A direct solver for the least‐squares problem arising from GMRES(k)

Abstract

This is a study about one of the core questions in the GMRES(k) method regarding the obtaining of vector yk for the least‐squares problem, argminy |Hky ‐ β(n)e1|2 (see Saad and Schultz1). We propose a simple but efficient approach to the resolution of this problem and a low cost computation of the residual and the residual norm, including both in a complete and detailed FGMRES(k) algorithm. The whole algorithm of minimization only involves two backward substitutions with triangular matrices and a dot product. The residual and the residual norm are computed, making use of results in the least‐squares problem.