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Given a symmetric positive‑definite matrix (\mathbfA), the Conjugate Gradient (CG) method converges in at most (N) iterations, with practical convergence governed by (\sqrt\kappa(\mathbfA)). Preconditioners (\mathbfM^-1) aim to cluster the spectrum of (\mathbfM^-1\mathbfA) around 1, reducing the effective condition number (\kappa_\texteff = \kappa(\mathbfM^-1\mathbfA)). Popular choices include Incomplete Cholesky (IC), Algebraic Multigrid (AMG), and Sparse Approximate Inverses (SAI) [5].