Test students' ability to analyze local and global convergence, order of convergence, stability under perturbations, and basins of attraction for iterative solvers of nonlinear equations (Newton's method, fixed-point iteration, secant, damped and quasi-Newton variants). Assess use of convergence theorems (Banach fixed-point, Kantorovich), derivation of error estimates, Jacobian/derivative conditions in one and multiple dimensions, construction and interpretation of basins of attraction (including fractal boundaries), and comparative evaluation of algorithmic choices (damping, step control, initial-guess selection) through theoretical proofs, counterexamples, and targeted numerical experiments. Scope: rigorous analysis, practical diagnostics, and method selection criteria for difficult nonlinear problems.