Assess students' ability to analyze convergence and stability of Newton's method and fixed-point iterations for solving nonlinear equations (scalar and systems). Tasks include: determine local convergence and order (using derivatives/Jacobian), verify contraction conditions and apply fixed-point theorems, analyze basins of attraction and sensitivity to initial guesses, identify causes of divergence or slow convergence, derive error estimates and stability criteria, and propose/justify modifications (damping, relaxation, modified Newton) to improve robustness; include short proofs, analytic checks, and one illustrative numerical example or counterexample.