Finding a Root of Nonlinear Equations Using Fixed-Point Iteration Method

This text presents various numerical methods for finding roots or solutions of nonlinear equation systems. These methods include fixed-point iteration, Newton's method, discrete Newton's method, Newton-Jacobi iteration, Newton-SOR iteration, Newton's descent method, two-point secant method, quasi-Newton method, symmetric rank-1 algorithm, D-F-P algorithm, B-F-S algorithm, numerical continuation method, steepest descent method, Gauss-Newton method, conjugate gradient method, and damped least squares method. These algorithms are applicable in different scenarios, particularly when dealing with nonlinear equation systems. Implementation typically involves iterative procedures where each method has distinct computational characteristics: Newton-based methods use Jacobian matrices or approximations, while optimization-inspired methods like conjugate gradients handle minimization formulations. The fixed-point iteration provides simple iterative updates, whereas quasi-Newton methods build approximate Hessians for faster convergence. Through these methods, we can computationally determine roots or solutions of nonlinear equations, enabling problem-solving and model optimization in scientific computing applications. Code implementations generally require function evaluations, derivative calculations (analytical or numerical), and convergence criteria checks within iterative loops.