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Abstract: Markov chains are commonly used in modeling many practical systems such as queuing systems, manufacturing systems and inventory systems. They are also effective in modeling categorical data sequences. In a conventional th order multivariate Markov chain model of chains, and each chain has the same set of states, the total number of parameters required to set up the model is . Such huge number of states discourages researchers or practitioners from using them directly. In this paper, we propose an th-order multivariate Markov chain model for modeling multiple categorical data sequences such that the total number of parameters are of . The proposed model requires significantly less parameters than the conventional one. We develop efficient estimation methods for the model parameters. An application to demand predictions in inventory control is also discussed. [Copyright &y& Elsevier]

This paper is concerned with the localization of the Perron root of a nonnegative irreducible matrix A. A new localization method that utilizes the relationship between the Perron root of a nonnegative matrix and the estimates of the row sums of its generalized Perron complement is presented. The method is efficient because it gives the bounds on ρ(A) only by computing the estimates of the row sums of the generalized Perron complement rather than the generalized Perron complement itself. Several numerical examples are given to illustrate the effectiveness of our method. [Copyright &y& Elsevier]

The symmetric Sinc–Galerkin method applied to a sparable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations(Ψx⊗Dy+Dx⊗Ψy)u=g,where⊗ is the Kronecker product symbol, Ψx and Ψy are Toeplitz-plus-diagonal matrices, and Dx and Dy are diagonal matrices. The main contribution of this paper is to present and analyze a two-step preconditioning strategy based on the banded matrix approximation (BMA) and the alternating direction implicit (ADI) iteration for these Sinc–Galerkin systems. In particular, we show that the two-step preconditioner is symmetric positive definite, and the condition number of the preconditioned matrix is bounded by the convergence factor of the involved ADI iteration. Numerical examples show that the new preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc–Galerkin linear system. [Copyright &y& Elsevier]

In 1992, Labahn and Shalom showed that the inverse of a Toeplitz matrix can be reconstructed by two of its columns and by some entries of the original Toeplitz matrix. We here present a modification of this result and another (shorter) proof. [Copyright &y& Elsevier]

The main aim of this paper is to generalize matrix numerical ranges to the tensor case based on tensor norms. We show that the basic properties of matrix numerical ranges such as compactness and convexity are valid for tensor numerical ranges. We make use of convexity property to propose an algorithm for approximating tensor numerical ranges in which tensor eigenvalues are contained. Also we consider tensor numerical ranges based on inner products, however, they may not be convex in general. [ABSTRACT FROM AUTHOR]

Primitivity is an important concept in the spectral theory of nonnegative matrices and tensors. It is well-known that an irreducible matrix is primitive if and only if the greatest common divisor of all the cycles in the associated directed graph is equal to 1. The main aim of this paper is to establish a similar result, i.e., we show that a nonnegative tensor is primitive if and only if the greatest common divisor of all the cycles in the associated directed hypergraph is equal to 1. [ABSTRACT FROM AUTHOR]

Abstract: We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately. [Copyright &y& Elsevier]

In this paper, we study the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors from multisensors. Preconditioned conjugate gradient methods with cosine transform based preconditioners and incomplete factorization based preconditioners are applied to solve this image reconstruction problem. Numerical examples are given to demonstrate the efficiency of these preconditioners. We find that cosine transform based preconditioners are effective when the number of shifted low-resolution frames are large, but are less effective when the number is small. However, incomplete factorization based preconditioners work quite well independent of the number of shifted low-resolution frames. [Copyright &y& Elsevier]

In this paper, we discuss discrete wavelet transforms for Toeplitz matrices and block–Toeplitz–Toeplitz–block matrices. The main contribution of this paper is to give the Toeplitz-like structure of the wavelet transformed Toeplitz matrices, and show that the computational cost for such structure is O(k3ln) where n is the size of the Toeplitz matrix, k is the order of the wavelet and l is the level used in the wavelet transform. The comparison between the wavelet transformed Toeplitz matrices and the Fourier transformed Toeplitz matrices is also given. [Copyright &y& Elsevier]