Quadratic Convergence Method for Complex and Real Root-Finding Without Higher-Order Derivatives

When it comes to dealing with nonlinear equations, numerical methods play a crucial role. Still, many of these methods come with limitations such as guaranteeing actual convergence, high computational costs, or strong dependence on derivatives. Traditional techniques, in particular, tend to struggle when the first derivative is close to zero or when they require second or third derivatives, which adds layers of complexity. The study presents a new iterative approach to overcome these challenges. It achieves a reliable second-order convergence and can handle both real and complex rootseven in situations where the first derivative approaches zero. The method starts with an initial guess, w0 ∈ C, and improves it step-by-step, gradually zeroing in on a solution. Its flexibility allows it to be applied to a broad range of equations. One of the key advantages is that it doesn’t depend on higher-order derivatives, which helps in maintaining a balance between computational efficiency and accuracy.. Interestingly, the method also manages to find complex roots even when the initial guess is entirely real, something many other methods struggle with. To evaluate how well the method works, experiments were conducted using Python version 3.10.12. The results shown in tables and graphs illustrate how the method converges over a set number of steps. Overall, this technique offers a reliable and practical alternative to conventional numerical methods, particularly for tackling nonlinear problems involving complex solutions