Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R >1

2012 ◽  
Vol 44 (3) ◽  
pp. 533-544 ◽  
Author(s):  
Robert D. Foley ◽  
David R. McDonald
Keyword(s):  
Harmonic Function ◽  
Nonnegative Matrix ◽  
Convergence Parameter ◽  
Irreducible Nonnegative Matrix

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376 ◽  
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

scholarly journals Fast verification for the Perron pair of an irreducible nonnegative matrix

2021 ◽  
Vol 37 ◽  
pp. 402-415
Author(s):  
Shinya Miyajima
Keyword(s):  
Linear System ◽  
Error Bounds ◽  
Computational Efficiency ◽  
Nonnegative Matrix ◽  
Fast Algorithms ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Perron Root ◽  
Irreducible Nonnegative Matrix

Fast algorithms are proposed for calculating error bounds for a numerically computed Perron root and vector of an irreducible nonnegative matrix. Emphasis is put on the computational efficiency of these algorithms. Error bounds for the root and vector are based on the Collatz--Wielandt theorem, and estimating a solution of a linear system whose coefficient matrix is an $M$-matrix, respectively. We introduce a technique for obtaining better error bounds. Numerical results show properties of the algorithms.

scholarly journals Quasistochastic matrices and Markov renewal theory

2014 ◽  
Vol 51 (A) ◽  
pp. 359-376
Author(s):  
Gerold Alsmeyer
Keyword(s):  
Probabilistic Approach ◽  
Nonnegative Matrix ◽  
Renewal Theory ◽  
Distribution Functions ◽  
Markov Renewal Equation ◽  
Markov Random Walk ◽  
The Matrix ◽  
Left And Right ◽  
Irreducible Nonnegative Matrix ◽  
Markov Renewal Theory

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

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