On Newton’s Method for Convex Functions and Operators and its Relationship with Contraction Principle

Octav Olteanu

Authors

Octav Olteanu Department of Mathematics-Informatics, University Politehnica of Bucharest, Romania

Keywords:

successive approximations, contraction principle, symmetric operators, convex operators, global Newton method

Abstract

In this review paper, generally known results on a version of global Newton’s method for convex increasing or decreasing functions and operators, as well as afferent examples and applications, are recalled. Connection with the contraction principle is discussed in detail and applied to approximate , Where is a positive invertible symmetric operator acting on a finite-dimensional Hilbert space, and is a real number. Two numerical examples for symmetric matrices with real coefficients are given. Some other nonlinear matrix or scalar equations are solved approximately.

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References

A. Aggarwal and S. Pant, Beyond Newton: A new root-finding fixed-point iteration for nonlinear equations, arXiv:1803.10156v2 [math.NA] 11 July 2018. arxiv.org/pdf/1803.10156.pdf

I.K. Argyros, A Semilocal convergence for directional Newton methods, Math. Comput., 80 (2011), 327-343. www.ams.org/journals/mcom/2011-80-273/S0025-5718-2010. https://doi.org/10.1090/S0025-5718-2010-02398-1

I.K. Argyros and S. George, Weak semilocal convergence conditions for a two-step Newton method in Banach space, Fundamental Journal of Mathematics and Applications, 1, 2(2018), 137-144. dergipark.org.tr/en/pub/fujma

V. Balan, A. Olteanu and O. Olteanu, On Newton’s me thod for convex operators with some applications. Rev. Roumaine Math. Pures Appl., 51, 3(2006), 277-290. imar.ro/journals/Revue_Mathematique/home_page.html

R. Cristescu, Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, and Abacus Press, Tunbridge Wells, Kent, 1976. b-ok.cc/book/3373479/d24f60

C.P. Niculescu, and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, 2nd Ed., CMS Books in Mathematics Vol. 23, Springer-Verlag, New York, 2018. www.springer.com/gp/book/9780387243009

O. Olteanu and Gh. Simion, A new geometric aspect of the implicit function principle and Newton’s method for operators, Mathematical Reports 5 (55), 1 (2003), 61-84.

www.imar.ro/journals/Mathematical_Reports/home_page.html

B. Saheya, Duo-qing Chen, Yun-kang Sui and Cai-ying Wu, A new Newton-like method for solving nonlinear equations, SpringerPlus 5:1269 (2016), 1-13. link.springer.com/journal/40064/volumes-and-issues https://doi.org/10.1186/s40064-016-2909-7