A DIRECT SOLVER FOR THE LEAST-SQUARES PROBLEM ARISING FROM GMRES(K)

This is a study about one of the core questions in the GMRES(k) method regarding the obtaining of vector Y(k) for the least-squares problem, argmin(y) parallel-to H(k)Y - beta(n)e1 parallel-to 2 (see Saad and S chultz1). We propose a simple but efficient approach to the resolution of this problem and a low cost computation of the residual and the re sidual norm, including both in a complete and detailed FGMRES(k) algor ithm. The whole algorithm of minimization only involves two backward s ubstitutions with triangular matrices and a dot product. The residual and the residual norm are computed, making use of results in the least -squares problem.