Skip to main content
SpringerLink
Log in

A derivative free iterative method for solving least squares problems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A derivative free iterative method for approximating a solution of nonlinear least squares problems is studied first in Shakhno and Gnatyshyn (Appl Math Comput 161:253–264, 2005). The radius of convergence is determined as well as usable error estimates. We show that this method is faster than its Secant analogue examined in Shakhno and Gnatyshyn (Appl Math Comput 161:253–264, 2005). Numerical example is also provided in this paper.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Argyros, I.K.: On a two–point Newton–like method of convergent order two. Int. J. Comput. Math. 82, 219–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Argyros, I.K.: A Kantorovich–type analysis for a fast iterative method for solving nonlinear equations. J. Math. Anal. Appl. 332, 97–108 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Argyros, I.K.: Computational theory of iterative methods. In: Chui, C.K., Wuytack, L. (eds.) Series: Studies in Computational Mathematics, vol. 15. Elsevier Publ. Co., New York (2007)

    Google Scholar 

  5. Argyros, I.K.: Convergence domains for some iterative processes in Banach spaces using outer or generalized inverses. J. Comput. Anal. Appl. 1, 87–104 (1999)

    MathSciNet  Google Scholar 

  6. Argyros, I.K., Ren, H.: Local convergence of a Secant–type method for solving least square problems. Appl. Math. Comput. 217, 3816–3824 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Israel, A., Greville, N.E.: Generalized Inverses: Theory and Applications, Pure and Applied Mathematics, Wiley-interscience. John Wiley and Sons, London (1974)

    Google Scholar 

  8. Brown, K.M., Dennis, J.M., Jr.: Derivative free analogues of the Levenberg–Marquardt and Gauss algorithms for nonlinear least squares approximation. Numer. Math. 18, 289–297 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, X., Nashed, M.Z.: Convergence of Newton–like methods for singular operator equations using outer inverses. Numer. Math. 66, 235–257 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dennis, J.M., Jr., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice–Hall, New York (1983)

    MATH  Google Scholar 

  11. More, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mühlig, H.: Lösung praktischer Approximationsaufgaben durch parameteridentifikation. ZAMM 73, T837–839 (1993)

    Article  Google Scholar 

  13. Ortega, J.M., Rheinbolt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  14. Potra, F.A.: On an iterative algorithm of order 1.839... for solving nonlinear operator equations. Numer. Funct. Anal. Optim. 7(1), 75–106 (1984–1985)

    Article  MathSciNet  Google Scholar 

  15. Ren, H.M., Wu, Q.B.: Convergence ball of a modified secant method with convergence order 1.839.... Appl. Math. Comput. 188, 281–285 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schwetlick, H.: Numerische Lösung nichtlinearer Gleichungen. R. Oldenburg Verlag, Munchen Wien (1979)

  17. Shakhno, S.M., Gnatyshyn, O.P.: Iterative-difference methods for solving nonlinear leastsquares problem. In: Arkeryd, L., Bergh, J., Brenner, P., Petersson, R. (eds.) Progress in Industrial Mathematics at ECMI’98, pp. 287–294. Teubner, Stuttgart (1999)

  18. Shakhno, S.M.: About the difference method with quadratic convergence for solving nonlinear operator equations. PAMM 4(1), 650–651 (2004)

    Article  MathSciNet  Google Scholar 

  19. Shakhno, S.M., Gnatyshyn, O.P.: On an iterative algorithm of order 1.839... for solving the nonlinear least squares problems. Appl. Math. Comput. 161, 253–264 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ulm, S.: Iteration methods with divided differences of the second order (in Russian). Dokl. Akad. Nauk SSSR. 158, 55–58 (1964)

    MathSciNet  Google Scholar 

  21. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon, Oxford (1965)

    MATH  Google Scholar 

  22. Yamamoto, T.: Uniqueness of the solution in a Kantorovich-type theorem of HauBler for the Gauss-Newton method. Jpn. J. Appl. Math. 6, 77–81 (1989)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information and Electronics, Hangzhou Radio and TV University, Hangzhou, 310012, Zhejiang, People’s Republic of China

    Hongmin Ren

  2. Department of Mathematics Sciences, Cameron University, Lawton, OK, 73505, USA

    Ioannis K. Argyros

  3. Laboratoire de Mathématiques et Applications, Poitiers University, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962, Futuroscope Chasseneuil Cedex, France

    Saïd Hilout

Authors
  1. Hongmin Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis K. Argyros
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Saïd Hilout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis K. Argyros.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ren, H., Argyros, I.K. & Hilout, S. A derivative free iterative method for solving least squares problems. Numer Algor 58, 555–571 (2011). https://doi.org/10.1007/s11075-011-9470-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-011-9470-9

Keywords

AMS 2000 Subject Classification

Search

Navigation