标题： 关于不定矩阵分裂和预处理的注记 显示英文标题

标题： A note on indefinite matrix splitting and preconditioning

摘要： 求解线性(化)方程组是科学计算中许多问题的核心。 特别是对于高维系统，迭代方法是一种主要的方法。 平稳迭代方法通常基于矩阵分裂，而多项式迭代方法如Krylov子空间迭代，则分裂矩阵是预条件矩阵。 多网格方法中的平滑器通常是一个平稳或多项式迭代。 这里我们考虑实对称不定和复共轭对称不定系数矩阵，并证明除非惯性被精确保持，否则没有分裂矩阵能导致收缩的平稳迭代。 这对我们进一步描述不定系统的预条件和多网格中的平滑有影响。 显示英文摘要

摘要： 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.