Convergence analysis of the Gauss-Newton-
Potra method for nonlinear least squares problems

S. M. Shakhno; H. P. Yarmola; Yu. V. Shunkin

doi:10.15330/ms.50.2.211-221

In this paper we study an iterative differential-difference method for solving nonlinear least squares problems with nondifferentiable residual function. We have proved theorems which establish the conditions of convergence, radius and the convergence order under Lipschitz and $\omega$-conditions for the first-order derivatives of the differentiable part and for the first and second orders divided differences of the nondifferentiable part of the nonlinear function. The carried numerical experiments demonstrate the efficiency of the proposed method.

1. I.K. Argyros, A.A. Magrenan, A Contemporary Study of Iterative Methods, Elsevier (Academic Press), New York, NY, USA, 2018.

2. I.K. Argyros, A.A. Magrenan, Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, 2011).

3. I.K. Argyros, S. Hilout, Newton-Kantorovich approximations under weak continuity conditions, J. Appl. Math. Comput., 37 (2011), 361-375.

4. I.K. Argyros, S. Hilout, On an improved convergence analysis of Newtons method, Appl. Math. Comput., 225 (2013), 372-386.

5. I.K. Argyros, H. Ren, A derivative free iterative method for solving least squares problems, Numer. Algorithms, 4(58) (2011), 555-571.

6. I.K. Argyros, S. Shakhno, Yu. Shunkin, Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems, Mathematics, 7 (2019), №1, Article id: 99 ; doi: 10.3390/math7010099.

7. E. Catinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Num?r. Th?or. Approx., 1(23) (1994), 47-53.

8. M.A. Hernandez, M.J.Rubio, A uniparametric family of iterative processes for solving nondifferentiable operators, J. Math. Anal. Appl., 275 (2002), 821-834.

9. M.A. Hernandez-Veron, M.J. Rubio, On the local convergence of a Newton.Kurchatov-type method for non-differentiable operators, Appl. Math. Comput., 304 (2017), 1-9.

10. J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996.

11. J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970.

12. M.A. Ren, I.K. Argyros, Local convergence of a secant type method for solving least squares problems, AMC (Appl. Math. Comp.), 217 (2010), 3816-3824.

13. S.M. Shakhno, A.-V.I. Babjak, H.P. Yarmola, Combined Newton-Potra method for solving nonlinear operator equations, J. Appl. Comput. Math. (Kyiv), 3(120) (2015), 170-178. (in Ukrainian)

14. S.M. Shakhno, O.P. Gnatyshyn, Iterative-difference methods for solving nonlinear least-squares problem, Progress in Industrial Mathematics at ECMI 98, Verlag B.G.Teubner GMBH, Stuttgart, (1999), 287–294.

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

16. S. Shakhno, Yu. Shunkin, One combined method for solving nonlinear least squares problems, Visn. Lviv Univ. Ser. Appl. Math. Informatics, 25 (2017), 38–48. (in Ukrainian).

17. S. Shakhno, Yu. Shunkin, H.P. Yarmola, Gauss-Newton-Potra Method for Nonlinear Least Squares Problems with Decomposition of Operator, XXXII Int. Conf. PDMU–2018 (Proceedings), (2018), 153– 159.

18. S.M. Shakhno, H.P. Yarmola, On the two-step method for solving nonlinear equations with nondifferentiable operator, Proc. Appl. Math. Mech., 1 (2012), 617–618.

19. S.M. Shakhno, H.P. Yarmola, Two-point method for solving nonlinear equations with nondifferentiable operator, Mat. Stud., 36 (2011), 213–220. (in Ukrainian)

20. S. Ulm, On generalized divided differences, Proc. Acad. Sci. Estonian SSR. Physics. Mathematic, 16 (1967), 13–26. (in Russian)

	Convergence analysis of the Gauss-Newton- Potra method for nonlinear least squares problems
Author	S. M. Shakhno¹, H. P. Yarmola², Yu. V. Shunkin² stepan.shakhno@lnu.edu.ua¹, halyna.yarmola@lnu.edu.ua², yuriy.shunkin@lnu.edu.ua³ Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	In this paper we study an iterative differential-difference method for solving nonlinear least squares problems with nondifferentiable residual function. We have proved theorems which establish the conditions of convergence, radius and the convergence order under Lipschitz and $\omega$-conditions for the first-order derivatives of the differentiable part and for the first and second orders divided differences of the nondifferentiable part of the nonlinear function. The carried numerical experiments demonstrate the efficiency of the proposed method.
Keywords	least squares problem; decomposition of operator; Gauss-Newton-Potra method; omega-conditions; divided differences
DOI	doi:10.15330/ms.50.2.211-221
Reference	1. I.K. Argyros, A.A. Magrenan, A Contemporary Study of Iterative Methods, Elsevier (Academic Press), New York, NY, USA, 2018. 2. I.K. Argyros, A.A. Magrenan, Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, 2011). 3. I.K. Argyros, S. Hilout, Newton-Kantorovich approximations under weak continuity conditions, J. Appl. Math. Comput., 37 (2011), 361-375. 4. I.K. Argyros, S. Hilout, On an improved convergence analysis of Newtons method, Appl. Math. Comput., 225 (2013), 372-386. 5. I.K. Argyros, H. Ren, A derivative free iterative method for solving least squares problems, Numer. Algorithms, 4(58) (2011), 555-571. 6. I.K. Argyros, S. Shakhno, Yu. Shunkin, Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems, Mathematics, 7 (2019), №1, Article id: 99 ; doi: 10.3390/math7010099. 7. E. Catinas, On some iterative methods for solving nonlinear equations, Rev. Anal. Num?r. Th?or. Approx., 1(23) (1994), 47-53. 8. M.A. Hernandez, M.J.Rubio, A uniparametric family of iterative processes for solving nondifferentiable operators, J. Math. Anal. Appl., 275 (2002), 821-834. 9. M.A. Hernandez-Veron, M.J. Rubio, On the local convergence of a Newton.Kurchatov-type method for non-differentiable operators, Appl. Math. Comput., 304 (2017), 1-9. 10. J.E. Dennis, R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996. 11. J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. 12. M.A. Ren, I.K. Argyros, Local convergence of a secant type method for solving least squares problems, AMC (Appl. Math. Comp.), 217 (2010), 3816-3824. 13. S.M. Shakhno, A.-V.I. Babjak, H.P. Yarmola, Combined Newton-Potra method for solving nonlinear operator equations, J. Appl. Comput. Math. (Kyiv), 3(120) (2015), 170-178. (in Ukrainian) 14. S.M. Shakhno, O.P. Gnatyshyn, Iterative-difference methods for solving nonlinear least-squares problem, Progress in Industrial Mathematics at ECMI 98, Verlag B.G.Teubner GMBH, Stuttgart, (1999), 287–294. 15. S.M. Shakhno, O.P. Gnatyshyn, On an iterative algorithm of order 1.839... for solving the nonlinear least squares problems, AMC (Appl. Math. Comp.), 161 (2005), 253–264. 16. S. Shakhno, Yu. Shunkin, One combined method for solving nonlinear least squares problems, Visn. Lviv Univ. Ser. Appl. Math. Informatics, 25 (2017), 38–48. (in Ukrainian). 17. S. Shakhno, Yu. Shunkin, H.P. Yarmola, Gauss-Newton-Potra Method for Nonlinear Least Squares Problems with Decomposition of Operator, XXXII Int. Conf. PDMU–2018 (Proceedings), (2018), 153– 159. 18. S.M. Shakhno, H.P. Yarmola, On the two-step method for solving nonlinear equations with nondifferentiable operator, Proc. Appl. Math. Mech., 1 (2012), 617–618. 19. S.M. Shakhno, H.P. Yarmola, Two-point method for solving nonlinear equations with nondifferentiable operator, Mat. Stud., 36 (2011), 213–220. (in Ukrainian) 20. S. Ulm, On generalized divided differences, Proc. Acad. Sci. Estonian SSR. Physics. Mathematic, 16 (1967), 13–26. (in Russian)
Pages	211-221
Volume	50
Issue	2
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Convergence analysis of the Gauss-Newton- Potra method for nonlinear least squares problems