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

İoannis K Argyros; Santhosh George

doi:10.33187/jmsm.432191

EN

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

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.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

İoannis K Argyros This is me
United States

Santhosh George ^*
India

Publication Date

September 30, 2018

Submission Date

June 8, 2018

Acceptance Date

September 13, 2018

Published in Issue

Year 2018 Volume: 1 Number: 2

DOI

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

IZ

https://izlik.org/JA45BN45JK

APA

Argyros, İ. K., & George, S. (2018). Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling, 1(2), 80-85. https://doi.org/10.33187/jmsm.432191

AMA

1.Argyros İK, George S. Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling. 2018;1(2):80-85. doi:10.33187/jmsm.432191

Chicago

Argyros, İoannis K, and Santhosh George. 2018. “Improved Semi-Local Convergence of the Gauss-Newton Method for Systems of Equations”. Journal of Mathematical Sciences and Modelling 1 (2): 80-85. https://doi.org/10.33187/jmsm.432191.

EndNote

Argyros İK, George S (September 1, 2018) Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling 1 2 80–85.

IEEE

[1]İ. K. Argyros and S. George, “Improved semi-local convergence of the Gauss-Newton method for systems of equations”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 2, pp. 80–85, Sept. 2018, doi: 10.33187/jmsm.432191.

ISNAD

Argyros, İoannis K - George, Santhosh. “Improved Semi-Local Convergence of the Gauss-Newton Method for Systems of Equations”. Journal of Mathematical Sciences and Modelling 1/2 (September 1, 2018): 80-85. https://doi.org/10.33187/jmsm.432191.

JAMA

1.Argyros İK, George S. Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling. 2018;1:80–85.

MLA

Argyros, İoannis K, and Santhosh George. “Improved Semi-Local Convergence of the Gauss-Newton Method for Systems of Equations”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 2, Sept. 2018, pp. 80-85, doi:10.33187/jmsm.432191.

Vancouver

1.İoannis K Argyros, Santhosh George. Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling. 2018 Sep. 1;1(2):80-5. doi:10.33187/jmsm.432191