2025/04/14 by Sugihara, Kouta, Ken Hayami, Hayami, Ken
Computer Science · Mathematics · Physics and Astronomy · #65F08 #65F10 #65F20 #65F50 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2504.09891
openalex publication_date 2025/04/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
GMRES is known to determine a least squares solution of A x = b where A ∈ Rn × n without breakdown for arbitrary b ∈ Rn , and initial iterate x0 ∈ Rn if and only if A is range-symmetric, i.e. R(AT) = R(A) , where A may be singular and b may not be in the range space R(A) of A . In this paper, we propose applying the Range Restricted GMRES (RRGMRES) to A C AT z = b , where C ∈ Rn × n is symmetric positive definite. This determines a least squares solution x = C AT z of A x = b without breakdown for arbitrary (singular) matrix A ∈ Rn × n and b, x0 ∈ Rn , and is much more stable and accurate compared to GMRES, RRGMRES and MINRES-QLP applied to A x = b for inconsistent problems when b ∉ R(A) . In particular, we propose applying the NR-SSOR as the inner iteration right preconditioner, which also works efficiently for least squares problems minx ∈ Rn ‖ b - A x‖2 for A ∈ Rm × n and arbitrary b ∈ Rm . Numerical experiments demonstrate the validity of the proposed method.