A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 2, Page 847-874, June 2024.
Abstract. Hierarchical matrices are dense but data-sparse matrices that use low-rank factorizations of suitable submatrices to reduce the storage and computational cost to linear-polylogarithmic complexity. In this paper, we propose a new approach to efficiently compute QR factorizations in the hierarchical matrix format based on block Householder transformations. To prevent unnecessarily high ranks in the resulting factors and to increase speed and accuracy, the algorithm meticulously tracks for which intermediate results low-rank factorizations are available. We also use a special storage scheme for the block Householder reflector to further reduce computational and storage costs. Numerical tests for two- and three-dimensional Laplacian boundary element matrices, different radial basis function kernel matrices, and matrices of typical hierarchical matrix structures but filled with random entries illustrate the performance of the new algorithm in comparison to some other QR algorithms for hierarchical matrices from the literature.