Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

Meng, Junmin; Seo, Sang-Hyup; Kim, Hyun‐Min

doi:10.1007/s10915-018-0641-x

Cited by 9 publications

(3 citation statements)

References 41 publications

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“…Moreover, we have the following theorem. The theorem is implied by [3,Theorem 3.1] or [22,Theorem 2.3].…”

Section: The Main Resultsmentioning

confidence: 99%

A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with $M$-matrix

Chen

2021

Preprint

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Consider the problem of finding the maximal nonpositive solvent Φ of the quadratic matrix equation (qme) X 2 + BX + C = 0 with B being a nonsingular M -matrix and C an M -matrix such that B −1 C ≥ 0, and B − C − I a nonsingular M -matrix. Such qme arises from an overdamped vibrating system. Recently, Yu et al. (Appl. Math. Comput., 218: 3303-3310, 2011) proved that ρ(Φ) ≤ 1 for this qme. In this paper, we slightly improve their result and prove ρ(Φ) < 1, which is important for the quadratic convergence of the structure-preserving doubling algorithm. Then, a new globally monotonically and quadratically convergent structure-preserving doubling algorithm to solve the qme is developed. Numerical examples are presented to demonstrate the feasibility and effectiveness of our method.

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“…Moreover, we have the following theorem. The theorem is implied by [3,Theorem 3.1] or [22,Theorem 2.3].…”

Section: The Main Resultsmentioning

confidence: 99%

A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with $M$-matrix

Chen

2021

Preprint

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“…To our knowledge, a numerical analysis of matrix equations of the kind (1.2) has not been performed in the literature, while there is wide literature on matrix equations of the kind ∞ i=0 A i G i = 0. We refer to the books [5] and [20] for a survey on matrix equations arising in structured Markov chains, to [16] and [17] for the solution of general quadratic matrix equations, to [23] for an analysis of the conditioning, and to [6], [8], [9], [13], [14], [24], [26], [28], and [29] for strategies to improve the accuracy and the convergence.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

Бини¹,

Latouche²,

Meini³

2021

Preprint

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Markov-modulated Lévy processes lead to matrix integral equations of the kind A 0 + A 1 X + A 2 X 2 + A 3 (X) = 0 where A 0 , A 1 , A 2 are given matrix coefficients, while A 3 (X) is a nonlinear function, expressed in terms of integrals involving the exponential of the matrix X itself.In this paper we propose some numerical methods for the solution of this class of matrix equations, perform a theoretical convergence analysis and show the effectiveness of the new methods by means of a wide numerical experimentation.

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“…A vast literature is devoted to methods based on cyclic reduction and on doubling algorithms; we refer the reader to the survey paper [2] and to the literature cited therein, to the paper [15] for the logarithmic reduction algorithm and to [6] for a convergence analysis of SDA-based algorithms. A conditioning analysis is performed in [19].…”

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confidence: 99%

A family of fast fixed point iterations for M/G/1-type Markov chains

Бини¹,

Latouche²,

Meini³

2020

Preprint

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We consider the problem of computing the minimal nonnegative solution G of the nonlinear matrix equation X = ∞ i=−1 A i X i+1 where A i , for i ≥ −1, are nonnegative square matrices such that ∞ i=−1 A i is stochastic. This equation is fundamental in the analysis of M/G/1type Markov chains, since the matrix G provides probabilistic measures of interest. We introduce here a new family of fixed point iterations that include the classical iterations for the numerical computation of G. We perform a detailed convergence analysis and prove that the iterations in the new class converge faster than the classical iterations. Numerical experiments show that the acceleration in terms of CPU time is substantial with a speed-up which is generally greater than 2 and, in several cases, reaches much larger values.

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Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

Cited by 9 publications

References 41 publications

A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with $M$-matrix

A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with $M$-matrix

Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

A family of fast fixed point iterations for M/G/1-type Markov chains

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