Computing Accurate Eigenvalues and Singular Values Using a Mixed-Precision (One-Sided) Jacobi Algorithm

We studied mixed-precision preconditioned Jacobi algorithms for computing eigenvalues and singular values. The main idea was to construct a preconditioner in low precision and apply it in higher precision to improve the accuracy and convergence of the Jacobi algorithm. We presented a relative forward error analysis showing that the proposed approaches can achieve significantly smaller relative forward errors than the Jacobi algorithm itself. We also discussed two practical constructions of low-precision preconditions and presented numerical experiments confirming the theoretical results. The dominant computational cost arose from a small number of high-precision matrix–matrix multiplications, suggesting strong potential for efficiency with improved hardware or software support.