A generalization of the optimal diagonal approximate inverse preconditioner

Abstract

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…