Structured preconditioners for nonsingular matrices of block two-by-two structures
Structured preconditioners for nonsingular matrices of block two-by-two structures
复制标题
DOI：
10.1090/s0025-5718-05-01801-6
复制
发表时间：
2005-11
期刊：
影响因子：
--
通讯作者：
Z. Bai
中科院分区：
文献类型：
--
作者：
Z. Bai
对于大型稀疏块2 × 2真实的非奇异矩阵，通过矩阵变换和矩阵逼近，建立了一个实用有效的结构预条件子的一般框架.对于修正的块Jacobi型、修正的块Gauss-Seidel型和修正的块非对称（对称）Gauss-Seidel型预条件子，我们精确地描述了它们的具体表达式，并分析了预条件矩阵的特征值分布和正定性.此外，我们表明，当这些结构化的预处理器预处理的Krylov子空间方法，如GMRES和重新启动GMRES，快速和有效的迭代求解器可以获得大型稀疏系统的线性方程组块2 × 2的系数矩阵。特别是，这些结构化的预处理器可以导致有效的和高质量的预处理矩阵的一些典型的矩阵从现实世界中的应用。
For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.