Adaptive width parabolic method

This paper introduces the Adaptive Width Parabolic Method (AWPM), a novel iterative algorithm designed for the numerical computation of roots in transcendental equations. The method approximates the function at each iteration point using a self-regulating parabolic curve, where the wideness of the parabola is dynamically adjusted based on the function's proximity to the x-axis and its local slope. A key feature of the AWPM is the integration of a shield parameter (𝕊(x_k)), which prevents the method from overshooting the solution when the derivative is excessively high, ensuring exceptional global stability. While the convergence speed may decrease in specific functional domains compared to traditional first-order methods, the AWPM demonstrates the ability to converge even when the initial starting point is significantly distant from the root. Computational results, implemented in Python, validate the method's robustness, particularly in cases where standard techniques like Newton-Raphson typically diverge. This paper also introduces analytical demonstrations regarding the convergence and rate of the method, alongside numerical examples compared with other methods, as well as an analysis of the conditions under which the method fails to converge (see the final sections). Every code will be uploaded on github. This method is conceived as a theoretical and numerical exploration and integration with the other methods and not as a substitution.