Accelerating preconditioned Jacobi methods via perturbation-inspired pivoting

Perturbation theory for symmetric matrices shows that eigenvalues with small spectral gaps are more sensitive to off-diagonal perturbation, implying that different entries affect the eigenvalues unevenly. Building on this insight, we incorporate spectral gap information into the Jacobi method for symmetric eigenvalue problems and propose a new pivoting strategy, which is completely different from classical ones governed solely by the magnitude of the off-diagonal entries. When combined with a mixed-precision preconditioner that diagonalizes the matrix to low precision, numerical experiments demonstrate that the resulting strategy can significantly outperform the classical greedy approach when the original matrix has clustered eigenvalues.