2024/01/22 by Du, Kui, Fan, Jia-Jun, Wang, Fang
#15A06 #15A09 #65F10 #65F25 #65F50 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2401.11788
It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum \mathbf A-residual) for range-symmetric linear systems \mathbf A\mathbf x=\mathbf b. At step k RSMAR minimizes ‖\mathbf A\mathbf rk‖ in the kth Krylov subspace generated with \\mathbf A, \mathbf r0\ rather than ‖\mathbf rk‖, where \mathbf rk is the kth residual vector and ‖⋅‖ denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.