QR Factorization of Block Low-rank Matrices with Weak Admissibility Condition

Abstract

The QR factorization of a matrix is a fundamental operation in linear algebra and it is widely utilized in scientific simulations. Although the QR factorization requires a memory storage of O(N²) and the computational cost of O(N³), they can be reduced if the matrix could be approximated using low-rank structured matrices, such as hierarchical matrices (H-matrices). In this paper, we study the QR factorization based on the block low-rank (BLR-) matrix representation, which is a simplified variant of the H-matrix. We demonstrate how the BLR block size should be defined in such matrices and confirm that the memory and computational complexities of the BLR are O(N^1.5) and O(N²) when using an appropriate block size. Furthermore, using the simple structure of BLR-matrices, we also present a parallel algorithm for the QR factorization of BLR-matrices on distributed memory systems. In numerical experiments performed, we observe that the accuracy of the QR factorization is controllable by the accuracy of the BLR-matrix approximation of the original matrix. Finally, we verify that our algorithm enables us to perform the QR factorization of matrices of several hundred thousand N within a reasonable amount of time using moderate computer resources.