On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method
Provides theoretical justification for using low-accuracy inner solves in SIRA and JD methods, benefiting researchers and practitioners in numerical linear algebra and eigenvalue computation.
The paper develops a general convergence theory for the Shift-Invert Residual Arnoldi (SIRA) method, proving that inexact SIRA with low-accuracy inner solves converges similarly to exact SIRA. The theory also applies to the Jacobi-Davidson method, and numerical experiments show both methods outperform inexact SIA.
Using a new analysis approach, we establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a given target $σ$ and the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimics the exact SIRA well, that is, the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with {\em low} or {\em modest} accuracy during outer iterations. Based on the theory, we design practical stopping criteria for inner solves. Our analysis is on one step expansion of subspace and the approach applies to the Jacobi--Davidson (JD) method with the fixed target $σ$ as well, and a similar general convergence theory is obtained for it. Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.