Explicit expression and fast algorithms for the inverse of some matrices arising from implicit time integration

TL;DR

This paper derives explicit formulas for inverting certain matrices from implicit time integration and introduces three efficient algorithms to compute their inverses, significantly reducing computational complexity.

Contribution

The paper provides explicit inverse formulas for matrices from implicit schemes and develops three fast algorithms with $O(n^2)$ complexity for their inversion.

Findings

Algorithms are highly efficient and accurate.

Explicit inverse formulas are derived for specific matrices.

Methods outperform traditional approaches in speed and precision.

Abstract

In this paper, we first present an explicit expression for the inverse\emph{} of a type of matrices. As special applications, the inverse of some matrices arising from implicit time integration techniques, such as the well-known implicit Runge-Kutta schemes and block implicit methods, can also be explicitly determined. Adiitionally, we introduce three fast algorithms for computing the elements of the inverse of these matrices in $O(n^2)$ arithmetic operations, i.e., the first one is based on Traub algorithm for fast inversion of Vandermonde matrices, while the other two utilize the special structure of the matrices. Finally, some symbolic and numerical results are presented to show that our algorithms are both highly efficient and accurate.