Solving Nonlinear Equations Using Newton's Method and Quasi-Newton Methods

Newton's method and quasi-Newton methods are widely used approaches for solving nonlinear equation systems. These methods have extensive applications across various engineering and scientific disciplines. Beyond nonlinear equations, they are also employed in optimization problems, parameter estimation, and solving differential equations. In our implementation, we verified the effectiveness of these algorithms through numerical experiments. The Newton method typically requires computing the Jacobian matrix and solving linear systems at each iteration, while quasi-Newton methods (like BFGS or DFP) approximate the Hessian to reduce computational cost. Our results demonstrate that both methods can efficiently locate solutions to nonlinear equations within reasonable timeframes, making them suitable for practical applications. Key implementation aspects include handling Jacobian matrix calculations, implementing backtracking line search for step size control, and setting appropriate convergence criteria based on residual norms or iteration limits.