Fast Inclusion and Residual Iteration for Solutions of Matrix Equations

This paper is concerned with the problem of verifying an accuracy of numerical solutions of matrix equations. In this paper, a fast algorithm of obtaining very sharp verified error bound is proposed. The new verification method is based on combination of the rounding mode controlled double precision computation and the high precision computation. The method can also be used to improve numerical solutions by the residual iteration. Numerical results have also been presented for illustrating that a very sharp verified error bound of numerical solutions can be obtained with a computational cost comparable to that for calculating an approximate solution. Moreover, it is shown using an example that a few residual iterations give extremely improved solutions with little computational cost.