Modified error bounds for approximate solutions of dense linear systems

We derive verified error bounds for approximate solutions of dense linear systems. There are verification methods using an approximate inverse of a coefficient matrix as a preconditioner, where the preconditioned coefficient matrix is likely to be anH-matrix (also known as a generalized diagonally dominant matrix). We focus on two inclusion methods of matrix multiplication for the preconditioning and propose verified error bounds adapted to the inclusion methods. These proposed error bounds are tighter than conventional ones, especially in critically ill-conditioned cases. Numerical results are presented showing the effectiveness of the proposed error bounds.