Jianhong Xu

On the stability of the computation of the stationary probabilities of Markov chains using Perron complements

For an n-state ergodic homogeneous Markov chain whose transition matrix is T∈ℝn,n, it has been shown by Meyer on the one hand and by Kirkland, Neumann, and Xu on the other hand that the stationary distribution vector and that the mean first passage matrix, respectively, can be computed by a divide and conquer parallel method from the Perron complements of T. This is possible due to the facts, shown by Meyer, that the Perron complements of T are themselves transition matrices for finite ergodic homogeneous Markov chains with fewer states and that their stationary distribution vectors are multiples of the corresponding subvectors of the stationary distribution vector of the entire chain. Here we examine various questions concerning the stability of computing the stationary distribution vectors of the Perron complements and compare them with the stability of computing the stationary distribution vector for the entire chain. In particular, we obtain that condition numbers which are related to the coefficients of ergodicity improve as we pass from the entire chain to the chains associated with its Perron complements. Copyright

A divide and conquer approach to computing the mean first passage matrix for Markov chains via Perron complement reductions

Let MT be the mean first passage matrix for an n-state ergodic Markov chain with a transition matrix T. We partition T as a 2×2 block matrix and show how to reconstruct MT efficiently by using the blocks of T and the mean first passage matrices associated with the non-overlapping Perron complements of T. We present a schematic diagram showing how this method for computing MT can be implemented in parallel. We analyse the asymptotic number of multiplication operations necessary to compute MT by our method and show that, for large size problems, the number of multiplications is reduced by about 1/8, even if the algorithm is implemented in serial. We present five examples of moderate sizes (of orders 20–200) and give the reduction in the total number of flops (as opposed to multiplications) in the computation of MT. The examples show that when the diagonal blocks in the partitioning of T are of equal size, the reduction in the number of flops can be much better than 1/8. Copyright

Proof of Stenger's conjecture on matrix I ( - 1 ) of Sinc methods

In this paper, we prove a conjecture, which was proposed by Frank Stenger in 1997, concerning the localization of eigenvalues of the Sinc matrix I^(^-^1^), a problem that is important in both the theory and the practice of Sinc methods. In 2003, Iyad Abu-Jeib and Thomas Shores established a partial answer to this unsolved problem. The techniques they have developed, however, turn out to be the key that finally leads to the settlement here of Stengers conjecture.

Convexity and Elasticity of the Growth Rate in Size-Classified Population Models

This paper investigates both the convexity and elasticity of the growth rate of size-classsified population models. For an irreducible population projection matrix, we discuss the convexity properties of its Perron eigenvalue under perturbation of the vital rates, extending work of Kirkland and Neumann on Leslie matrices. We also provide nonnegative attainable lower bounds on the derivatives of the elasticity of the Perron eigenvalue under perturbation of the vital rates, sharpening, in the context of population projection matrices, the main result of Kirkland, Neumann, Ormes, and Xu.

On the Elasticity of the Perron Root of a Nonnegative Matrix

Let A=(ai,j) be an n × n nonnegative irreducible matrix whose Perron root is

Technical communique: A characterization of the generalized spectral radius with Kronecker powers

\lambda

MARKOV CHAIN SMALL-WORLD MODEL WITH ASYMMETRIC TRANSITION PROBABILITIES ∗

. The quantity

ON THE TRACE CHARACTERIZATION OF THE JOINT SPECTRAL RADIUS

e_{i,j}=\frac{a_{i,j}}{\lam}\frac{\partial \lambda}{\partial a_{i,j}}

On the roots of certain polynomials arising from the analysis of the Nelder–Mead simplex method

is known as the elasticity of

Unified, improved matrix upper bound on the solution of the continuous coupled algebraic Riccati equation

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