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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=BEE T C 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:
65F08Preconditioners for iterative methods
65F10Iterative methods for linear systems
65F50Sparse matrices (numerical linear algebra)
15B57Hermitian, skew-Hermitian, and related matrices