Constructive solution of the common invariant cone problem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915270413516800 |
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| author | Mejstrik, Thomas Protasov, Vladimiar Yu. |
| author_facet | Mejstrik, Thomas Protasov, Vladimiar Yu. |
| contents | Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be algorithmically undecidable for general sets of matrices. We show that it can nevertheless be efficiently solved in practice. An algorithm that for a given finite set of matrices, either finds a common invariant cone or proves its non-existence is presented. Numerical results demonstrate that it works for a vast majority of matrix sets. The structure and properties of the minimal and maximal invariant cones are analyzed. Applications to dynamical systems and combinatorics are considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_01229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive solution of the common invariant cone problem Mejstrik, Thomas Protasov, Vladimiar Yu. Numerical Analysis 15B48, 52B05 Sets of $d\times d$ matrices sharing a common invariant cone enjoy special properties, which are widely used in applications. However, finding this cone or even proving its existence/non-existence is hard. This problem is known to be algorithmically undecidable for general sets of matrices. We show that it can nevertheless be efficiently solved in practice. An algorithm that for a given finite set of matrices, either finds a common invariant cone or proves its non-existence is presented. Numerical results demonstrate that it works for a vast majority of matrix sets. The structure and properties of the minimal and maximal invariant cones are analyzed. Applications to dynamical systems and combinatorics are considered. |
| title | Constructive solution of the common invariant cone problem |
| topic | Numerical Analysis 15B48, 52B05 |
| url | https://arxiv.org/abs/2505.01229 |