A generalization of the optimal diagonal approximate inverse preconditioner

The classical optimal (in the Frobenius sense) diagonal preconditioner for large sparse linear systems Ax=bAx=b is generalized and improved. The new proposed approximate inverse preconditioner NN is based on the minimization of the Frobenius norm of the residual matrix AM−IAM−I, where MM runs over a certain linear subspace of n×nn×n real matrices, defined by a prescribed sparsity pattern. The number of nonzero entries of the n×nn×n preconditioning matrix NN is less than or equal to 2n2n, and nn of them are selected as the optimal positions in each of the nn columns of matrix NN. All theoretical results are justified in detail. In particular, the comparison between the proposed preconditioner NN and the optimal diagonal one is theoretically analyzed. Finally, numerical experiments reported confirm the theory and illustrate that our generalization of the optimal diagonal preconditioner improves (in general) its efficiency, when they do not coincide.