Abstract
The famous Newton—Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton—Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton— Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.
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References
I. K. Argyros,On a new Newton—Mysovskii-type theorem with applications to inexact Newton-like methods and their discretizations, IMA J. Numer. Anal.18 (1997), 37–56.
I. K. Argyros,Local convergence of inexact Newton-like iterative methods and applications, Computers and Mathematics with Application39 (2000), 69–75.
I. K. Argyros,Advances in the Efficiency of Computational Methods and Applications, World Scientific Publ. Co., River Edge, NJ, 2000.
I. K. Argyros and F. Szidarovszky,The Theory and Applications of Iteration Methods, C.R.C. Press, Boca Raton, Florida, 1993.
X. Chen and T. Yamamoto,Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimiz.10, (1989), Nos. 1 and 2, 37–48.
J. E. Dennis,Toward a unified convergence theory for Newton-like methods, inNonlinear Functional Analysis and Applications (L.B. Rall, ed.), Academic Press, New York, 1971, 425–472.
P. Deuflhard and G. Heindl,Affine invariant convergence theorems for Newton's method and extensions to related methods, SIAM J. Numer. Anal.16 (1979), 1–10.
L. V. Kantorovich and G. P. Akilov,Functional Analysis, Pergamon Press, Oxford, 1982.
G. J. Miel,Majorizing sequences and error bounds for iterative methods, Math. Comput.34(149) (1980), 185–202.
I. Moret,A note on Newton-type iterative methods, Computing33 (1984), 65–73.
F. A. Potra,On the convergence of a class of Newton-like methods. In:Iterative Solution of Nonlinear Systems of Equations (R. Ansarge and W. Toening, eds.), Lecture Notes in Math.953, Springer, Berlin, Heidelberg, New York, 1982, 125–137.
W. C. Rheinboldt,A unified convergence theory for a class of iterative processes, SIAM J. Numer. Anal.5 (1968), 42–63.
W. C. Rheinboldt,An adaptive continuation process for solving systems of nonlinear equations, Polish Academy of Science, Banach Ctr. Publ.3 (1977), 129–142.
T. Yamamoto,A convergence theorem for Newton-like methods in Banach spaces, Numer. Math.51 (1987), 545–557.
T. J. Ypma,Local convergence of inexact Newton methods, SIAM J. Numer. Anal.21 (3) (1984), 583–590.
P. P. Zabrejko and D. F. Nguen,The majorant method in the theory of Newton approximations and the Ptak error estimates, Numer. Funct. Anal. and Optimiz.9 (1987), 671–684.
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Dr. Ioannis K. Argyros is a Full Professor of Mathematics at Cameron University, Lawton OK U.S.A. Research Interests: Numerical Analysis, Numerical Functional Analysis, Functional Analysis, Optimization, Mathematical Economics, Applied Mathematics, Applied Analysis, and Approximation Theory. Publications: Over 250 manuscripts, and 4 books in Mathematics. Editor: (1)Advances in nonlinear variational inequalities; (2) Computational analysis and Applications; (3)Journal of Applied Mathematics and Computing; (4)Southwest Journal of Pure and Applied Mathematics. Citizenship: U.S.A. City of Birth: Athens, Greece. Degrees: B.Sci.University of Athens Greece; M.Sci University of Georgia, Athens GA,U.S.A., Ph. D. University of Georgia Athens, GA, U.S.A.
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Argyros, I.K. Weak sufficient convergence conditions and applications for newton methods. JAMC 16, 1–17 (2004). https://doi.org/10.1007/BF02936147
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DOI: https://doi.org/10.1007/BF02936147