SuperDC: Superfast Divide-And-Conquer Eigenvalue Decomposition With Improved Stability for Rank-Structured Matrices

Abstract

For dense symmetric matrices with small off-diagonal (numerical) ranks and in a hierarchically semiseparable form, we give a divide-and-conquer eigendecomposition method with nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in [J. Vogel, et al., SIAM J. Sci. Comput., 38 (2016), PP. A1358--A1382]. Some stability risks in the original algorithm are analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank updating strategy with balancing that eliminates the exponential norm growth and also minimizes the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, the original algorithm directly uses the standard fast multipole method (FMM) to accelerate function evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate eigenvalues, we design a novel local shifting strategy to integrate FMM accelerations into the solution of shifted secular equations. This helps achieve both the efficiency and the reliability. We also provide a deflation strategy with a user-supplied tolerance and give a precise description of the structure of the resulting eigenvector matrix. The SuperDC eigensolver has significantly improved stability while keeping the nearly linear complexity for finding the entire eigenvalue decomposition. Extensive numerical tests are used to show the efficiency and accuracy of SuperDC.