Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices

Xue, Jing; Ye, Qiang

doi:10.1007/s00211-008-0167-5

Cited by 6 publications

(11 citation statements)

References 17 publications

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“…The problem of the computation of the exponential of a generator has been considered in Xue and Ye [21,20] and by Shao et al [17], where the authors propose component-wise accurate algorithms for the computation. These algorithms are efficient for matrices of small size.…”

Section: Resultsmentioning

confidence: 99%

Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues

Бини

Dendievel

Latouche

et al. 2016

Linear Algebra and its Applications

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The Erlangian approximation of Markovian fluid queues leads to the problem of computing the matrix exponential of a subgenerator having a block-triangular, block-Toeplitz structure. To this end, we propose some algorithms which exploit the Toeplitz structure and the properties of generators. Such algorithms allow us to compute the exponential of very large matrices, which would otherwise be untreatable with standard methods. We also prove interesting decay properties of the exponential of a generator having a block-triangular, block-Toeplitz structure

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Section: Resultsmentioning

confidence: 99%

Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues

Бини

Dendievel

Latouche

et al. 2016

Linear Algebra and its Applications

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“…Theorem A.3. Let P be a n × n stochastic matrix, and define E, F , G, and H as in (11), and A + , A = , and A − as in (30). Then, for k ≥ 0 the matrix F k (P ) is obtained by censoring the top (2 k − 1)n × (2 k − 1)n block of the 2 k n × 2 k n matrix…”

Section: A Doubling and Censoringmentioning

confidence: 99%

Componentwise accurate fluid queue computations using doubling algorithms

Nguyen

Poloni

2014

Numer. Math.

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Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations.Xue et al. (Accurate solutions of M-matrix algebraic Riccati equations, Numer. Math., 2012) provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense.In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.

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“…For the exponential of an essentially non-negative matrix A, we have recently obtained an entrywise perturbation analysis in [28] showing that, if E is a small perturbation to A such that |E| ≤ |A|, then we have…”

Section: Introductionmentioning

confidence: 99%

“…Here the absolute value and inequalities on matrices are entrywise. With the upper bound (1.2) attainable, κ exp (A) is a condition number for the entrywise perturbation; see [28]. Indeed, it is intrinsic to the exponential function itself in the sense that it is present in the perturbation of the exponential function of a real variable when n = 1 (see [28]).…”

Section: Introductionmentioning

confidence: 99%

“…With the upper bound (1.2) attainable, κ exp (A) is a condition number for the entrywise perturbation; see [28]. Indeed, it is intrinsic to the exponential function itself in the sense that it is present in the perturbation of the exponential function of a real variable when n = 1 (see [28]). The perturbation bound suggests that it is possible to compute all entries of the exponential of an essentially non-negative matrix with a relative accuracy in the order of machine precision up to a scaling by the condition number κ exp (A).…”

Section: Introductionmentioning

confidence: 99%

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Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

Xue

2013

Math. Comp.

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A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. It has recently been shown that the exponential of an essentially non-negative matrix is determined entrywise to high relative accuracy by its entries up to a condition number intrinsic to the exponential function (Numer. Math. 110 (2008), 393-403). Thus the smaller entries of the exponential may be computed to the same relative accuracy as the bigger entries. This paper develops algorithms to compute exponentials of essentially non-negative matrices entrywise to high relative accuracy.

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Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices

Cited by 6 publications

References 17 publications

Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues

Computing the exponential of large block-triangular block-Toeplitz matrices encountered in fluid queues

Componentwise accurate fluid queue computations using doubling algorithms

Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy

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