A generalization of the optimal diagonal approximate inverse preconditioner

The classical optimal (in the Frobenius sense) diagonal preconditioner for large sparse linear systems Ax = b is generalized and improved. The new proposed approximate inverse preconditioner N is based on the minimization of the Frobenius norm of the residual matrix AM − I, where M runs over a certain linear subspace of n × n real matrices, defined by a prescribed sparsity pattern. The number of nonzero entries of the n×n preconditioning matrix N is less than or equal to 2n, and n of them are selected as the optimal positions in each of the n columns of matrix N . All theoretical results are justified in detail. In particular, the comparison between the proposed preconditioner N 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.