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A Convergence Analysis of Newton-Like Method for Singular Equations Using recurrent functions

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2010年第2期31卷 112-130页

作者： Argyros, Ioannis K. Hilout, Said Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France Cameron Univ Dept Math Sci Lawton OK 73505 USA

We provide new semilocal convergence results for Newton-like method involving outer or generalized inverses in a Banach space setting. Using our new idea of recurrent functions and the same or weaker conditions than before [5-195, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], we provide more precise information on the location of the solution and finer bounds on the distances involved. Moreover, since our Newton-Kantorovich-type hypothesis is weaker than before, we can now cover cases not previously possible. Applications and numerical examples involving a nonlinear integral equation of Chandrasekhar-type and a differential equation with Green's function are also provided in this study.

关键词： Banach space Generalized inverse Newton-Kantorovich hypothesis Newton-like methods Outer inverse recurrent functions Semilocal convergence analysis

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On the weakening of the convergence of Newton's method using recurrent functions

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JOURNAL OF COMPLEXITY 2009年第6期25卷 530-543页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France

We use Newton's method to approximate a locally unique solution of an equation in a Banach space setting. We introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [J. Appell, E. De Pascale, J.V. Lysenko, P.P. Zabrejko, New results on Newton-Kantorovich approximations with applications to nonlinear integral equations, Numer. Funct. Anal. Optim. 18 (1997) 1-17;I.K. Argyros, The theory and application of abstract polynomial equations, in: Mathematics Series, St. Lucie/CRC/Lewis Publ., Boca Raton, Florida, USA, 1998;I.K. Argyros, Concerning the "terra incognita" between convergence regions of two Newton methods, Nonlinear Anal. 62 (2005) 179-194;I.K. Argyros, Convergence and Applications of Newton-Type Iterations, Springer-Verlag Publ., New York, 2008: S. Chandrasekhar, Radiative Transfer, Dover Publ., New York. 1960;F. Cianciaruso, E. De Pascale, Newton-Kantorcivich approximations when the derivative is Holderian: Old and new results, Numer. Funct. Anal. Optim. 24 (2003) 713-723;N.T. Demidovich, P.P. Zabrejko, Ju.V. Lysenko, Some remarks on the Newton-Kantorovich method for nonlinear equations with Holder continuous linearizations, Izv. Akad. Nauk Belorus 3 (1993) 22-26. (in Russian): E. De Pascale, P.P. Zabrejko, Convergence of the Newton-Kantorovich method under Vertgeim conditions: A new improvement, Z. Anal, Anwendvugen 17 (1998) 271-280;LV. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982;J.V. Lysenko, Conditions for the convergence of the Newton-Kantorovich method for nonlinear equations with Hblder linearizations, Dokl. Akad. Nauk BSSR 38 (1994) 20-24. (in Russian);B.A. Vertgeim, On conditions for the applicability of Newton's method, (Russian), Dokl. Akad. Nauk., SSSR 110 (1956) 719-722: B.A. Vertgeim, On some methods for the approximate solution of nonlinear functional equations in Banach spaces, Uspekhi Mat. Nauk 12 (1957) 166-169. (in Russian): English trans].:: Amer. Math.

关键词： Newton's method Banach space recurrent functions Holder continuity Lipschitz continuity Semilocal convergence Newton-Kantorovich hypothesis Integral equation Differential equation

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On the convergence of Newton-type methods using recurrent functions

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2010年第14期87卷 3273-3296页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases [X. Chen, On the convergence of Broyden-like methods for nonlinear equations with nondifferentiable terms, Ann. Inst. Statist. Math. 42 (1990), pp. 387-401;X. Chen and T. Yamamoto, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optim. 10 (1989), pp. 37-48;Y. Chen and D. Cai, Inexact overlapped block Broyden methods for solving nonlinear equations, Appl. Math. Comput. 136 (2003), pp. 215-228;J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in Nonlinear Functional Analysis and Applications, L.B. Rall, ed., Academic Press, New York, 1971, pp. 425-472;P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, Springer-Verlag, Berlin, 2004;P. Deuflhard and G. Heindl, Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal. 16 (1979), pp. 1-10;Z. Huang, A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993), pp. 211-217;L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982;D. Li and M. Fukushima, Globally Convergent Broyden-like Methods for Semismooth Equations and Applications to VIP, NCP and MCP, Optimization and Numerical Algebra (Nanjing, 1999), Ann. Oper. Res. 103 (2001), pp. 71-97;C. Ma, A smoothing Broyden-like method for the mixed complementarity problems, Math. Comput. Modelling 41 (2005), pp. 523-538;G.J. Miel, Unified error analysis for Newton-type methods, Numer. Math. 33 (1979), pp. 391-396;G.J. Miel, Majorizing sequences and error bounds for iterative methods, Math. Comp. 34 (1980), pp. 185-202;I. Moret, A

关键词： Newton-type method recurrent functions

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On the Convergence of Broyden-Like Methods Using recurrent functions

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2011年第1期32卷 26-40页

作者： Argyros, Ioannis K. Cho, Yeol Je Hilout, Said Gyeongsang Natl Univ Dept Math Educ Chinju 660701 South Korea Gyeongsang Natl Univ RINS Chinju 660701 South Korea Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat Futuroscope France

In this article, we provide a finer semilocal convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the p-dimensional Euclidean space (p epsilon 1, a natural number) [2, 4, 5, 8, 9]. We achieve the goal by using our new idea of recurrent functions. Special cases and applications are also provided.

关键词： Banach space Broyden-like method 2-norm Majorant principle Frechet-derivative Newton-Kantorovich hypothesis recurrent functions Semilocal convergence

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A unified approach for the convergence of certain numerical algorithms, using recurrent functions

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COMPUTING 2010年第3-4期90卷 131-164页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France

The Kantorovich analysis (Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008;Argyros and Hi lout in Efficient methods for solving equations and variational inequalities, Polimetrica Publisher, Milano, 2009;Kantorovich and Akilov in Functional analysis, Pergamon Press, Oxford, 1982), and recurrent relation's approach (Gutierrez et al. in J Comput Appl Math 115:181-192,2000) are the most popular ways for generating sufficient conditions for the convergence of numerical algorithms to a solution of a nonlinear equations as well as providing the corresponding error estimates on the distances involved. We introduce the new approach of recurrent functions to show that a finer convergence analysis can be provided under the same hypotheses, and computational cost. Numerical examples are provided where our results apply, hut not earlier ones.

关键词： Numerical algorithms Banach space Majorizing sequence Newton-Kantorovich hypothesis Semilocal convergence recurrent functions Boundary value problem with a Green's kernel

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On Newton-like methods of "bounded deterioration" using recurrent functions

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AEQUATIONES MATHEMATICAE 2010年第1-2期79卷 61-82页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France

We provide a semilocal convergence analysis for Newton-like methods of "bounded deterioration" in a Banach space setting. We establish tighter error bounds on the distances involved, and a more precise information on the location of the solution, under the same or weaker hypotheses than before (Argyros, Acta Math. Sin. (Engl. Ser.), 23: 2087-2096, 2007;Deuflhard, Newton methods for nonlinear problems. Affine invariance and adaptive algorithms, Springer Series in Computational Mathematics, vol. 35. Springer, Berlin, 2004;Ezquerro and Hernandez, IMA J. Numer. Anal., 22: 187-205, 2002) using recurrent functions. Numerical examples are also provided involving polynomial, integral, and differential equations.

关键词： Newton-like method Banach space bounded deterioration majorizing sequence Frechet-derivative Newton-Kantorovich hypothesis recurrent functions integral equation of Chandrasekhar type differential equation with a Green's function

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A Kantorovich-type analysis of Broyden's method using recurrent functions

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2010年第2期32卷 353-368页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Poitiers Univ Lab Math & Applicat F-86962 Futuroscope France

We provide a semilocal convergence analysis for Broyden's method for approximating locally unique solutions of nonlinear operator equations. Using the majorant principle we show that under the same or weaker hypotheses, in combination with our new idea of recurrent functions, we can find weaker sufficient conditions for the convergence of Broyden's method as well as finer error bounds on the distances involved, and a more precise information on the location of the solution than before (Broyden, Math. Comput. 19: 577-593, 1965;Chen, Ann. Inst. Stat. Math. 42: 387-401, 1990;Dennis, Nonlinear Functional Analysis and Applications, pp. 425-472, Academic Press, San Diego, 1971;Li and Fukushima, Ann. Oper. Res. 103: 71-97, 2001). Numerical examples are also provided involving polynomial, integral, and differential equations.

关键词： Broyden's method Banach space Majorant principle Frechet-derivative Radius of convergence Jacobian recurrent functions Integral equation of Chandrasekhar type Differential equation with a Green's function

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Extended sufficient semilocal convergence for the Secant method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2011年第2期62卷 599-610页

作者： Cho, Yeol Je Argyros, Ioannis K. Hilout, Said Gyeongsang Natl Univ Dept Math Educ Chinju 660701 South Korea Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France Gyeongsang Natl Univ RINS Chinju 660701 South Korea

We establish new sufficient convergence conditions for the Secant method to a locally unique solution of a nonlinear equation in a Banach space. Using our new concept of recurrent functions, and combining Lipschitz and center-Lipschitz conditions on the divided difference operator, we obtain a new semilocal convergence analysis of the Secant method. Moreover, our sufficient convergence conditions expand the applicability of the Secant method in cases not covered before (Dennis, 1971 [9], Hernandez et al., 2005 [8], Laasonen, 1969 [15], Ortega and Rheinboldt, 1970 [11], Potra, 1982 [5], Potra, 1985 [7], Schmidt, 1978 [18], Yamamoto, 1987 [12], Wolfe, 1978 [19]). Numerical examples are also provided in this study. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： recurrent functions Semilocal convergence Secant method Banach space Majorizing sequence Divided difference

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An amplitude spectral tonometer

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2003年第5-6期46卷 959-969页

作者： Haidar, NHS Hawaii Univ Ctr Dept Basic Sci Beirut Lebanon Univ Massachusetts Dept Math Boston MA 02125 USA

Variations in harmonic amplitudes of one-dimensional recurrent signals of the types encountered in medical tonometry are analyzed over finite length records by single Fourier series. Two regularizational algorithms, one differentiation-invoking and the other differentiation-free, are advanced for the design of a universal amplitude spectral "tonometer" for such signals. (C) 2003 Elsevier Ltd. All rights reserved.

关键词： harmonic amplitudes variations recurrent functions finite time records inverse filtration regularization weighted Sobolev spaces

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Inexact Newton-type methods

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JOURNAL OF COMPLEXITY 2010年第6期26卷 577-590页

作者： Argyros, Ioannis K. Hilout, Said Cameron Univ Dept Math Sci Lawton OK 73505 USA Univ Poitiers Lab Math & Applicat F-86962 Futuroscope France

We introduce the new idea of recurrent functions to provide a semilocal convergence analysis for an inexact Newton-type method, using outer inverses. It turns out that our sufficient convergence conditions are weaker than in earlier studies in many interesting cases (Argyros, 2004 [5,6], Argyros, 2007 [7], Dennis, 1971 [14], Deuflhard and Heindl, 1979 [15], Gutierrez, 1997 [16], Gutierrez et al., 1995 [17], Haubler, 1986 [18], Huang, 1993 [19], Kantorovich and Akilov, 1982 [20], Nashed and Chen, 1993 [21], Potra, 1982 [22], Potra, 1985[23]). (C) 2010 Elsevier Inc. All rights reserved.

关键词： Inexact Newton-type method Newton's method recurrent functions Banach space Semilocal convergence

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