The solution of systems of linear(ized) equations lies at the heart of many problems in Scientific Computing. In particular for systems of large dimension, iterative methods are a primary approach. Stationary iterative methods are generally based on a matrix splitting, whereas for polynomial iterative methods such as Krylov subspace iteration, the splitting matrix is the preconditioner. The smoother in a multigrid method is generally a stationary or polynomial iteration. Here we consider real symmetric indefinite and complex Hermitian indefinite coefficient matrices and prove that no splitting matrix can lead to a contractive stationary iteration unless the inertia is exactly preserved. This has consequences for preconditioning for indefinite systems and smoothing for multigrid as we further describe.

翻译：线性(线性化)方程组的求解是科学计算中众多问题的核心。对于大规模方程组，迭代方法是一种主要途径。定常迭代方法通常基于矩阵分裂，而对于多项式迭代方法（如Krylov子空间迭代），分裂矩阵即预条件子。多重网格方法中的光滑子通常采用定常迭代或多项式迭代。本文针对实对称不定与复Hermitian不定系数矩阵，证明了除非惯性被精确保持，否则任何分裂矩阵都无法产生压缩的定常迭代。这一结论对不定系统的预条件处理及多重网格光滑子构造具有重要影响，我们将进一步阐述其具体内涵。