Assess learners' ability to evaluate convergence behavior, numerical stability, and preconditioning strategies for large sparse linear systems solved with Krylov methods (Conjugate Gradient, GMRES). Topics include spectral properties and condition-number effects on convergence, eigenvalue clustering and convergence bounds, stability issues (loss of orthogonality, round-off, breakdowns), stopping criteria and GMRES restart strategies, and design/selection of preconditioners (Jacobi, SSOR, ILU, algebraic multigrid) with trade-offs in fill-in, robustness, and parallel scalability. Items may require diagnosing convergence failures from spectra or residual histories, comparing preconditioners quantitatively, and recommending solver/preconditioner choices given matrix characteristics and resource constraints.