We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear control system, both to arbitrary accuracy. Previous state-of-the-art methods for these two quantities rely on 2D level-set tests that are based on solving large eigenvalue problems. Consequently, these methods are costly, i.e., $\mathcal{O}(n^6)$ work using dense eigensolvers, and often multiple tests are needed before convergence. Divide-and-conquer techniques have been proposed that reduce the work complexity to $\mathcal{O}(n^4)$ on average and $\mathcal{O}(n^5)$ in the worst case, but these variants are nevertheless still very expensive and can be numerically unreliable. In contrast, our new interpolation-based globality certificates perform level-set tests by building interpolant approximations to certain one-variable continuous functions that are both relatively cheap and numerically robust to evaluate. Our new approach has an $\mathcal{O}(kn^3)$ work complexity and uses $\mathcal{O}(n^2)$ memory, where $k$ is the number of function evaluations necessary to build the interpolants. Not only is this interpolation process mostly “embarrassingly parallel," but also low-fidelity approximations typically suffice for all but the final interpolant, which must be built to high accuracy. Even without taking advantage of the aforementioned parallelism, $k$ is sufficiently small that our new approach is generally orders of magnitude faster than the previous state of the art.


  1. transient growth
  2. robust stability
  3. controllability
  4. 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PLEASE NOTE: These supplementary files have not been peer-reviewed.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: Fast Interpolation-Based Globality Certificates for Computing Kreiss Constants and the Distance to Uncontrollability

Author: Tim Mitchell

File: kreiss_dtu_code.zip

Type: Compressed code files

Contents: MATLAB code.

File: kreiss_dtu_mitchell_supp.pdf

Type: PDF

Contents: Detailed descriptions of both our implementation and experimental setup for reproducibility of all results, tables, and figures in the paper.


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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 578 - 607
ISSN (online): 1095-7162


Submitted: 3 October 2019
Accepted: 11 January 2021
Published online: 8 April 2021


  1. transient growth
  2. robust stability
  3. controllability
  4. pseudospectra

MSC codes

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



Funding Information

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

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