A note to the gmres method

Abstract

GMRES method [4] is an effective conjugate gradient-like iterative method for solving linear systems of equations. In each loop of the GMRES we have to solve a special least squares problem (1). A classical way of solving such least squares problems is to factor H into QR using Givens plane rotations. In [4] it is shown that the factorization can be updated progressively as each column of H appears (i.e., at every setup of the Arnoldi process), therefore it enables us to obtain the residual norm of the approximate solution without computing Xj, thus allowing us to decide when to stop the process without wasting needless operations. In this note we give a direct method for the least squares problems, which uses less computational work compared with Givens rotation method and has the nice property mentioned above.

Keywords:

• Least squares problems
• linear systems
• GMRES method
• Givens plane rotation and direct methods

AMS(MOS) Subject Classification:

• 65F10
• 65F20

C.R. Categories:

• G.1.3