Theoretical advances in two-step iterative schemes for nonlinear problems
DOI:
https://doi.org/10.65112/tcmis.10011Keywords:
Local convergence, Taylor series, computational time, nonlinear problemAbstract
This study develops a two-step iterative method for solving nonlinear equations, focusing on its convergence properties. Using Taylor series expansion, the method assumes the nonlinear function is at least four times continuously differentiable with bounded derivatives, though this can be restrictive for unbounded higher-order derivatives. To overcome this, a convergence radius is introduced, allowing initial approximations without exact root knowledge. This expands the convergence region without extra smoothness assumptions. Ball convergence analysis establishes error bounds and uniqueness, while local convergence analysis ensures stability and robustness near the root. By combining these perspectives, the framework enhances the theoretical foundation and extends applicability to complex nonlinear problems.
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