On Newton’s Method for Singular Problems

Abstract

Previous article Next article On Newton’s Method for Singular ProblemsG. W. ReddienG. W. Reddienhttps://doi.org/10.1137/0715064PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractSufficient conditions are given for Newton’s method to converge when the derivative is singular at the root.[1] R. C. Cavanagh, Masters Thesis, Difference equations and iterative processes, Thesis, Computer Sci. Dept., Univ. of Maryland, College Park, 1970 Google Scholar[2] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966xix+592 MR0203473 0148.12601 CrossrefGoogle Scholar[3] Herbert B. Keller, P. H. Rabinowitz, Numerical solution of bifurcation and nonlinear eigenvalue problemsApplications of bifurcation theory (Proc. Advanced Sem., Univ. Wisconsin, Madison, Wis., 1976), Academic Press, New York, 1977, 359–384. Publ. Math. Res. Center, No. 38 MR0455353 0581.65043 Google Scholar[4] L. B. Rall, Convergence of the Newton process to multiple solutions, Numer. Math., 9 (1966), 23–37 10.1007/BF02165226 MR0210316 0163.38702 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Improved two-step Newton’s method for computing simple multiple zeros of polynomial systemsNumerical Algorithms, Vol. 91, No. 1 | 2 February 2022 Cross Ref Benchmarking results for the Newton–Anderson methodResults in Applied Mathematics, Vol. 8 | 1 Nov 2020 Cross Ref Derivative-free method for singular systems2019 IEEE 17th International Symposium on Intelligent Systems and Informatics (SISY) | 1 Sep 2019 Cross Ref Finite-difference method for singular nonlinear systemsNumerical Algorithms, Vol. 79, No. 1 | 19 October 2017 Cross Ref Computing Multiple Zeros of Polynomial Systems: Case of Breadth One (Invited Talk)Computer Algebra in Scientific Computing | 30 August 2017 Cross Ref Verified error bounds for singular solutions of nonlinear systemsNumerical Algorithms, Vol. 70, No. 2 | 18 December 2014 Cross Ref Attraction of Newton method to critical Lagrange multipliers: fully quadratic caseMathematical Programming, Vol. 152, No. 1-2 | 10 April 2014 Cross Ref Verified Error Bounds for Isolated Singular Solutions of Polynomial SystemsNan Li and Lihong ZhiSIAM Journal on Numerical Analysis, Vol. 52, No. 4 | 10 July 2014AbstractPDF (300 KB)Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth oneTheoretical Computer Science, Vol. 479 | 1 Apr 2013 Cross Ref Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth OneNan Li and Lihong ZhiSIAM Journal on Numerical Analysis, Vol. 50, No. 1 | 28 February 2012AbstractPDF (265 KB)Practical Quasi-Newton algorithms for singular nonlinear systemsNumerical Algorithms, Vol. 55, No. 4 | 6 February 2010 Cross Ref Convergence analysis of a variant of the Newton method for solving nonlinear equationsComputers & Mathematics with Applications, Vol. 59, No. 6 | 1 Mar 2010 Cross Ref Computing the Least Fixed Point of Positive Polynomial SystemsJavier Esparza, Stefan Kiefer, and Michael LuttenbergerSIAM Journal on Computing, Vol. 39, No. 6 | 26 March 2010AbstractPDF (846 KB)An accelerated Newton method for equations with semismooth Jacobians and nonlinear complementarity problemsMathematical Programming, Vol. 117, No. 1-2 | 10 August 2007 Cross Ref Local dynamics of a family of quasilinear ODEs with folded singular equilibriaNonlinear Analysis: Theory, Methods & Applications, Vol. 68, No. 8 | 1 Apr 2008 Cross Ref On the Performance of Tensor Methods for Solving Ill-Conditioned ProblemsBrett W. 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