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Abstract
We study the problem of finding the nearest Ωstable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Ω. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on smallscale and mediumscale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.
Original language  English 

Pages (fromto)  817–851 
Number of pages  35 
Journal  Numerische Mathematik 
Volume  148 
Issue number  4 
Early online date  2021 
DOIs  
Publication status  Published  Aug 2021 
MoE publication type  A1 Journal articlerefereed 
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 1 Active

Noferini_Vanni_AoF_Project: Noferini Vanni Academy Project
Noferini, V., Quintana Ponce, M., Barbarino, G. & Wood, R.
01/09/2020 → 31/08/2024
Project: Academy of Finland: Other research funding