Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXTENDING THE APPLICABILITY OF INEXACT GAUSS-NEWTON METHOD FOR SOLVING UNDERDETERMINED NONLINEAR LEAST SQUARES PROBLEMS

  • Received : 2018.02.15
  • Accepted : 2018.11.29
  • Published : 2019.03.01
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Abstract

The aim of this paper is to extend the applicability of Gauss-Newton method for solving underdetermined nonlinear least squares problems in cases not covered before. The novelty of the paper is the introduction of a restricted convergence domain. We find a more precise location where the Gauss-Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local as well the semilocal convergence of Gauss-Newton method. The new developmentes are obtained under the same computational cost as in earlier studies, since the new Lipschitz constants are special cases of the constants used before. Numerical examples further justify the theoretical results.

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References

  1. I. K. Argyros, Inexact Newton methods and nondifferentiable operator equations on Banach spaces with a convergence structure, Approx. Theory Appl. (N.S.) 13 (1997), no. 3, 91-103.
  2. I. K. Argyros, On a new Newton-Mysovskii-type theorem with applications to inexact Newton-like methods and their discretizations, IMA J. Numer. Anal. 18 (1998), no. 1, 37-56. https://doi.org/10.1093/imanum/18.1.37
  3. I. K. Argyros and S. Hilout, Weaker conditions for the convergence of Newton's method, J. Complexity 28 (2012), no. 3, 364-387. https://doi.org/10.1016/j.jco.2011.12.003
  4. I. K. Argyros and A. Magrenan, Iterative Methods and Their Dynamics with Applications, CRC Press, Boca Raton, FL, 2017.
  5. I. K. Argyros and F. Szidarovszky, The Theory and Applications of Iteration Methods, Systems Engineering Series, CRC Press, Boca Raton, FL, 1993.
  6. J.-F. Bao, C. Li, W.-P. Shen, J.-C. Yao, and S.-M. Guu, Approximate Gauss-Newton methods for solving underdetermined nonlinear least squares problems, Appl. Numer. Math. 111 (2017), 92-110. https://doi.org/10.1016/j.apnum.2016.08.007
  7. A. Ben-Israel and T. N. E. Greville, Generalized Inverses, second edition, CMS Books in Mathematics/Ouvrages de Mathematiques de la SMC, 15, Springer-Verlag, New York, 2003.
  8. A. Bjorck, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
  9. P. N. Brown, A local convergence theory for combined inexact-Newton/finite-difference projection methods, SIAM J. Numer. Anal. 24 (1987), no. 2, 407-434. https://doi.org/10.1137/0724031
  10. R. S. Dembo, S. C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), no. 2, 400-408. https://doi.org/10.1137/0719025
  11. J. E. Dennis, Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, corrected reprint of the 1983 original, Classics in Applied Mathematics, 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
  12. J. A. Ezquerro Fernandez and M. Hernandez Veron, Newton's method: an updated approach of Kantorovich's theory, Frontiers in Mathematics, Birkhauser/Springer, Cham, 2017.
  13. X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2007), no. 2, 231-242.
  14. J. M. Gutierrez and M. A. Hernandez, Newton's method under weak Kantorovich conditions, IMA J. Numer. Anal. 20 (2000), no. 4, 521-532. https://doi.org/10.1093/imanum/20.4.521
  15. W. M. Haussler, A Kantorovich-type convergence analysis for the Gauss-Newton-method, Numer. Math. 48 (1986), no. 1, 119-125. https://doi.org/10.1007/BF01389446
  16. N. Josephy, Newton's Method for Generalized Equations and the PIES Energy Model, University of Wisconsin-Madiso, 1979.
  17. L. V. Kantorovich and G. P. Akilov, Functional analysis in normed spaces, Translated from the Russian by D. E. Brown. Edited by A. P. Robertson. International Series of Monographs in Pure and Applied Mathematics, Vol. 46, The Macmillan Co., New York, 1964.
  18. A. Magrenan, Different anomalies in a Jarratt family of iterative root-finding methods, Appl. Math. Comput. 233 (2014), 29-38. https://doi.org/10.1016/j.amc.2014.01.037
  19. A. Magrenan, A new tool to study real dynamics: the convergence plane, Appl. Math. Comput. 248 (2014), 215-224. https://doi.org/10.1016/j.amc.2014.09.061
  20. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, reprint of the 1970 original, Classics in Applied Mathematics, 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
  21. G. W. Stewart and J. G. Sun, Matrix Perturbation Theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990.
  22. T. J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (1984), no. 3, 583-590. https://doi.org/10.1137/0721040