Differential-difference method with approximation of the inverse operator: Fìz.-mat. model. ìnf. tehnol. 2021, 33:186-190 - Shakhno, Stepan; Yarmola, Halyna (PHYSICO-MATHEMATICAL MODELLING AND INFORMATIONAL TECHNOLOGIES, 2021)
The problem of finding an approximate solution of a nonlinear equation with operator decomposition is considered. For equations of this type, a nonlinear operator can be represented as the sum of two operators – differentiable and nondifferentiable. For numerical solving such an equation, a differential-difference method, which contains the sum of the derivative of the differentiable part and the divided difference of the nondifferentiable part of the nonlinear operator, is proposed. Also, the proposed iterative process does not require finding the inverse operator. Instead of inverting the operator, its one-step approximation is used. The analysis of the local convergence of the method under the Lipschitz condition for the first-order divided differences and the bounded second derivative is carried out and the order of convergence is established. References Argyros, I. K. (2008). Convergence and applications of Newton’s-type iterations. New York: Springer-Verlag. Hernandez, M. A., Rubio, M. J. (2002). The Secant method for nondiﬀerentiable operators. Appl. Math. Lett., 15, 395-399. Cătinaş, E. (1994). On some iterative methods for solving nonlinear equations. Rev. Anal. Numer. Theorie Approximation, 23(1), 47-53. Shakhno, S. M., Yarmola, H. P. (2011). Two-point method for solving nonlinear equation with nondiﬀerentiable operator. Matematychni Studii, 36(2), 213-220. (in Ukrainian). DOI https://doi.org/10.15330/ms.48.1.97-107 Ulm, S.Yu. (1967). On iterative methods with successive approximation of the inverse operator. Izv. Acad. Nauk Est. SSR. Physics. Mathematics, 16(4), 403-411. (in Russian). Rooze, A. F. (1982). An iterative method for solving nonlinear equations using parallel inverse operator approximation. Izv. Acad. Nauk Est. SSR. Physics. Mathematic., 31(1), 32-37. (in Russian). Dennis, J. E., Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. Philadelphia: SIAM. Argyros, I. K., Regmi, S. (2019). Majorizing sequences for single step iterative processes and restricted convergence regions. PanAmerican Mathematical Journal, 29, 93-102. Shakhno, S. M., Grab, S. I., Yarmola, H. P. (2009). Twoparametric secant type methods for solving nonlinear equations. Visnyk of the Lviv University. Series Applied Mathematics and Computer Science 15. (in Ukrainian).