Hilbert’s projective metric and iterated nonlinear maps

Nussbaum, Roger D.

doi:10.1090/memo/0391

Cited by 204 publications

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“…Hence, Algorithm 4.1 is also well-defined for weakly irreducible nonnegative tensors. The following theorem establishes convergence of Algorithm 4.1 if the underlying tensor T is weakly primitive, where we need to use the concept of Hilbert's projective metric [9]. We first recall such a concept.…”

Section: Global R-linear Convergence Of a Power Methods For Weakly Irrmentioning

confidence: 99%

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Strictly nonnegative tensors and nonnegative tensor partition

Huang

2013

Sci. China Math.

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In this paper, we introduce a new class of nonnegative tensors -strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T , there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

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Section: Global R-linear Convergence Of a Power Methods For Weakly Irrmentioning

confidence: 99%

“…. , n} and x ∈ ℜ n + , then (1) is strongly related to the eigenvalue problem for the nonlinear map F T discussed in [9]. Denote by ρ(T ) := max{|λ| | λ ∈ σ(T )} where σ(T ) is the set of all eigenvalues of T .…”

Section: Introductionmentioning

confidence: 99%

Strictly nonnegative tensors and nonnegative tensor partition

Huang

2013

Sci. China Math.

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“…Remark that, when g maps from C to itself and C is a pointed cone, all the eigenvalues are nonnegative. The existence of nonlinear eigenvectors is guaranteed by standard fixed point arguments [Nus88].…”

Section: Basic Notionsmentioning

confidence: 99%

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

Adjé¹,

Gaubert²,

Goubault³

2014

Journal of Mathematical Analysis and Applications

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The problem of computing the smallest fixed point of an orderpreserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.

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“…The latest result on the Perron-Frobenius Theorem is that the eigenvalues with modulus ρ(A) have the same geometric multiplicity in [4]. Some other results of nonnegative tensors were established in [5][6][7][8][9][10][11][12]. What is more, Ng, Qi, and Zhou proposed the NQZ method for finding spectral radius of a nonnegative irreducible tensor in [13].…”

Section: Introductionmentioning

confidence: 99%

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

Yang

2016

J. Oper. Res. Soc. China

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In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090-1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

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Hilbert’s projective metric and iterated nonlinear maps

Cited by 204 publications

References 0 publications

Strictly nonnegative tensors and nonnegative tensor partition

Strictly nonnegative tensors and nonnegative tensor partition

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

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