Approximate inverse computation using Frobenius inner product

Abstract

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, minM∈𝒮∥AM−I∥ F and matrix M0∈𝒮 corresponding to this minimum (𝒮 being any vectorial subspace of ℳ︁n(ℝ)) 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.