Hierarchical Orthogonal Factorization: Sparse Square Matrices

In this work, we develop a new fast algorithm, spaQR -- sparsified QR, for solving large, sparse linear systems. The key to our approach is using low-rank approximations sparsify the separators in Nested Dissection based Householder QR factorization. First, modified version of used identify interiors/separators and reorder matrix. Then, classical factorize interiors, going from leaves root elimination tree. After every level interior factorization, remaining by approximations. This operation reduces size without introducing any fill-in However, it introduces small approximation error which can be controlled user. resulting approximate factorization stored as sequence orthogonal upper-triangular factors. Hence, applied efficiently solve Additionally, further improve algorithm block diagonal scaling. show systematic analysis effectiveness Finally, perform numerical tests on benchmark unsymmetric problems evaluate performance algorithm. time scales $\mathcal{O}(N \log N)$ $\mathcal{O}(N)$.