Homotopy Curve Method for Solving Nonlinear Equation Systems

The homotopy curve method, also known as homotopy iteration method, is a numerical computation technique for solving nonlinear equation systems. Based on topological concepts, this method constructs a continuous curve connecting a simple equation with known analytical solutions to the target nonlinear equation system. Implementation typically involves parameterizing the homotopy function H(x,t) = (1-t)*G(x) + t*F(x), where t evolves from 0 to 1, G(x) represents the simple system with known solutions, and F(x) is the target nonlinear system. The algorithm tracks solution paths using predictor-corrector methods, often employing Euler-Newton continuation with adaptive step size control. Key computational advantages include overcoming limitations of traditional methods like Newton-Raphson iteration, particularly in handling singular Jacobians and poor initial guesses. The method's robustness makes it suitable for multivariate nonlinear equation systems, with applications in engineering and physics domains such as computational fluid dynamics (CFD simulations), power system analysis (load flow calculations), and chemical reaction kinetics (rate equation solutions). Code implementation typically requires ODE solvers for path tracking and sparse matrix handling for large-scale systems.