The generalized joint spectral radius. A geometric approach

Protasov, Vladimir Yu.

doi:10.1070/im1997v061n05abeh000161

Cited by 77 publications

(49 citation statements)

References 2 publications

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“…In recent years most research efforts have concentrated on finding reasonable approximations for the JSR (see, e.g., [3,4,19,21,25]). But all these approximations suffer from an intrinsic limitation since it is known that the problem of computing the JSR of two matrices with binaries entries is NP-hard and that, unless P = NP, there is no approximation algorithm that is polynomial with respect to the accuracy [24].…”

Section: Introduction a Discrete-time Switching Linear System Generamentioning

confidence: 99%

Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

Blondel¹,

Nesterov²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We propose two simple upper bounds for the joint spectral radius of sets of nonnegative matrices. These bounds, the joint column radius and the joint row radius, can be computed in polynomial time as solutions of convex optimization problems. We show that these bounds are within a factor 1/n of the exact value, where n is the size of the matrices. Moreover, for sets of matrices with independent column uncertainties or with independent row uncertainties, the corresponding bounds coincide with the joint spectral radius. In these cases, the joint spectral radius is also given by the largest spectral radius of the matrices in the set. As a by-product of these results, we propose a polynomial-time technique for solving Boolean optimization problems related to the spectral radius. We also describe economics and engineering applications of our results.

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Section: Introduction a Discrete-time Switching Linear System Generamentioning

confidence: 99%

Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

Blondel¹,

Nesterov²

2010

SIAM J. Matrix Anal. & Appl.

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“…This idea was put to good use in [22,30] (polyhedral norms), [8,31,41] (Kronecker lifting), and [29] (sums of squares). The computational complexity of these methods grows exponentially with the dimension of the matrices.…”

Section: K}mentioning

confidence: 99%

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

Protasov¹,

Jungers²,

Blondel³

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We propose a new method to compute the joint spectral radius and the joint spectral subradius of a set of matrices. We first restrict our attention to matrices that leave a cone invariant. The accuracy of our algorithm, depending on geometric properties of the invariant cone, is estimated. We then extend our method to arbitrary sets of matrices by a lifting procedure, and we demonstrate the efficiency of the new algorithm by applying it to several problems in combinatorics, number theory, and discrete mathematics.

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“…Fortunately, from [19,24] it follows that ρ 2 ({A m }) = λ 1/2 can be found more efficiently by determining the largest eigenvalue λ > 0 of the associated matrix transfer operator…”

Section: Proof Of Theoremmentioning

confidence: 99%

Optimality of multilevel preconditioning for nonconforming P1 finite elements

Oswald

2008

Numer. Math.

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We prove the optimality of hierarchical and BPX-type preconditioners for finite element discretizations with nonconforming P1 finite elements. Such preconditioners were proposed about 15 years ago, and until now only suboptimal estimates of their preconditioning properties have been available. The main new tool is an improved Bernstein type inequality for an associated subdivision process generated by the prolongations which allows us to give an asymptotically optimal upper bound for the spectrum of the preconditioned systems.

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The generalized joint spectral radius. A geometric approach

Cited by 77 publications

References 2 publications

Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

Polynomial-Time Computation of the Joint Spectral Radius for Some Sets of Nonnegative Matrices

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

Optimality of multilevel preconditioning for nonconforming P1 finite elements

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