Expanding the applicability of the Gauss-Newton method for a certain class of systems of equations

Ioannis K. Argyros; Santhosh George

doi:10.33993/jnaat451-1102

Authors

Ioannis K. Argyros Cameron University, USA
Santhosh George National Institute of Technology Karnataka, India

DOI:

https://doi.org/10.33993/jnaat451-1102

Keywords:

Gauss-Newton method, Newton method, semilocal convergence, least squares problem

Abstract views: 212

Abstract

We present a new semilocal convergence analysis of the Gauss-Newton method in order to solve a certain class of systems of equations under a majorant condition. Using a center majorant function as well as a majorant function and under the same computational cost as in earlier studies such as [11]-[13], we present a semilocal convergence analysis with the following advantages: weaker sufficient convergence conditions; tighter error estimates on the distances involved and an at least as precise information on the location of the solution. Special cases and applications complete this study.

Downloads

Download data is not yet available.

Author Biography

Ioannis K. Argyros, Cameron University, USA

Full tenured Professor of Mathematics.

References

I. Argyros, On the semilocal convergence of the Gauss-Newton method, Adv. Nonlinear Var. Inequal., 8 (2005) 2, pp. 93-99.

I. Argyros and S. Hilout, On the local convergence of the Gauss-Newton method, Punjab Univ. J. Math., 41 (2009), pp. 23-33.

I. Argyros and S. Hilout, On the Gauss-Newton method, J. Appl. Math. Comput., 35 (2011), pp. 537–550, http://dx.doi.org/10.1007/s12190-010-0377-8 DOI: https://doi.org/10.1007/s12190-010-0377-8

I. Argyros and S. Hilout, Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions, Numer. Algor., 58 (2011) 1, pp. 23-52, http://dx.doi.org/10.1007/s11075-011-9446-9 DOI: https://doi.org/10.1007/s11075-011-9446-9

I. Argyros and S. Hilout, Improved local convergence of Newton’s method under weak majorant condition, J. Comp. Appl. Math., 236 (2012) 7, pp. 1892-1902, http://dx.doi.org/10.1016/j.cam.2011.10.021 DOI: https://doi.org/10.1016/j.cam.2011.10.021

I. Argyros and S. Hilout, Numerical methods in nonlinear analysis, World Scientific Publ. Comp. New Jersey, USA, 2013. DOI: https://doi.org/10.1142/8475

A. Ben-Israel and T.N.E. Greville, Generalized inverses, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 15. Springer-Verlag, New York, 2nd ed., Theory and Applications, 2003.

E. Catinas, The inexact, inexact perturbed, and quasi-Newton methods are equivalent models, Math. Comput., 74 (2005) 249, pp. 291-301, http://dx.doi.org/10.1090/S0025-5718-04-01646-1 DOI: https://doi.org/10.1090/S0025-5718-04-01646-1

J.P. Dedieu and M.H. Kim, Newton’s method for analytic systems of equations with constant rank derivatives, J. Complexity, 18, (2002) 1, pp. 187-209, http://dx.doi.org/10.1006/jcom.2001.0612 DOI: https://doi.org/10.1006/jcom.2001.0612

O.P. Ferreira, M.L.N. Goncalves and P.R. Oliveira, Local convergence analysis of inexact Gauss-Newton like methods under majorant condition, J. Complexity, 27 (2011) 1, pp. 111-125, http://dx.doi.org/10.1016/j.jco.2010.09.001 DOI: https://doi.org/10.1016/j.jco.2010.09.001

O.P. Ferreira and B.F. Svaiter, Kantorovich’s majorants principle for Newton’s method, Comput. Optim. Appl., 42 (2009) 2, pp. 213-229, http://dx.doi.org/10.1007/s10589-007-9082-4 DOI: https://doi.org/10.1007/s10589-007-9082-4

W.M. Haussler, A Kantorovich-type convergence analysis for the Gauss-Newton Method, Numer. Math., 48 (1986) 1, pp. 119-125, http://dx.doi.org/10.1007/BF01389446 DOI: https://doi.org/10.1007/BF01389446

M.L.N. Goncalves and P.R. Oliveira, Convergence of the Gauss-Newton method for a special class of systems of equations under a majorant condition, Optimization, 64 (2015) 3, pp. 577-594, http://dx.doi.org/10.1080/02331934.2013.778854 DOI: https://doi.org/10.1080/02331934.2013.778854

N. Hu, W. Shen and C. Li, Kantorovich’s type theorems for systems of equations with constant rank derivatives , J. Comput. Appl. Math., 219 (2008) 1, pp. 110-122, http://dx.doi.org/10.1016/j.cam.2007.07.006 DOI: https://doi.org/10.1016/j.cam.2007.07.006

L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.

C. Li, N. Hu and J. Wang, Convergence behavior of Gauss-Newton’s method and extensions of the Smale point estimate theory. J. Complexity, 26 (2010) 3, pp. 268-295, http://dx.doi.org/10.1016/j.jco.2010.02.001 DOI: https://doi.org/10.1016/j.jco.2010.02.001

F.A. Potra and V. Ptak, Nondiscrete induction and iterative processes, Research notes in Mathematics, 103, Pitman (Advanced Publishing Program), Boston, MA, 1984.

S. Smale, Newton’s method estimates from data at one point. The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pp. 185-196, Springer, New York, 1986. DOI: https://doi.org/10.1007/978-1-4612-4984-9_13

X.H. Wang, Convergence of Newton’s method and uniqueness of the solution of equations in Banach spaces, IMA J. Numer. Anal., 20 (2000), pp. 123-134, http://dx.doi.org/10.1093/imanum/20.1.123 DOI: https://doi.org/10.1093/imanum/20.1.123