Hybrid two-point methods using only one inverse for solving nonlinear equations

• Date: 2026
• Main Authors: Argyros, I. K., Shakhno, S., Havdiak, M., Argyros, M. I.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2026
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/9002
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.6, 517.98 The task of solving of nonlinear equations often involves the use of iterative methods that require multiple inversions of linear operators, which can be computationally costly, especially for large matrices or complex operators. We propose hybrid two-point methods that overcome the indicated limitation by requiring only a single inversion of a fixed linear operator throughout the entire process. This approach preserves the rapid convergence and avoids the repeated inversions of $Q'(y_n)$ in each step. We provide both semilocal and local convergence analyses, utilizing a majorant function to control the Fréchet derivative of the operator. Numerical experiments confirm that the performance of our methods is comparable with the classical approaches but is characterized by much lower computational costs. These findings highlight the potentials of this technique as a practical alternative to Newton's method and other iterative methods requiring operator inversions.