# Complexity estimates for triangular hierarchical matrix algorithms

## TL;DR

This paper provides theoretical complexity estimates for hierarchical matrix algorithms, demonstrating that LR factorizations are computationally efficient and comparable to matrix multiplication, thereby supporting their practical advantages in solving integral and differential equations.

## Contribution

It offers the first theoretical complexity bounds for $ ext{H}$-matrix LR factorizations, inversion, and multiplication, confirming their efficiency over direct inversion.

## Key findings

- LR factorization requires no more operations than matrix multiplication.
- An improved upper bound for the complexity of matrix multiplication is established.
- Theoretical estimates support the efficiency of $ ext{H}$-matrix algorithms in practical applications.

## 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.