Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 386-395
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Matrix Equation ◽  
Nonnegative Solution ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

scholarly journals Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes

2011 ◽  
Vol 48 (04) ◽  
pp. 925-937 ◽  
Author(s):  
Yuanyuan Liu
Keyword(s):  
Markov Chains ◽  
Central Limit ◽  
Discrete Time ◽  
Nonnegative Solution ◽  
Return Time ◽  
Additive Functionals ◽  
The Minimal Nonnegative Solution ◽  
First Return Time ◽  
Do So ◽  
Birth Death

In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.

scholarly journals Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes

2011 ◽  
Vol 48 (4) ◽  
pp. 925-937 ◽  
Author(s):  
Yuanyuan Liu
Keyword(s):  
Markov Chains ◽  
Central Limit ◽  
Discrete Time ◽  
Nonnegative Solution ◽  
Return Time ◽  
Additive Functionals ◽  
The Minimal Nonnegative Solution ◽  
First Return Time ◽  
Do So ◽  
Birth Death

In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.

scholarly journals A Fast Newton-Shamanskii Iteration for a Matrix Equation Arising from M/G/1-Type Markov Chains

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Nonnegative Solution ◽  
Initial Guess ◽  
Monotone Convergence ◽  
Nonnegative Matrices ◽  
Schur Decomposition ◽  
Minimal Nonnegative Solution ◽  
Convergence Results ◽  
The Minimal Nonnegative Solution ◽  
Nonlinear Matrix

For the nonlinear matrix equations arising in the analysis of M/G/1-type and GI/M/1-type Markov chains, the minimal nonnegative solution G or R can be found by Newton-like methods. We prove monotone convergence results for the Newton-Shamanskii iteration for this class of equations. Starting with zero initial guess or some other suitable initial guess, the Newton-Shamanskii iteration provides a monotonically increasing sequence of nonnegative matrices converging to the minimal nonnegative solution. A Schur decomposition method is used to accelerate the Newton-Shamanskii iteration. Numerical examples illustrate the effectiveness of the Newton-Shamanskii iteration.

Geometric Markov chains

1995 ◽  
Vol 32 (02) ◽  
pp. 349-374
Author(s):  
William Rising
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Nearest Neighbor ◽  
First Passage Times ◽  
First Passage ◽  
Death Rates ◽  
Passage Times ◽  
Birth Death

A generalization of the familiar birth–death chain, called the geometric chain, is introduced and explored. By the introduction of two families of parameters in addition to the infinitesimal birth and death rates, the geometric chain allows transitions beyond the nearest neighbor, but is shown to retain the simple computational formulas of the birth–death chain for the stationary distribution and the expected first-passage times between states. It is also demonstrated that even when not reversible, a reversed geometric chain is again a geometric chain.

scholarly journals Newton's method for solving a quadratic matrix equation with special coefficient matrices

2014 ◽  
Vol 490 ◽  
pp. 012001
Author(s):  
Sang-Hyup Seo ◽  
Jong Hyun Seo ◽  
Hyun-Min Kim
Keyword(s):  
Newton’S Method ◽  
Matrix Equation ◽  
Newton's Method ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains

2001 ◽  
Vol 38 (2) ◽  
pp. 519-541 ◽  
Author(s):  
Qi-Ming He ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chains ◽  
Nonnegative Solution ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Boundary States ◽  
The Minimal Nonnegative Solution ◽  
Stochastic Solution ◽  
Matrix Geometric Solution ◽  
Technical Issues ◽  
Nonlinear Matrix

We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

Newton’s method for the quadratic matrix equation

2006 ◽  
Vol 182 (2) ◽  
pp. 1772-1779 ◽  
Author(s):  
Yong-Hua Gao
Keyword(s):  
Newton’S Method ◽  
Matrix Equation ◽  
Newton's Method ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix
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