Point estimation of root finding methods

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The challenge of solving nonlinear equations and systems is significant in applied mathematics and various engineering fields, physics, computer science, astronomy, and finance. The extensive bibliography and contributions from prominent mathematicians highlight the contemporary interest in this area. While digital computers have enabled effective implementation of numerous numerical methods, practical applications face challenges such as computational efficiency, the development of iterative methods with rapid convergence in the presence of multiple solutions, control of rounding errors, and establishing error bounds for approximate solutions. Additionally, it is crucial to determine computationally verifiable initial conditions that ensure reliable convergence. This study focuses on identifying and analyzing initial conditions that guarantee the convergence of iterative methods for equations of the form f(z) = 0. The traditional approach relies on asymptotic convergence analysis, which typically requires strong assumptions about differentiability and derivative bounds across a broad domain.