero, N.: A modification of Cauchy’s method for quadratic equations. J. Math. Anal. Appl. 339, 954–969 (2008)
Hernández, M.A., Romero, N.: On a characterization of some Newton-like methods of R-order at least three. J. Comput. Appl. Math. 183(1), 53–66 (2005)
Househölder, A.S.: The Numerical Treatment of a Single Nonlinear Equation. McGraw-Hill, New York (1970)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Kou, J.: Some new sixth-order methods for solving non-linear equations. Appl. Math. Comput. 189, 647–651 (2007)
Kou, J., Li, Y.: A family of modified super-Halley methods with fourth-order convergence. Appl. Math. Comput. 189, 366–370 (2007)
Kou, J., Wang, X.: Some variants of Chebyshev-Halley methods for solving nonlinear equations. Appl. Math. Comput. 189, 1839–1843 (2007)
Marquina, A., Osher, S.: Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal. SIAM J. Sci. Comput. 22(2), 387–405 (2000)
Noor, M.A., Khan, W.A., Hussain, A.: A new modified Halley method without second derivatives for nonlinear equation. Appl. Math. Comput. 189(2), 1268–1273 (2007)
Potra, F.A., Pták, V.: Nondiscrete Induction and Iterative Processes. Research Notes in Mathematics, vol. 103. Pitman, Boston (1984)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Pose Problems. Wiley, New York (1997)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs (1964)
Vogel, C.R., Oman, M.E.: Iterative methods for total variation denoising. SIAM J. Sci. Stat. Comput. 17, 227–238 (1996)
Acknowledgments
S. Amat: Research supported in part by the Spanish grants MINECO-FEDER MTM2010-17508 and 08662/PI/08. The second and third authors are partly supported by MINECO-FEDER MTM 2011-28636-C02-01.
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Amat, S., Hernández, M.A. & Romero, N. On a family of high-order iterative methods under gamma conditions with applications in denoising. Numer. Math. 127, 201–221 (2014). https://doi.org/10.1007/s00211-013-0589-6
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DOI: https://doi.org/10.1007/s00211-013-0589-6