Forward Error-Oriented Iterative Refinement for Eigenvectors of a Real Symmetric Matrix

Abstract: In this paper, we discuss numerical methods for the eigenvalue decomposition of real symmetric matrices. While many existing methods can compute approximate eigenpairs with sufficiently small backward errors, the magnitude of the resulting forward errors is often unknown. Consequently, when high-precision numerical solutions are required, the computational cost tends to increase significantly because backward errors must be reduced to an excessive degree. To address this issue, we propose an efficient approximation algorithm that aims to achieve a prescribed forward error, together with a high-accuracy numerical algorithm based on the Ozaki scheme -- an emulation technique for matrix multiplication -- adapted to this problem. Since the proposed method is not primarily focused on reducing backward errors, the computational cost can be significantly reduced. Finally, we present numerical experiments to evaluate the efficiency of the proposed method.