Solving Nonlinear Equations Using Simulated Annealing Method

In this document, we employ the simulated annealing method to solve nonlinear equations. This approach not only minimizes computational overhead but also achieves rapid convergence. By transforming the problem into an optimization process, simulated annealing effectively locates solutions for nonlinear equations. The method mimics the metallurgical annealing process, utilizing random search and gradual temperature reduction to iteratively refine solution quality. Its effectiveness stems from the algorithm's ability to perform global exploration within the search space while avoiding local optima. Key implementation aspects include defining an appropriate energy function (representing equation residuals), configuring cooling schedules, and managing acceptance probabilities for suboptimal solutions. The algorithm typically involves initialization of parameters, iterative state transitions, and dynamic temperature updates using logarithmic or geometric cooling strategies. As a versatile numerical computation technique, simulated annealing finds broad applications in optimization problems and machine learning algorithms, particularly where traditional gradient-based methods face limitations with non-convex or discontinuous functions.