Hybrid approach to the joint spectral radius computation

TL;DR

This work addresses exact joint spectral radius computation for a finite matrix set by fusing the finite tree algorithm and the invariant polytope algorithm into two hybrid, tree-flavoured methods. The key idea is to use V-closed and strongly V-closed trees to define Minkowski norms that certify $\operatorname{JSR}(\mathcal{A})\le1$ in broader scenarios, including cases where IPA alone fails. The proposed approaches seed from spectrum-maximizing products, leverage leading eigenvectors, and adaptively grow a polytope norm through subtree checks and pseudo-spectral radius bounds to manage growth and termination. The results indicate broader applicability and potential speed-ups, with implementation details for bounding growth and extensions to complex cases, though comparative performance awaits further study.

Abstract

In this paper we propose a modification to the invariant polytope algorithm (ipa) using ideas of the finite expressible tree algorithm (feta) by Möller and Reif. We show that our new feta-flavoured-ipa applies to a wider range of matrix families.

Figures (5)

• Figure 1: $({\mathcal{A}},\mathbf{G})$-tree for ${\mathcal{A}} = \{A_1,A_2\}$ and $\mathbf{G} = \{[1,2]\}$ with covered ($\square$) and uncovered ($\circ$) leafs. The covered leaf $\{A_2 A_1 (A_2 A_1)^n A_1, n \in {\mathbb N}_0\} = \{(A_2 A_1)^n A_1: n \in {\mathbb N}\}$ is a (proper) subset of its grandparent $\{(A_2 A_1)^n A_1: n \in {\mathbb N}_0\}$.
• Figure 2: $V\!$-closed tree ${\mathbf{T}}_1$(left) and strongly $V\!$-closed tree ${\mathbf{T}}_\gamma$(right) for the example in the proof of Theorem \ref{['thm:terminate']}.
• Figure 3: Tree constructed by the tree-flavoured-invariant-polytope-algorithm for the problem considered in Example \ref{['ex:infinitypath']}. Dots $(\bullet)$ mark the vertices of the polytope $P$; Empty bullets $(\circ)$ mark vertices which are mapped into $\operatorname{co}_s P$; Squares $(\square)$ mark covered nodes. Note that the starting vector $v_0$ use to construct the polytope $P$ is an eigenvector of $A_2$, and thus the node $\left\lbrace A_2^n v_0 \right\rbrace$ only consists of one vector;
• Figure 4: Comparison between the original invariant polytope algorithm, and the algorithm described in Remark \ref{['rem_extrapath']}. For each dimension 20 tests were done. First row: Ratio of number of vertices of computed polytope (values smaller than one mean, the new algorithm produces polytopes with less vertices). Second row: Ratio between the computation times (values smaller than one mean, the new algorithm is faster). One can see, that the new algorithm produces consistently smaller polytopes, but the introduced overhead does not pay of for small matrices.
• Figure 5: Symmetric and elliptic convex hull. $v_1=12$, $v_2=21$, $v_3=2i$, $v_4=i2$,

Theorems & Definitions (27)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: BW1992
• Definition 1.4
• Lemma 1.5: Jung2009
• Definition 1.6
• Definition 1.7
• Lemma 1.8
• Definition 1.9
• Definition 2.1: $({\mathcal{A}},\mathbf{G})$-tree