Year: 2019
Type: preprint
Abstract: Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to solve a system of linear questions is superior to direct inversion both with respect to accuracy and efficiency, but so far theoretical estimates quantifying these advantages were missing. Due to a lack of symmetry in $\mathcal{H}$-matrix algorithms, we cannot hope to prove that the LR factorization takes one third of the operations of the inversion or the matrix multiplication, as in standard linear algebra. We can, however, prove that the LR factorization together with two other operations of similar complexity, i.e., the inversion and multiplication of triangular matrices, requires not more operations than the matrix multiplication. We can complete the estimates by proving an improved upper bound for the complexity of the matrix multiplication, designed for recently introduced variants of classical $\mathcal{H}$-matrices.
Source: arXiv (Cornell University)
Author: Börm, Steffen
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2945870734