Assess students' ability to analyze and compare the convergence behavior of Newton's method and fixed-point iteration for solving nonlinear equations. Scope: local and global convergence conditions, order of convergence and error estimates, nonzero derivative/Jacobian assumptions, contraction-mapping criteria, basins of attraction, multiple roots, and damping/line search modifications. Tasks may include proving local quadratic vs linear convergence, deriving error constants, testing convergence for given functions/iterations, and explaining method selection and stabilization strategies.