Solving Nonlinear Equations using Fixed-Point Iteration Methods

This document introduces multiple numerical methods for finding roots of nonlinear equation systems, including but not limited to: - Fixed-point iteration method (typically implemented with a convergence tolerance check and maximum iteration limit) - Newton's method (utilizes Jacobian matrix computation and linear system solving at each iteration) - Discrete Newton method (approximates derivatives using finite differences to avoid explicit Jacobian calculation) - Newton-Jacobi iteration method (combines Newton's method with Jacobi iterative scheme for large systems) - Newton-SOR iteration method (incorporates Successive Over-Relaxation for improved convergence) - Newton descent method (includes a damping factor to ensure global convergence properties) - Two-point secant method - first variant (uses previous two approximations to estimate derivatives) - Two-point secant method - second variant (alternative derivative approximation approach) - Quasi-Newton methods (including rank-1 update algorithms that approximate inverse Hessian matrices) - D-F-P algorithm (Davidon-Fletcher-Powell method for unconstrained optimization) - B-F-S algorithm (Broyden-Fletcher-Goldfarb-Shanno method with efficient Hessian updates) - Numerical continuation method (tracks solution paths through parameter space) - Euler method in parametric differentiation (solves parameterized systems using first-order integration) - Midpoint integration method in parametric differentiation (second-order accurate path following) - Steepest descent method (uses gradient information for direction selection) - Gauss-Newton method (optimized for nonlinear least squares problems) - Conjugate gradient method (effective for large-scale symmetric systems) - Damped least squares method (incorporates regularization for ill-conditioned problems) Each method possesses distinct advantages and applicable scenarios, requiring selection based on specific problem characteristics such as system size, smoothness, and available derivative information. These algorithms serve as essential tools for solving nonlinear equation roots or solution sets, contributing significantly to mathematical research and computational mathematics development. Implementation typically involves setting appropriate convergence thresholds, handling singularity cases, and optimizing computational efficiency.