Abstract
We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)
Argyros, I.K.: A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space. J. Math. Anal. Appl. 298, 374–397 (2004)
Argyros, I.K.: Convergence and applications of Newton-type iterations. Springer, New York (2008)
Argyros, I.K., Cho, Y.J., Hilout, S.: Numerical Methods for Equations and Its Applications. Science Publishers, New Hampshire (2012)
Argyros, I.K., Hilout, S.: Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions. Numer. Algorithms 58, 23–52 (2011)
Argyros, I.K., Hilout, S.: Improved local convergence of Newton’s method under weak majorant condition. J. Comput. Appl. Math. 236, 1892–1902 (2012)
Argyros, I.K., Hilout, S.: Weaker conditions for the convergence of Newton’s method. J. Complex. 28, 364–387 (2012)
Chen, J.: The convergence analysis of inexact Gauss-Newton methods for nonlinear problems. Comput. Optim. Appl. 40, 97–118 (2008)
Chen, J., Li, W.: Convergence of Gauss-Newton’s method and uniqueness of the solution. Appl. Math. Comput. 170, 686–705 (2005)
Chen, J., Li, W.: Local convergence results of Gauss-Newton’s like method in weak conditions. J. Math. Anal. Appl. 324, 1381–1394 (2006)
Dennis, J.E.Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Corrected reprint of the 1983 original), Classics in Appl. Math., 16, SIAM, Philadelphia (1996)
Ferreira, O.P.: Local convergence of Newton’s method under majorant condition. J. Comput. Appl. Math. 235, 1515–1522 (2011)
Ferreira, O.P., Gonçalves, M.L.N.: Local convergence analysis of inexact-Newton like methods under majorant condition. Comput. Optim. Appl. 48, 1–21 (2011)
Ferreira, O.P., Gonçalves, M.L.N., Oliveira, P.R.: Local convergence analysis of Gauss-Newton’s method under majorant condition. J. Complex. 27(1), 111–125 (2011)
Gutiérrez, J.M., Hernández, M.A.: Newton’s method under weak Kantorovich conditions. IMA J. Numer. Anal. 20, 521–532 (2000)
Häubler, W.M.: A Kantorovich-type convergence analysis for the Gauss-Newton-method. Numer. Math. 48, 119–125 (1986)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms (two volumes). I. Fundamentals. II. Advanced theory and bundle methods, vol. 305–306. Springer, Berlin (1993)
Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press, Oxford (1982)
Li, C., Hu, N., Wang, J.: Convergence bahavior of Gauss-Newton’s method and extensions to the Smale point estimate theory. J. Complex. 26, 268–295 (2010)
Li, C., Zhang, W.-H., Jin, X.-Q.: Convergence and uniqueness properties of Gauss-Newton’s method. Comput. Math. Appl. 47, 1057–1067 (2004)
Potra, F.A.: Sharp error bounds for a class of Newton-like methods. Lib. Math. 5, 71–84 (1985)
Proinov, P.D.: General local convergence theory for a class of iterative processes and its applications to Newton’s method. J. Complex. 25, 38–62 (2009)
Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R., Gross, K., Martin, C. (eds.) The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)
Traub, J.F.: Iterative Methods for the Solution of Equations, Englewood Cliffs. Prentice Hall, New Jersey (1964)
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)
Acknowledgments
The research of the second author has been supported by the project MTM2011-28636- C02-01 of the Spanish Ministry of Science and Innovation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Argyros, I.K., Magreñán, Á.A. Improved local convergence analysis of the Gauss–Newton method under a majorant condition. Comput Optim Appl 60, 423–439 (2015). https://doi.org/10.1007/s10589-014-9704-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-014-9704-6