Optimal Error Bounds for the Newton–Kantorovich Theorem

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Previous article Next article Optimal Error Bounds for the Newton–Kantorovich TheoremW. B. Gragg and R. A. TapiaW. B. Gragg and R. A. Tapiahttps://doi.org/10.1137/0711002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractBest possible upper and lower bounds for the error in Newton's method are established under the hypotheses of the Kantorovich theorem.[1] J. E. Dennis, Jr., Masters Thesis, Variations on Newton's method, Doctoral dissertation, Univ. of Utah, Salt Lake City, 1966 Google Scholar[2] L. V. Kantorovich, Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.), 3 (1948), 89–185, English transl., Rep. 1509, National Bureau of Standards, Washington, D.C., 1952. MR0027947 Google Scholar[3] L. V. Kantorovich and , G. P. Akilov, Functional Analysis in Normed Spaces, Fizmatgiz, Moscow, 1959, English transl., Pergamon Press, Oxford, 1964. Google Scholar[4] James M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658–660 MR0231218 0183.43004 CrossrefISIGoogle Scholar[5] J. M. Ortega and , W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970xx+572 MR0273810 0241.65046 Google Scholar[6] A. M. Ostrowski, Solution of equations and systems of equations, Second edition. Pure and Applied Mathematics, Vol. 9, Academic Press, New York, 1966xiv+338 MR0216746 0222.65070 Google Scholar[7] Alexandre Ostrowski, La méthode de Newton dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1251–A1253 MR0285110 0228.65041 Google Scholar[8] Louis B. Rall, Computational solution of nonlinear operator equations, With an appendix by Ramon E. Moore, John Wiley & Sons Inc., New York, 1969viii+225 MR0240944 0175.15804 Google Scholar[9] R. A. Tapia, Classroom Notes: The Kantorovich Theorem for Newton's Method, Amer. Math. Monthly, 78 (1971), 389–392 MR1536290 0215.27404 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Asymptotic theory in network models with covariates and a growing number of node parametersAnnals of the Institute of Statistical Mathematics, Vol. 9 | 2 September 2022 Cross Ref On semilocal convergence analysis for two-step Newton method under generalized Lipschitz conditions in Banach spacesNumerical Algorithms, Vol. 90, No. 2 | 6 October 2021 Cross Ref Asymptotic in the Ordered Networks with a Noisy Degree SequenceJournal of Systems Science and Complexity, Vol. 35, No. 3 | 8 August 2021 Cross Ref Certification for polynomial systems via square subsystemsJournal of Symbolic Computation, Vol. 109 | 1 Mar 2022 Cross Ref Dimensionality of Iterative Methods: The Adimensional Scale Invariant Steffensen (ASIS) MethodMathematics, Vol. 10, No. 6 | 12 March 2022 Cross Ref Asymptotic in undirected random graph models with a noisy degree sequenceCommunications in Statistics - Theory and Methods, Vol. 51, No. 3 | 24 April 2020 Cross Ref Specification and Validation of Numerical Algorithms with the Gradual Contracts PatternTesting Software and Systems | 10 May 2022 Cross Ref A network Poisson model for weighted directed networks with covariatesCommunications in Statistics - Theory and Methods, Vol. 26 | 28 November 2021 Cross Ref A note on undirected random graph models parameterized by the strengths of verticesCommunications in Statistics - Theory and Methods, Vol. 50, No. 22 | 18 February 2020 Cross Ref Asymptotics in the β -model for networks with a differentially private degree sequenceCommunications in Statistics - Theory and Methods, Vol. 49, No. 18 | 3 April 2019 Cross Ref Nonlinear Fredholm integral equations and majorant functionsNumerical Algorithms, Vol. 82, No. 4 | 11 January 2019 Cross Ref Improved semilocal convergence analysis in Banach space with applications to chemistryJournal of Mathematical Chemistry, Vol. 56, No. 7 | 10 November 2017 Cross Ref New improved convergence analysis for Newton-like methods with applicationsJournal of Mathematical Chemistry, Vol. 55, No. 7 | 18 January 2017 Cross Ref Improving Newton's Method Performance by Parametrization: The Case of the Richards EquationKonstantin Brenner and Clément CancèsSIAM Journal on Numerical Analysis, Vol. 55, No. 4 | 18 July 2017AbstractPDF (1318 KB)A Superquadratic Variant of Newton's MethodFlorian A. 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