Assess students' ability to analyze and compare convergence and numerical stability of Conjugate Gradient (CG) and GMRES for large sparse linear systems. Scope: algorithmic assumptions (symmetry, definiteness), spectral influence on convergence, finite-precision effects (loss of orthogonality, breakdowns), role and design of preconditioners, storage and cost trade-offs, stopping criteria, and interpretation of small numerical experiment results to recommend method and preconditioner choices.