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

Year 2018, , 80 - 85, 30.09.2018
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.

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.
Year 2018, , 80 - 85, 30.09.2018
https://doi.org/10.33187/jmsm.432191

Abstract

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.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İoannis K Argyros This is me

Santhosh George

Publication Date September 30, 2018
Submission Date June 8, 2018
Acceptance Date September 13, 2018
Published in Issue Year 2018

Cite

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 Argyros İK, George S. Improved semi-local convergence of the Gauss-Newton method for systems of equations. Journal of Mathematical Sciences and Modelling. September 2018;1(2):80-85. doi:10.33187/jmsm.432191
Chicago 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 1, no. 2 (September 2018): 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 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, 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 2018), 80-85. https://doi.org/10.33187/jmsm.432191.
JAMA 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, 2018, pp. 80-85, doi:10.33187/jmsm.432191.
Vancouver 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):80-5.

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