Nearest Ω -stable matrix via Riemannian optimization

Vanni Noferini*, Federico Poloni

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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We study the problem of finding the nearest Ω-stable 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 small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.

Original languageEnglish
Pages (from-to)817–851
Number of pages35
JournalNumerische Mathematik
Issue number4
Early online date2021
Publication statusPublished - Aug 2021
MoE publication typeA1 Journal article-refereed


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