Mj. Galan et al., A DIRECT SOLVER FOR THE LEAST-SQUARES PROBLEM ARISING FROM GMRES(K), Communications in numerical methods in engineering, 10(9), 1994, pp. 743-749
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.