Constraint Preconditioners for Symmetric Indefinite Matrices
Constraint Preconditioners for Symmetric Indefinite Matrices
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DOI：
10.1137/080720243
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发表时间：
2009-03
期刊：
影响因子：
--
通讯作者：
Z. Bai;M. Ng;Zeng-Qi Wang
中科院分区：
文献类型：
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作者：
Z. Bai;M. Ng;Zeng-Qi Wang
研究了块2 × 2非奇异对称不定矩阵的特征值界，其中$（1，1）$块是对称正定的，关于$（2，2）$块的Schur补是对称不定的.通过简单地用对称正定逼近代替$（1，1）$块，构造了该矩阵的约束预条件子，并讨论了预条件矩阵的谱性质。数值结果表明，对于适当选取的$（1，1）$块矩阵，该约束预条件子用于加速求解块2 × 2对称正不定线性方程组的GMRES方法时，在迭代步数和计算时间上优于块对角和块三对角预条件子.新结果将对称半正定$（2，2）$块的块2 × 2矩阵的已有结果推广到一般对称$（2，2）$块的情形。
We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose $(1,1)$ block is symmetric positive definite and Schur complement with respect to its $(2,2)$ block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the $(1,1)$ block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen $(1,1)$ block-matrix, this constraint preconditioner outperforms the block-diagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite $(2,2)$ blocks to those of general symmetric $(2,2)$ blocks.