Algorithms of predictor-corrector type with convergence and stability analysis for solving nonlinear systems

Many researchers have proposed iterative algorithms for nonlinear equations and systems of nonlinear equations; similarly, in this paper, we developed two two-step algorithms of the predictor-corrector type. A combination of Taylor's series and the composition approach was used. One of the algorithms had an eighth order of convergence and a high-efficiency index of approximately 1.5157, which was higher than that of some existing algorithms, while the other possessed fourth-order convergence. The convergence analysis was carried out in both senses, that is, local and semi-local convergence. Various complex polynomials of different degrees were considered for visual analysis via the basins of attraction. We analyzed and compared the proposed algorithms with other existing algorithms having the same features. The visual results showed that the modified algorithms had a higher convergence rate compared to existing algorithms. Real-life systems related to chemistry, astronomy, and neurology were used in the numerical simulations. The numerical simulations of the test problems revealed that the proposed algorithms surpassed similar existing algorithms established in the literature.