. This paper presents a novel fourth-order two-step iterative method for solving nonlinear equations f (x) = 0, derived by optimizing a key parameter in a combination of established third-order methods. The proposed method enhances efficiency and accuracy, outperforming existing iterative techniques. Extensive numerical experiments on diverse polynomial and transcendental functions demonstrate superior convergence speed and precision. Additionally, the method shows significant promise for applications in polynomiography and broader mathematical analysis, as evidenced by its robust performance in visualization and root-finding tasks.
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