### SECANT TYPE METHOD WITH APPROXIMATION OF THE INVERSE OPERATOR FOR THE NONLINEAR LEAST SQUARE PROBLEM

Stepan Shakhno, Mykhailo Havdiak, Roman Iakymchuk, Halyna Yarmola

#### Анотація

In this article, we propose a difference method with successive approximation of the inverse operator for finding an approximate solution of the nonlinear least squares problem. Classical methods are effective for such problems, but still there are types of problems for which they cannot be applied. Besides, these methods require the calculation of the inverse matrix or solving a system of linear equations at each iteration, which complicates the task. Hence, the considered method consists of two parts: the first part is for finding an approximation to the solution, while the second part is for approximation of the inverse operator instead of finding the inverse matrix. This part uses the first-order divided difference of the function instead of the Jacobian matrix. The analysis of local convergence of the proposed method is carried out under the classical Lipschitz conditions. This method was also applied for solving test problems to show the effectiveness of the method and to demonstrate its properties. These test functions also contain non-differentiable parts. For comparison, the number of iterations for the Secant method and the method with the approximation of the inverse operator are shown. Additionally, methods are compared for different initial approximations.
Furthermore, the proposed method can be used for regression analysis problems and in the study of some physical processes if there are difficulties with calculating the derivatives of a nonlinear function and with finding the inverse operator of the divided difference.

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#### Посилання

1. Argyros I.K. Convergence and applications of Newton’s-type iterations / I.K. Argyros. – New York: –Springer-Verlag, 2008.

2. Dennis J.E. Numerical Methods for Unconstrained Optimization and Nonlinear Equations / J.E. Dennis, R.B. Schnabel. – Philadelphia: SIAM, 1996.

3. Iakymchuk R. Methods with Successive and Parallel Approximations of Inverse Operator for the Nonlinear Least Squares Problem / R. Iakymchuk, S. Shakhno // PAMM-Proc. Appl. Math. Mech. – 2015. – Vol.15. – P.569-570.

4. More J.J. Testing Unconstrained Optimization Software / J.J. More, B.S. Garbow, K.E. Hillstrom // ACM Transactionson Mathematical Software. – 1981. – Vol.7, No1. – P.17-41.

5. Ren H. Local convergence of a secant type method for solving least squares problems / H. Ren, I.K. Argyros // Applied Mathematics and Computation. – 2010. – Vol.217. – P.3816-3824.

6. Rooze A.F. An iterative method for solving nonlinear equations using parallel inverse operator approximation / A.F.Rooze // Izv. Acad. Nauk Est. SSR. Fiz. Mat. – 1982. – V.31, №1. – P.32-37. (In Russian)

7. Shakhno S.M. Iterative methods for solving nonlinear least squares problems / S.M. Shakhno // Computational and Applied Mathematics. – 2013. – Vol. 1(111). – P.154-169. (In Ukrainian)

8. Shakhno S.M. Method with successive approximation of the inverse operator for nonlinear squares problem / S. Shakhno, M. Havdiak, H. Yarmola // APAMCS. – 2021. – C.181-184. (In Ukrainian)

9. Shakhno S.M. On a two-step iterative process under generalized Lipschitz conditions for first-order divided differences / S.M. Shakhno // Journal of Mathematical Sciences. – 2010. – Vol.168. – P.576 – 584.

10. Shakhno S.M. On an iterative algorithm of order 1.839... for solving the nonlinear least squares problems / S.M. Shakhno, O.P. Gnatyshyn // Applied Mathematics and Computation. – 2005. – Vol.161. – P.253-264.

11. Ulm S.U. On iterative methods with successive approximation of the inverse operator / S.U. Ulm // Izv. Acad. Nauk Est. SSR. Fiz. Mat. – 1967. – Vol.6, №4. – P.403-411. (In Russian)

DOI: http://dx.doi.org/10.30970/vam.2022.30.11568

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