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Approximate inverse computation using Frobenius inner product. (English) Zbl 1071.65534
Summary: Parallel preconditioners are presented for the solution of general linear systems of equations. The computation of these preconditioners is achieved by orthogonal projections related to the Frobenius inner product. So, ${min}_{M\in 𝒮}{\parallel AM-I\parallel }_{F}$ and matrix ${M}_{0}\in 𝒮$ corresponding to this minimum ($𝒮$ being any vectorial subspace of ${ℳ}_{n}\left(ℝ\right)$) are explicitly computed using accumulative formulae in order to reduce computational cost when subspace $𝒮$ is extended to another one containing it. Every step, the computation is carried out taking advantage of the previous one, what considerably reduces the amount of work. These general results are illustrated with the subspace of matrices $M$ such that $AM$ is symmetric. The main application is developed for the subspace of matrices with a given sparsity pattern which may be constructed iteratively by augmenting the set of non-zero entries in each column. Finally, the effectiveness of the sparse preconditioners is illustrated with some numerical experiments.
##### MSC:
 65F35 Matrix norms, conditioning, scaling (numerical linear algebra) 65F10 Iterative methods for linear systems 65Y05 Parallel computation (numerical methods) 65Y20 Complexity and performance of numerical algorithms