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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 \(A=\begin{pmatrix} B&E\cr E^T&C\end{pmatrix}\) 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:

65F08 Preconditioners for iterative methods
65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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