Corrigendum 2 to: A geometric view of Krylov subspace methods on singular systems

Hayami, Ken; Sugihara, Kota

doi:10.1002/nla.2368

Cited by 2 publications

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“…If all computations are done in exact arithmetic, GMRES using pseudo-inverse determines a solution of min x∈R n b − Ax 2 when h k+1,k = 0. When h k+1,k = 0 holds, H k,k is singular (See [7], Theorem 1, point a, b; [8], Theorem 4). On the other hand, numerical experiments on some semi-definite inconsistent systems indicate that Ar 2 / Ab 2 becomes very small when the smallest singular value of H k+1,k is very small.…”

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confidence: 99%

GMRES using pseudoinverse for range symmetric singular systems

Sugihara,

Hayami,

Zeyu

2022

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Consider solving large sparse range symmetric singular linear systems Ax = b which arise, for instance, in the discretization of convection diffusion equations with periodic boundary conditions, and partial differential equations for electromagnetic fields using the edge-based finite element method.In theory, the Generalized Minimal Residual (GMRES) method converges to the least squares solution for inconsistent systems if the coefficient matrix A is range symmetric, i.e. R(A) = R(A T ), where R(A) is the range space of A.However, in practice, GMRES may not converge due to numerical instability. In order to improve the convergence, we propose using the pseudo-inverse for the solution of the severely ill-conditioned Hessenberg systems in GMRES. Numerical experiments on semi-definite inconsistent systems indicate that the method is efficient and robust. Finally, we further improve the convergence of the method, by reorthogonalizing the Modified Gram-Schmidt procedure.

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confidence: 99%