Local convergence analysis of the Gauss-Newton-Kurchatov method

We present a local convergence analysis of the Gauss-Newton-Kurchatov method  for solving nonlinear least squares problems with the decomposition of the operator.  The method uses the sum of the derivative of the differentiable part of the operator and the divided difference of the nondifferentiable part instead of computing  the full Jacobian.  A theorem, which establishes the conditions of convergence, radius, and the convergence order of the proposed method, is proved [1].  However, the radius of convergence is small in general limiting the choice of initial points.  Using tighter estimates on the distances, under weaker hypotheses [1], we provide an analysis of the Gauss--Newton--Kurchatov method with the following advantages over the corresponding results [1]: extended convergence region; finer error distances, and an at least as precise information on the location of the solution. The numerical examples illustrate the theoretical results.

Gauss-Newton-Kurchatov method
local convergence
Frechet-derivative
Lipschitz condition
Ccenter-Lipschitz condition
convergence domain
  1. Shakhno S. M.  Gauss–Newton–Kurchatov method for solving nonlinear least squares problems.  Mat. metody phys.-mech. polia. 60, 52–62 (2017), (in Ukrainian).
  2. Argyros I. K., Hilout S.  On an improved convergence analysis of Newton's method.  Applied Mathematics and Computation. 225, 372–386 (2013).
  3. Argyros I. K.  Convergence and applications of Newton–type iterations.  Springer–Verlag, New York (2008).
  4. Dennis J. E. (Jr.) Schnabel R. B.  Numerical methods for unconstrained optimization and nonlinear equations.  SIAM, Philadelphia (1996).
  5. Ortega J. M., Rheinboldt W. C.  Iterative solution of nonlinear equations in several variables.  Academic Press, New York etc. (1970).
  6. Shakhno S.  Some numerical methods for nonlinear least squares problems.  In: Alefeld G., Rohn J., Rump S., Yamamoto T. (eds.)  Symbolic-algebraic Methods and Verification Methods, Springer, Vienna. Pp.235–243 (2001).
  7. Ren H., Argyros I. K.  Local convergence of a secant type method for solving least squares problems.  Appl. Math. Comp. 217 (8), 3816–3824 (2010).
  8. Ren H., Argyros I. K., Hilout S.  A derivative free iterative method for solving least squares problems.  Numer. Algor. 58, 555–571 (2011).
  9. Shakhno S. M., Gnatyshyn O. P.  Iterative-difference methods for solving nonlinear least-squares problem.  In: Arkeryd L., Bergh J., Brenner P. et al. (eds.) Progress in Industrial Mathematics at ECMI 98, B. G. Teubner, Stuttgart. Pp. 287–294 (1999).
  10. Shakhno S. M., Gnatyshyn O. P.  On an iterative algorithm of order $1.839\ldots$ for solving the nonlinear least squares problems.  Appl. Math. Comp. 161 (1), 253–264 (2005).
  11. Cătinaş E.  On some iterative methods for solving nonlinear equations.  Rev. Anal. Numér. Théor. Approx. 23 (1), 47–53 (1994).
  12. Hernández-Verón M. A., Rubio M. J.  On the local convergence of Newton–Kurchatov–type method for non-differentiable operators.  Appl. Math. Comp. 304, 1–9 (2017).
  13. Iakymchuk R., Shakhno S., Yarmola H.  Combined Newton-Kurchatov method for solving nonlinear operator equations.  Proc. Appl. Math. Mech. 16, 719–720 (2016).
  14. Shakhno S. M., Mel'nyk I. V., Yarmola H. P.  Analysis of the convergence of a combined method for the solution of nonlinear equations.  J. Math. Sci. 201, 32–43 (2014).
  15. Shakhno S. M., Yarmola H. P.  Two-point method for solving nonlinear equation with nondifferentiable operator.  Matematychni Studii. 36, 213–220 (2011), (in Ukrainian).
  16. Shakhno S. M.  Combined Newton–Kurchatov method under the generalized Lipschitz conditions for the derivatives and divided differences.  J. Comp. Appl. Math. 2 (119), 78–89 (2015).
  17. Magrenan A. A., Argyros I. K.  A contemporary study of iterative methods. Convergence, Dynamics and applications.  Academic Press,  London (2018).
  18. Deuflhard P.  Newton methods for nonlinear problems. Affine invariance and adaptive algorithms.  Springer–Verlag, Berlin (2004).
  19. Shakhno S. M.  On the difference method with quadratic convergence for solving nonlinear operator equations.  Matematychni Studii. 26, 105–110 (2006), (in Ukrainian).
  20. Shakhno S.  Secant method under the generalized Lipschitz conditions for the first-order divided differences.  Mathematical bulletin of the Shevchenko scientific society. 4, 293–303 (2007), (in Ukrainian).
  21. Ulm S.  On generalized divided differences. I, II.  Izv. Akad. Nauk Est. SSR. Fiz. mat. 16, 13–26, 146–156 (1967), (in Russian).