Improved semi-local convergence of the Gauss-Newton method for systems of equations

Year 2018, , 80 - 85, 30.09.2018

İoannis K Argyros Santhosh George

https://doi.org/10.33187/jmsm.432191

Abstract

Our new technique of restricted convergence domains is employed to provide a finer convergence analysis of the Gauss-Newton method in order to solve a certain class of systems of equations under a majorant condition. The advantages are obtained under the same computational cost as in earlier studies such as [5, 14]. Special cases and a numerical example are also given in this study.

Keywords

Gauss- Newton method, Newton's method, Semi-local convergence, Least squares problem

References

• [1] Argyros, I.: On the semilocal convergence of the Gauss-Newton method. Adv. Nonlinear Var.Inequal. 8(2), 93–99, 2005.
• [2] Argyros, I., Hilout, S.: On the local convergence of the Gauss-Newton method. Punjab Univ. J.Math. 41, 23–33, 2009.
• [3] Argyros, I., Hilout, S.: On the Gauss-Newton method. J. Appl. Math. Comput. 1–14, 2010.
• [4] Argyros, I. K, Hilout, S.: Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions. Numer. Algorithms 58(1), 23–52, 2011.
• [5] Argyros, I. K, S. George, Expanding the applicability of the Gauss-Newton method for a certain class of systems of equations, J. Numer.Anal. Approx. Theory, vol., 45(1), 3–13, (2016).
• [6] Argyros, I. K., Hilout, S.: Improved local convergence of Newton’s method under weak majorant condition, Journal of Computational and Applied Mathematics, 236(7), 1892–1902, 2012.
• [7] Argyros, I., Hilout, S.: Numerical methods in nonlinear analysis, World Scientific Publ. Comp. New Jersey, USA, 2013.
• [8] Ben-Israel, A., Greville, T.N.E.: Generalized inverses. CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 15. Springer-Verlag, New York, second edition, Theory and Applications, 2003.
• [9] Catinas, E.: The inexact, inexact perturbed, and quasi-Newton methods are equivalent models, Math. Comput. 74, 249, (2005), 291-301.
• [10] Dedieu, J.P., Kim, M.H.: Newton’s method for analytic systems of equations with constant rank derivatives. J. Complexity, 18(1): 187-209, 2002.
• [11] Ferreira, O.P., Gonc¸alves, M.L.N, Oliveira, P.R.:, Local convergence analysis of inexact Gauss–Newton like methods under majorant condition, J. Complexity, 27(1), 111-125, 2011.
• [12] Ferreira, O.P., Svaiter, B.F.: Kantorovich’s majorants principle for Newton’s method. Comput.Optim. Appl. 42(2), 213–229, 2009.
• [13] Haussler, W.M.: A Kantorovich-type convergence analysis for the Gauss-Newton-method. Numer. Math. 48(1), 119–125, 1986.
• [14] Gonc¸alves, M.L.N, Oliveira, P.R.:, Convergence of the Gauss- Newton method for a special class of systems of equations under a majorant condition, Optimization, 64, 3(2015), 577–594.
• [15] Hu, N., Shen, W. and Li, C.: Kantorovich’s type theorems for systems of equations with constant rank derivatives, J. Comput. Appl.Math., 219(1): 110-122, 2008.
• [16] Kantorovich, L.V., Akilov, G.P., Functional Analysis, Pergamon Press, Oxford, 1982.
• [17] Li, C., Hu, N., Wang, J.: Convergence behavior of Gauss-Newton’s method and extensions of the Smale point estimate theory. J. Complex. 26(3), 268–295, 2010.
• [18] Potra, F.A., Ptak, V.: Nondiscrete induction and iterative processes. Research notes in Mathematics, 103, Pitman(A´ovanced Publishing Program), Boston, MA, 1984.
• [19] Smale, S., Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), 185-196, Springer, New York, 1986.
• [20] Wang, X.H., Convergence of Newton’s method and uniqueness of the solution of equations in Banach spaces, IMA J. Numer. Anal., 20 , 123–134, 2000.