Published Paper Details:

Nonlinear equations are integral to a wide range of real-world problems in fields such as physics, engineering, and economics. Solving these equations presents challenges due to the potential for multiple solutions and sensitivity to initial conditions. This paper explores advanced numerical methods for solving nonlinear equations, focusing on Newton's method, the secant method, and homotopy continuation. These methods are evaluated based on their convergence rates, accuracy, and applicability to different types of nonlinear problems. Our findings indicate that while Newton's method offers fast convergence near the solution, its performance can degrade with poor initial guesses. The secant method, which does not require derivative information, provides a robust alternative with slower convergence. Homotopy continuation, though computationally intensive, excels in finding multiple solutions to nonlinear systems. The paper demonstrates that these advanced methods significantly improve the efficiency and reliability of solving nonlinear equations, making them invaluable tools in both theoretical and applied mathematics. Their applications extend to fields such as optimization, cryptography, and engineering, where solving nonlinear systems is crucial.