Test knowledge of convergence theory and practical analysis of Krylov subspace methods (Conjugate Gradient, GMRES) for large sparse linear systems, including spectral properties, condition number effects, convergence bounds, and the role and design of preconditioners; require interpretation of residual and error behavior, assessment of preconditioner effectiveness, and diagnosing stagnation or breakdown in iterative solves. Includes application to symmetric positive-definite and non-symmetric systems, polynomial and spectral preconditioning concepts, and simple numerical diagnostics and stopping criteria.