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.

Keywords

  1. discontinuity of Kreiss constants
  2. inverses of Kronecker sums
  3. distance to uncontrollability algorithms
  4. transient growth
  5. pseudospectra

MSC codes

  1. 15A16
  2. 37C75
  3. 39A22
  4. 39A30
  5. 65F30
  6. 65F60

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Supplementary Material


PLEASE NOTE: These supplementary files have not been peer-reviewed.


Index of Supplementary Materials

Title of paper: Computing the Kreiss Constant of a Matrix

Author: Tim Mitchell

File: kreiss_code.zip

Type: Compressed files

Contents: MATLAB code.


File: kreiss_mitchell_supp.pdf

Type: PDF

Contents: A PDF to describe and reproduce the experiments.

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Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1944 - 1975
ISSN (online): 1095-7162

History

Submitted: 16 July 2019
Accepted: 8 September 2020
Published online: 15 December 2020

Keywords

  1. discontinuity of Kreiss constants
  2. inverses of Kronecker sums
  3. distance to uncontrollability algorithms
  4. transient growth
  5. pseudospectra

MSC codes

  1. 15A16
  2. 37C75
  3. 39A22
  4. 39A30
  5. 65F30
  6. 65F60

Authors

Affiliations

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1620083

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