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On eigenvalue estimates for block incomplete factorization methods. (English) Zbl 0837.65022

Let \(A\) be a symmetric matrix which is expressed as the matrix sum \(A= D_A+ L_A+ L^T_A\), where \(D_A\) and \(L_A\) are, respectively, the block diagonal part and strictly lower block triangular part of \(A\). The authors consider the preconditioner \(C= (X+ L)X^{- 1}(X+ L^T)\), where \(X\) and \(L\) are the block diagonal and the block lower triangular matrices partitioned in blocks consistently with \(D_A\) and \(L_A\), respectively. They present an upper and lower bound of eigenvalues of the preconditioned matrix \(C^{- 1} A\) depending on \(X\), \(L\) and two parameters which are discussed. The authors, moreover, generalize the well-known inequality \(\varrho(A)\leq \text{tr}(A)\) to block form. The theory is applied to a symmetric successive overrelaxation preconditioner which has the form given above.
Reviewer: J.Zítko (Praha)

MSC:

65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
15A42 Inequalities involving eigenvalues and eigenvectors
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