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Zbl 1195.65033
Bai, Zhong-Zhi; Ng, Michael K.; Wang, Zeng-Qi
Constraint preconditioners for symmetric indefinite matrices.
(English)
[J] SIAM J. Matrix Anal. Appl. 31, No. 2, 410-433 (2009). ISSN 0895-4798; ISSN 1095-7162/e

The authors consider the preconditioning of a matrix $A$ that has the form $A=\pmatrix B&E\cr E^T&C\endpmatrix$ with $B$ symmetric positive definite of size $p\times p$ and $C$ symmetric of size $q\times q$ and a nonsingular Schur complement $S=C-K$ with $K=E^TB^{-1}E$. In the literature one usually deals with a saddle point context where $S$ is negative definite. The preconditioner $P$ considered is exactly like $A$, but $B$ is replaced by a symmetric positive definite approximant $G$, hence the name constrained preconditioner. First, a positive and a negative interval are found that contain the eigenvalues of the symmetric matrix $A$ which depends on the locations of the intervals containing the eigenvalues of $B$, $S$, and $K$. Next the spectrum of $P^{-1}A$ is investigated, mainly dealing with the multiplicity of the eigenvalue 1, but also with the structure of the eigenvectors. These properties depend on the (dimension of the) null space of $B-G$. Numerical examples illustrate the method.
MSC 2000:
*65F08
65F10 Iterative methods for linear systems
65F50 Sparse matrices
15B57

Keywords: symmetric indefinite systems; constraint preconditioners; Schur complement; eigenvalues; symmetric matrix; numerical examples

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