Local convergence of a two-step Gauss-Newton Werner-type method for solving least squares problems

Ioannis K. Argyros; Santhosh George

doi:10.33993/jnaat531-1165

Authors

Ioannis K. Argyros Department of Mathematical Sciences, Cameron University, USA https://orcid.org/0000-0002-9189-9298
Santhosh George Department of Mathematical and Computational Sciences, NIT Karnataka, India https://orcid.org/0000-0002-3530-5539

DOI:

https://doi.org/10.33993/jnaat531-1165

Keywords:

Gauss-Newton method, Werner's method, local convergence, least squares problem, average Lipschitz condition

Abstract views: 89

Abstract

The aim of this paper is to extend the applicability of a two-step Gauss-Newton-Werner-type method (TGNWTM) for solving nonlinear least squares problems. The radius of convergence, error bounds and the information on the location of the solution are improved under the same information as in earlier studies. Numerical examples further validate the theoretical results.

Downloads

Download data is not yet available.

References

S. Amat, M.A. Hernández, N. Romero, Semilocal convergence of a sixth order iterative method for quadratic equations, Appl. Numer. Math., 62 (2012), pp.~833–841. https://doi.org/10.1016/j.apnum.2012.03.001 DOI: https://doi.org/10.1016/j.apnum.2012.03.001

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

I.K. Argyros, S. Hilout, On the semilocal convergence of Werner's method for solving equations using recurrent functions, Punj. Univ. J. Math., 43 (2011), pp.~19–28. https://pu.edu.pk/images/journal/maths/PDF/vol43/paper3_Argyros_WMethod.pdf

I.K. Argyros, A. Magreñan, Iterative methods and their dynamics with applications, CRC Press, New York, 2017. DOI: https://doi.org/10.1201/9781315153469

J. Chen, W. Li, Convergence of Gauss-Newton method's and uniqueness of the solution, Appl. Math. Comput., 170 (2005), pp.~686--705. https://doi.org/10.1016/j.amc.2004.12.055 DOI: https://doi.org/10.1016/j.amc.2004.12.055

J.M. Dennis, R.B. Schnabel, Numerical methods for Unconstrained optimization and nonlinear equations, Prentice-Hall, N.Y, 1983.

W.M. Häubler, A Kantorovich-type convergence analysis for the Gauss-Newton method, Numer. Math., 48 (1986), pp.~119--125. https://doi.org/10.1007/BF01389446 DOI: https://doi.org/10.1007/BF01389446

R. Iakymchuk, S.M. Shakhno, On the convergence analysis of a two-step modification of the Gauss-Newton method, PAMM, 14 (2014), pp.~813--814. https://doi.org/10.1002/pamm.201410387 DOI: https://doi.org/10.1002/pamm.201410387

R.P. Iakymchuk, S.M. Shakhno, H.P. Yarmola, Convergence analysis of a two-step modification of the Gauss-Newton method and its applications, J. Numer. Appl. Math., 3(126) (2017), pp.~61--74. http://jnam.lnu.edu.ua/pdf/y2017_no3(126)_art05_iakymchuk_shakhno_yarmola.pdf

A.A. Magreñan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput., 248 (2014), pp.~29--38. https://doi.org/10.1016/j.amc.2014.09.061 DOI: https://doi.org/10.1016/j.amc.2014.09.061

G.N. Silva, Local convergence of Newton's method for solving generalized equations with monotone operator, Appl. Anal., 97 (2018), pp.~1094--1105. https://doi.org/10.1080/00036811.2017.1299860 DOI: https://doi.org/10.1080/00036811.2017.1299860

X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal., 20 (2000), pp.~123--134. https://doi.org/10.1007/s10114-002-0238-y DOI: https://doi.org/10.1093/imanum/20.1.123

W. Werner, Über ein Verfahren der Ordnung 1=√2 zur Nullstellenbestimmung, Numer. Math., 32 (1979), pp.~333--342. https://doi.org/10.1007/BF01397005 DOI: https://doi.org/10.1007/BF01397005