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Constraint preconditioners for symmetric indefinite matrices. (English) Zbl 1195.65033

The authors consider the preconditioning of a matrix $A$ that has the form $A=\left(\begin{array}{cc}B& E\\ {E}^{T}& C\end{array}\right)$ with $B$ symmetric positive definite of size $p×p$ and $C$ symmetric of size $q×q$ and a nonsingular Schur complement $S=C-K$ with $K={E}^{T}{B}^{-1}E$. 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:
 65F08 Preconditioners for iterative methods 65F10 Iterative methods for linear systems 65F50 Sparse matrices (numerical linear algebra) 15B57 Hermitian, skew-Hermitian, and related matrices