On algebraic Riccati equations associated with M -matrices

Guo, Chen

doi:10.1016/j.laa.2013.08.018

Cited by 25 publications

(19 citation statements)

References 22 publications

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“…M-matrices that admit a triplet representation are called regular M-matrices. Non-regular M-matrices must necessarily be singular and reducible [Guo13], so most M-matrices appearing in applications (and, in particular, all those appearing in the rest of this paper) are indeed regular.…”

Section: Triplet Representations and Accurate Matrix Exponentialsmentioning

confidence: 99%

Componentwise accurate Brownian motion computations using Cyclic Reduction

Nguyen,

Poloni

2016

Preprint

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Markov-modulated Brownian motion is a popular tool to model continuous-time phenomena in a stochastic context. The main quantity of interest is the invariant density, which satisfies a differential equation associated with the quadratic matrix polynomial P (z) = V z 2 −Dz +Q, where the matrices V and D are diagonal and Q is the transition matrix of a discrete-time Markov chain. Its solution is typically constructed by computing an invariant pair of P (z) associated with its eigenvalues in the left half-plane, or by solving the matrix equation X 2 V − XD + Q = 0. We show that these tasks can be solved using a componentwise accurate algorithm based on Cyclic Reduction, generalizing the recently appeared algorithms for the linear case (V = 0). We give a proof of the numerical stability of our algorithm in the componentwise sense; the same proof applies to Cyclic Reduction in a more general M-matrix setting which appears in other applications such as the modelling of QBD processes.

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Section: Triplet Representations and Accurate Matrix Exponentialsmentioning

confidence: 99%

Componentwise accurate Brownian motion computations using Cyclic Reduction

Nguyen,

Poloni

2016

Preprint

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“…Let A be a Z-matrix. Then the following statements are equivalent: For the minimal nonnegative solution of the MARE, we have the following important result [5,7,8,12]. Lemma 1.6.…”

Section: Introductionmentioning

confidence: 99%

“…The solution of practical interest is the minimal nonnegative solution. For theoretical background we refer to [5,7,8,[10][11][12]15].…”

Section: Introductionmentioning

confidence: 99%

New Alternately Linearized Implicit Iteration for M-Matrix Algebraic Riccati Equations

Lu¹

2017

JMS

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Research on the theories and the efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years due to its broad applications. In this paper, based on the alternately linearized implicit iteration method (ALI) [Z.-Z. Bai et al., Numer. Linear Algebra Appl., 13(2006), 655-674.], we propose a new alternately linearized implicit iteration method (NALI) for computing the minimal nonnegative solution of M-matrix algebraic Riccati equation. Convergence of the NALI method is proved by choosing proper parameters for the MARE associated with nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the NALI method is more efficient than the ALI method in some cases.

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“…Thereby, it is important to discuss the minimal nonnegative solutions of the NCARE (2) and NARE (2), and there are still many problems worth further study. The computation of the minimal nonnegative solution of the NARE has been investigated by many authors, and various direct and iteration methods have been proposed [1][2][3][4][5], [8][9][10][11][12][13][14][15][16][17][18][19]. Such as the classical Newton iteration method (Newton), fixed point iteration (FXP) [1], alternating implicit iteration (ALI) [2], structure-preserving doubling algorithm (SDA) [3] and inexact Newton iteration method (INewton) [4].…”

Section: Introductionmentioning

confidence: 99%

Numerical methods for the minimal non‐negative solution of the non‐symmetric coupled algebraic Riccati equation

Zhang

Tan

2019

Asian Journal of Control

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In this paper, base on the theory of the non‐symmetric algebraic Riccati equation and the coupled Riccati equation, we give the general form of the non‐symmetric coupled algebraic Riccati equation (NCARE). In order to effectively solve the minimal non‐negative solution of the NCARE, two numerical iteration methods are improved: inexact Newton method (INewton) and alternate linear implicit method (ALI). Further, we give the convergence theory of the two iteration methods. Finally, we offer numerical examples to show the effectiveness of the derived methods.

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On algebraic Riccati equations associated with M -matrices

Cited by 25 publications

References 22 publications

Componentwise accurate Brownian motion computations using Cyclic Reduction

Componentwise accurate Brownian motion computations using Cyclic Reduction

New Alternately Linearized Implicit Iteration for M-Matrix Algebraic Riccati Equations

Numerical methods for the minimal non‐negative solution of the non‐symmetric coupled algebraic Riccati equation

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