Improved componentwise verified error bounds for least squares problems and underdetermined linear systems

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results.