Fast Methods for Computing the<i>p</i>-Radius of Matrices

Jungers, Raphaël M.; Protasov, Vladimir Yu.

doi:10.1137/090777906

Cited by 13 publications

(12 citation statements)

References 28 publications

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“…Thus, without being explicitly stated, we assume that probability distributions appearing in this paper have a bounded support. Though in general the computation of p-radius is a difficult problem [11], the next simple formula for p-radius is available under certain assumptions. .…”

Section: Joint Spectral Characteristicsmentioning

confidence: 99%

“…In this case there exists > 0 such that H(M, M r ) > for every r > 0. By the definition of the Hausdorff metric (11)…”

Section: Proofmentioning

confidence: 99%

“…where · denotes any matrix norm. Firstly introduced [8,9] for p = 1 and then extended [10] for a general p, L p -norm joint spectral radius has found many applications in various areas of applied mathematics (see [11] and references therein). In particular p-radius has an application to the stability theory of stochastic switched systems [12,13,14], which is a dynamical system whose structure randomly experiences abrupt changes [15,16].…”

Section: Introductionmentioning

confidence: 99%

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A limit formula for joint spectral radius with p-radius of probability distributions

Ogura

Martin

2014

Linear Algebra and its Applications

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In this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the p-radius of an associated probability distribution when p tends to ∞. Allowing the set to have infinitely many matrices, the obtained formula extends the results in the literature. Based on the formula, we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic Lyapunov functions of any higher degrees. Numerical examples are presented to illustrate the results.

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Section: Joint Spectral Characteristicsmentioning

confidence: 99%

“…In this case there exists > 0 such that H(M, M r ) > for every r > 0. By the definition of the Hausdorff metric (11)…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A limit formula for joint spectral radius with p-radius of probability distributions

Ogura

Martin

2014

Linear Algebra and its Applications

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“…In this paper we propose novel lower bounds on the p-radius for integer values of p with no assumptions on the given set of matrices on the contrary to [2,8]. The lower bounds are given as the spectral radius of a weighted average of the given matrices and the weights are realized by Kronecker product of matrices.…”

Section: Introductionmentioning

confidence: 99%

Efficient method for computing lower bounds on the p-radius of switched linear systems

Ogura

Preciado

Jungers

2016

Systems & Control Letters

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This paper proposes lower bounds on a quantity called L p -norm joint spectral radius, or in short, p-radius, of a finite set of matrices. Despite its wide range of applications to, for example, stability analysis of switched linear systems and the equilibrium analysis of switched linear economical models, algorithms for computing the p-radius are only available in a very limited number of particular cases. The proposed lower bounds are given as the spectral radius of an average of the given matrices weighted via Kronecker products and do not place any requirements on the set of matrices. We show that the proposed lower bounds theoretically extend and also can practically improve the existing lower bounds. A Markovian extension of the proposed lower bounds is also presented.

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“…For instance, for a two‐regime model, if the transition probabilities are symmetric (

p_{11} = p_{22} = 1 false/ 2

), the limit of

u_{k}

when k tends to infinity exactly corresponds to the 1‐radius of

false{{normalΓ}_{1}^{−1}, {normalΓ}_{2}^{−1} false}

. We refer to Jungers and Protasov () for a detailed presentation of this quantity. The complexity of these concepts is therefore well known in control theory.…”

mentioning

confidence: 99%

Monetary policy switching and indeterminacy

Barthélemy¹,

Marx²

2019

Quantitative Economics

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This paper determines conditions for the existence of a unique rational expectations equilibrium-determinacy-in a monetary policy switching economy. We depart from the existing literature by providing such conditions considering all bounded equilibria. We then apply these conditions to a new Keynesian model with switching Taylor rules. First, deviation from the Taylor principle in one regime does not necessarily cause indeterminacy. Second, very different responses to inflation may trigger indeterminacy even if both regimes satisfy the Taylor principle. Determinacy thus results from the adequacy between monetary regimes rather than the determinacy of each of them taken in isolation.

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Fast Methods for Computing thep-Radius of Matrices

Cited by 13 publications

References 28 publications

A limit formula for joint spectral radius with p-radius of probability distributions

A limit formula for joint spectral radius with p-radius of probability distributions

Efficient method for computing lower bounds on the p-radius of switched linear systems

Monetary policy switching and indeterminacy

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