# Computing the Kreiss Constant of a Matrix

## TL;DR

This paper introduces the first globally convergent algorithms for accurately computing the Kreiss constant of a matrix, with improved complexity and scalable optimization methods for large matrices.

## Contribution

It presents novel algorithms for computing the Kreiss constant with proven convergence and reduced computational complexity, including continuous and discrete-time variants.

## Key findings

- Algorithms achieve arbitrary accuracy in computing Kreiss constants.
- Complexity reduced from $ ext{O}(n^6)$ to $ ext{O}(n^4)$ on average.
- Efficient local optimization methods for large-scale matrices.

## Abstract

We establish the first globally convergent algorithms for computing the Kreiss constant of a matrix to arbitrary accuracy. We propose three different iterations for continuous-time Kreiss constants and analogues for discrete-time Kreiss constants. With standard eigensolvers, the methods do $\mathcal{O}(n^6)$ work, but we show how this theoretical work complexity can be lowered to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case via divide-and-conquer variants. Finally, locally optimal Kreiss constant approximations can be efficiently obtained for large-scale matrices via optimization.