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.

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

2001 ◽  
Vol 38 (02) ◽  
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.

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.

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 Sensitivity Analysis of the Matrix Equation from Interpolation Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Li ◽  
Yuhai Zhang
Keyword(s):  
Sensitivity Analysis ◽  
Approximate Solution ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
Backward Error ◽  
Perturbation Bound ◽  
Interpolation Problems ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Theoretical Results

This paper studies the sensitivity analysis of a nonlinear matrix equation connected to interpolation problems. The backward error estimates of an approximate solution to the equation are derived. A residual bound of an approximate solution to the equation is obtained. A perturbation bound for the unique solution to the equation is evaluated. This perturbation bound is independent of the exact solution of this equation. The theoretical results are illustrated by numerical examples.

An iterative method to solve a nonlinear matrix equation

2016 ◽  
Vol 31 ◽  
pp. 620-632
Author(s):  
Peng Jingjing ◽  
Liao Anping ◽  
Peng Zhenyun
Keyword(s):  
Approximate Solution ◽  
Iterative Method ◽  
Matrix Equation ◽  
Closed Ball ◽  
Open Ball ◽  
Nonlinear Matrix Equation ◽  
Initial Matrix ◽  
Convergence Results ◽  
The Matrix ◽  
Nonlinear Matrix

n this paper, an iterative method to solve one kind of nonlinear matrix equation is discussed. For each initial matrix with some conditions, the matrix sequences generated by the iterative method are shown to lie in a fixed open ball. The matrix sequences generated by the iterative method are shown to converge to the only solution of the nonlinear matrix equation in the fixed closed ball. In addition, the error estimate of the approximate solution in the fixed closed ball, and a numerical example to illustrate the convergence results are given.

scholarly journals Comment on “Perturbation Analysis of the Nonlinear Matrix Equation ”

2013 ◽  
Vol 2013 ◽  
pp. 1-2 ◽  
Author(s):  
Maher Berzig ◽  
Erdal Karapınar
Keyword(s):  
Perturbation Analysis ◽  
Matrix Equation ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Nonlinear Matrix

We show that the perturbation estimate for the matrix equation due to J. Li, is wrong. Our discussion is supported by a counterexample.

scholarly journals Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

Iteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation

2018 ◽  
Vol 34 ◽  
pp. 217-230
Author(s):  
Syed M. Raza Shah Naqvi ◽  
Jie Meng ◽  
Hyun-Min Kim
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Condition Numbers ◽  
Symmetric Positive Definite Matrix ◽  
Nonlinear Matrix Equation ◽  
Symmetric Positive Definite ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

In this paper, the nonlinear matrix equation $X^p+A^TXA=Q$, where $p$ is a positive integer, $A$ is an arbitrary $n\times n$ matrix, and $Q$ is a symmetric positive definite matrix, is considered. A fixed-point iteration with stepsize parameter for obtaining the symmetric positive definite solution of the matrix equation is proposed. The explicit expressions of the normwise, mixed and componentwise condition numbers are derived. Several numerical examples are presented to show the efficiency of the proposed iterative method with proper stepsize parameter and the sharpness of the three kinds of condition numbers.

scholarly journals Existence and Uniqueness of the Positive Definite Solution for the Matrix EquationX=Q+A∗(X^−C)−1A

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Dongjie Gao
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Positive Semidefinite ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Block Diagonal Matrix ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

We consider the nonlinear matrix equationX=Q+A∗(X^−C)−1A, whereQis positive definite,Cis positive semidefinite, andX^is the block diagonal matrix defined byX^=diag(X,X,…,X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

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