Over the past several years a number of hybrid algorithms have been proposed for solving large sparse systems of linear algebraic equations. In this paper we consider the hybrid GMRES algorithm by Nachtigal, Reichel and Trefethen (1992) and show that in the floating-point arithmetic there exist some cases in which the properties of this algorithm are lost, e.g., the result is false, or the coefficients of the GMRES residual polynomial are non-significant and lead to serious round-off errors. The subject of this paper is to show how by using the CADNA library, it is possible during the run of the hybrid GMRES code to detect the numerical instabilities, to stop correctly the process, and to evaluate the accuracy of the results provided by the computer. Numerical examples are used to show the good numerical properties.

Keywords