|1Chuiko, SM |
1Donbass State Pedagogical University, Slovyansk
|Dopov. Nac. akad. nauk Ukr. 2020, 3:3-9|
Constructive conditions for the solvability and an iterative scheme of finding the solutions of systems of nonli near real equations in the case of a Jacobian with constant rank are obtained. The results are a generalization of the Newton-Kantorovich method for systems of nonlinear real equations, the number of components of which does not coincide with the number of the unknowns.
|Keywords: half-inverse matrix, matrix pseudoinverse by Moore—Penrose, modification of Newton—Kantorovich method, nonlinear real equations|
1. Kantorovich, L. V. & Akilov, G. P. (1977). Functional analysis. Moscow: Nauka (in Russian).
2. Boichuk, A. A. & Samoilenko, A. M. (2016). Generalized inverse operators and Fredholm boundary-value problems. 2th ed. Berlin, Boston: De Gruyter. Doi: https://doi.org/10.1515/9783110378443
3. Chuiko, S. M. (2018). A generalization of the Newton—Kantorovich theorem in a Banakh space. Dopov. Nac. akad. nauk Ukr., No. 6, pp. 22-31 (in Ukrainian). Doi: https://doi.org/10.15407/dopovidi2018.06.022
4. Dennis, J. E. & Schnabel, R. B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. Colorado: Society for Industrial and Applied Mathematics. Doi: https://doi.org/10.1137/1.9781611971200
5. Polyak, B. T. (2006). Newton’s method and its role in optimization and computational mathematics. Trudy ISA RAN, 28, pp. 48-66 (in Russian).
6. Chuiko, S. M. & Boichuk, I. A. (2009). Autonomous Noetherian boundary-value problem in the critical case. Nonlinear Oscillations, 12, No. 3, pp. 417-428. Doi: https://doi.org/10.1007/s11072-010-0085-1
7. Chuiko, S. M., Boichuk, I. A. & Pirus, O. E. (2013). On the approximate solution of an autonomous boundary-value problem by the Newton—Kantorovich method. J. Math. Sci., 189, No. 5, pp. 867-881. Doi: https://doi.org/10.1007/s10958-013-1225-9
8. Chuiko, S. M. & Pirus O. E. (2013). On the approximate solution of autonomous boundary-value problems by the Newton method. J. Math. Sci., 191, No. 3, pp. 449-463. Doi: https://doi.org/10.1007/s10958-013-1329-2
9. Ben-Israel, A. (1966). A Newton—Raphson method for the solution of systems of equations. J. Math. Anal. Appl., 15, pp. 243-252. Doi: https://doi.org/10.1016/0022-247X(66)90115-6
10. Chuiko, S. M. (2017). To the generalization of the Newton—Kantorovich theorem. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics And Mechanics. 85, No. 1, pp. 62-68.
11. Arnold, V. I., Gusein-Zade, S. M. & Varchenko, A. N. (2012). Singularities of differentiable maps. Boston: Birkhauser. Vols. 1, 2. Doi: https://doi.org/10.1007/978-0-8176-8343-6
12. Chuiko, S. M. (2018). On a reduction of the order in a differential-algebraic system. J. Math. Sci., No. 1, pp. 2-14. Doi: https://doi.org/10.1007/s10958-018-4054-z
13. Ben-Israel, A. & Greville, Th.N.E. (2003). Generalized inverses : theory and applications. 2nd ed. New York: Springer.
14. Chuiko, S. M., Chuiko, O. S. & Chechetenko, V. O. (2018). On of solving nonlinear Noether integral-differential boundary value problems by the of Newton—Kantorovich method. Nauk. Visnyk Uzhgorod Univ. Ser. Matematyka i Informatyka, No. 1, pp. 147-158.