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 5 hierarchically semiseparable form, we give a divide-and-conquer eigendecomposition method with 6 nearly linear complexity (called SuperDC) that significantly improves an earlier basic algorithm in 7 [Vogel, Xia, et al., SIAM J. Sci. Comput., 38 (2016)]. Some stability risks in the original algorithm are 8 analyzed, including potential exponential norm growth, cancellations, loss of accuracy with clustered 9 eigenvalues or intermediate eigenvalues, etc. In the dividing stage, we give a new structured low-rank 10 updating strategy with balancing that eliminates the exponential norm growth and also minimizes 11 the ranks of low-rank updates. In the conquering stage with low-rank updated eigenvalue solution, 12 the original algorithm directly uses the standard fast multipole method (FMM) to accelerate function 13 evaluations, which has the risks of cancellation, division by zero, and slow convergence. Here, we 14 design a triangular FMM to avoid cancellation. Furthermore, when there are clustered intermediate 15 eigenvalues, we design a novel local shifting strategy to integrate FMM accelerations into the solution 16 of shifted secular equations. This helps achieve both the efficiency and the reliability. We also provide 17 a deflation strategy with a user-supplied tolerance and give a precise description of the structure of 18 the resulting eigenvector matrix. The SuperDC eigensolver has significantly improved stability while 19 keeping the nearly linear complexity for finding the entire eigenvalue decomposition. Extensive 20 numerical tests are used to show the efficiency and accuracy of SuperDC.