Search

In this paper, we propose several improved uniqueness conditions of solution for multilinear PageRank with order-3 irreducible stochastic tensors. By considering upper and lower bounds of solutions, the refined uniqueness conditions are more effective than the existing ones. Furthermore, as an application, sharp perturbation bounds for multilinear PageRank are also developed basing on the new uniqueness conditions. Numerical examples are given to verify the efficiency of the proposed uniqueness conditions. [ABSTRACT FROM AUTHOR]

This paper concerns with measures of the sensitivity of a multiple nonzero finite generalized singular value of two matrices having the same number of columns. Our results show that a multiple nonzero generalized singular value generally has multiple condition numbers and their explicit expressions are derived. A numerical example is used to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized eigenvalue problem Ax = λBx arising from economic models. Under certain conditions, there is a simple generalized eigenvalue ρ(A, B) in the interval (0, 1) with a positive eigenvector. Based on the Noda iteration, a modified Noda iteration (MNI) and a generalized Noda iteration (GNI) are proposed for finding the generalized eigenvalue ρ(A, B) and the associated unit positive eigenvector. It is proved that the GNI method always converges and has a quadratic asymptotic convergence rate. So GNI has a similar convergence behavior as MNI. The efficiency of these algorithms is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we introduce the variant tensor splittings, and present some equivalent conditions for a strong M -tensor based on the tensor splitting. Secondly, the existence and uniqueness conditions of the solution for multi-linear systems are given. Thirdly, we propose some tensor splitting algorithms for solving multi-linear systems with coefficient tensor being a strong M -tensor. As an application, a tensor splitting algorithm for solving the multi-linear model of higher order Markov chains is proposed. Numerical examples are given to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we consider the eigenvalue variation for any perturbation of Hermitian matrices, and we obtain two perturbation bounds. The first bound always improves the existing bound, and the second bound also improves the existing one under a suitable condition. A simple example is given for comparing these bounds. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give some structured perturbation bounds for generalized saddle point matrices and Hermitian block tridiagonal matrices. Our bounds improve some existing ones. In particular, the proposed bounds reveal the sensitivity of the eigenvalues with respect to perturbations of different blocks. Numerical examples confirm the theoretical results. [Copyright &y& Elsevier]

We consider a discrete-time orthogonal spline collocation scheme for solving Schrödinger equation with wave operator. The scheme is proposed recently by Wang et al. (J Comput Appl Math 235 (2011), 1993-2005) and is showed to have high-order convergence rate when a parameter θ in the scheme is not less than \documentclass{article} \usepackage{amsmath,amsfonts, amssymb}\pagestyle{empty}\begin{document}$\frac{1}{4}$\end{document}. In this article, we show that the result can be extended to include \documentclass{article} \usepackage{amsmath,amsfonts, amssymb}\pagestyle{empty}\begin{document}$\theta\in(0,\frac{1}{4})$\end{document} under an assumption. Numerical example is given to justify the theoretical result. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

Abstract: Error estimate plays an important role for numerical integration. Some Ostrowski-like type inequalities concerning a new type of quadrature formula are established recently by Huy and Ngô (2009, 2010) . In this note, improvement and extension to these inequalities are given. [Copyright &y& Elsevier]

Abstract: It is known that for any nonzero complex matrices and the quotient of Frobenius normsdoes not exceed . However, except for some special cases, only necessary conditions for attaining this bound have been found so far. We will completely characterize the pairs of matrices that satisfy equality with the quotient’s maximum. [Copyright &y& Elsevier]

Abstract: Recently, Lu and Hurvich [Y. Lu, C. Hurvich, On the complexity of the preconditioned conjugate gradient algorithm for solving toeplitz systems with a Fisher–Hartwig singularity, SIAM J. Matrix Anal. Appl. 27 (2005) 638–653] used the preconditioned conjugate gradient method with the optimal circulant preconditioner proposed in Chan [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988) 766–771] for solving the Toeplitz system where the generating function f is given bywith . The function is positive continuous on and differentiable on . In this paper, we will use the superoptimal circulant preconditioner proposed by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459–473] to solve the same problem when . Our convergence analysis shows that the number of iterations is bounded by and therefore the complexity of our algorithm is . We notice that the numerical performance of superoptimal preconditioner is almost the same as that of the optimal preconditioner. [Copyright &y& Elsevier]

Abstract: The Euler equations with frictional force have been extensively studied. Since the Boltzmann equation is closely related to the equations of gas dynamics, we study, in this paper, the Boltzmann equation with frictional force when the external force is proportional to the macroscopic velocity. It is shown that smooth initial perturbation of a given global Maxwellian leads to a unique global-in-time classical solution which approaches to the global Maxwellian time asymptotically. The analysis is based on the macro–micro decomposition for the Boltzmann equation introduced in Liu et al. [Energy method for the Boltzmann equation, Physica D 188 (3–4) (2004) 178–192] and Liu and Yu [Boltzmann equation: micro–macro-decompositions and positivity of shock profiles, Comm. Math. Phys. 246(1) (2004) 133–179] through energy estimates. [Copyright &y& Elsevier]

This is a continuation of the paper (Comm. Math. Phys. 230 (2002) 329) on the study of the compressible Navier–Stokes equations for isentropic flow when the initial density connects to vacuum continuously. The degeneracy appears in the initial data and has effect on the viscosity coefficient because the coefficient is assumed to be a power function of the density. This assumption comes from physical consideration and it also gives the well-posedness of the Cauchy problem. A new global existence result is established by some new a priori estimates so that the interval for the power of the density in the viscosity coefficient is enlarged to (0,1/3). [Copyright &y& Elsevier]

Well-conditioned matrices are often required in science and engineering, such as signal processing and finance. Problem to find the nearest positive definite matrix by explicitly imposing a constraint on the condition number are considered in this paper. A new algorithm based on geometric perspective is proposed for getting the required well-conditioned matrix. Based on these, a condition number-constrained matrix minimization problem is further considered, where the constraints are imposed for avoiding degenerate solutions in which parameter matrices become rank deficient. An inexact version of alternating direction method with truly implementable inexactness criteria is proposed for solving this problem. Numerical experiments, including comparison with some existing methods, are performed to illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the multilinear PageRank problem. Under the framework of tensor, we establish several new and tighter uniqueness conditions for the multilinear PageRank vector. Meanwhile, a refined error bound for the inverse iteration as well as the new perturbation bounds under different norms, which improve the existing ones in the current literature, are developed with feasible computations. Several numerical examples are given to validate the significant effectiveness of the proposed bounds. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new preconditioned SOR method for solving the multi-linear systems whose coefficient tensor is an M -tensor. The corresponding comparison for spectral radii of iterative tensors is given. Numerical examples demonstrate the efficiency of the proposed preconditioned methods. [ABSTRACT FROM AUTHOR]

It is known that the spectral radius of the iterative tensor can be seen as an approximate convergence rate for solving multi-linear systems by tensor splitting iterative methods. So in this paper, first we give some spectral radius comparisons between two different iterative tensors. Then, we propose the preconditioned tensor splitting method for solving multi-linear systems, which provides an alternative algorithm with the choice of a preconditioner. In particular, also we give some spectral radius comparisons between the preconditioned iterative tensor and the original one. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the H -eigenpairs and Z -eigenpairs of a tensor. By estimating the ratio of the smallest and largest entries in the Perron vector, we present some sharper bounds for the H -spectral radius of a nonnegative irreducible tensor. Similarly, we also obtain some Z -spectral radius bounds of an irreducible weakly symmetric tensor. In addition, by using the technique of matricizing, several bounds for the Z -spectrum are derived. These proposed bounds improve some existing ones, and some numerical examples are given to show the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the parameter matrix compared to the recent results. Numerical examples show that the proposed method is efficient with the two-step technique and confirm the improvement of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Z -eigenpair of a tensor, in particular, an irreducible nonnegative tensor. We present some bounds for the eigenvector and Z -spectral radius. The proposed bounds improve some existing ones. An example with practical applications is given to show the proposed bound. [ABSTRACT FROM AUTHOR]

An implicit second-order finite difference scheme, which is unconditionally stable, is employed to discretize fractional advection–diffusion equations with constant coefficients. The resulting systems are full, unsymmetric, and possess Toeplitz structure. Circulant and skew-circulant splitting iteration is employed for solving the Toeplitz system. The method is proved to be convergent unconditionally to the solution of the linear system. Numerical examples show that the convergence rate of the method is fast. [ABSTRACT FROM PUBLISHER]

Abstract: We use the normalized preconditioned conjugate gradient method with Strang’s circulant preconditioner to solve a nonsymmetric Toeplitz system , which arises from the discretization of a partial integro-differential equation in option pricing. By using the definition of family of generating functions introduced in , we prove that Strang’s circulant preconditioner leads to a superlinear convergence rate under certain conditions. Numerical results exemplify our theoretical analysis. [Copyright &y& Elsevier]

Abstract: In this short note, it is proved that given any positive definite Hermitian matrix, the eigenvalues of the superoptimal preconditioned matrix do not exceed the corresponding eigenvalues of the optimal preconditioned matrix. [Copyright &y& Elsevier]

In this paper, we present an inexact generalized Noda iteration (IGNI) for finding the Perron pair of the generalized eigenproblem arising from economic models. We prove that the approximate eigenvalue sequence generated by IGNI converges globally linearly; Furthermore, we also propose an enhanced inexact generalized Noda iteration(EIGNI), and prove that the approximate eigenvector sequence generated by EIGNI algorithms converges superlinearly. Numerical examples illustrate that the proposed IGNI and EIGNI algorithms are efficient. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we propose several relaxation algorithms for solving the tensor equation arising from the higher‐order Markov chain and the multilinear PageRank. The semi‐symmetrization technique on the original equation is also employed to modify the proposed algorithms. The convergence analysis is given for the proposed algorithms. It is shown that the new algorithms are more efficient than the existing ones by some numerical experiments when relaxation parameters are chosen suitably. [ABSTRACT FROM AUTHOR]

We discuss the uniqueness and the perturbation analysis for sparse non-negative tensor equations arriving from data sciences. By two different techniques, we may get better ranges of parameters to guarantee the uniqueness of the solution of the tensor equation. On the other hand, we present some perturbation bounds for the tensor equation. Numerical examples are given to show the efficiency of the theoretical results. [ABSTRACT FROM AUTHOR]

Summary: The finite‐difference method applied to the time‐fractional subdiffusion equation usually leads to a large‐scale linear system with a block lower triangular Toeplitz coefficient matrix. The approximate inversion method is employed to solve this system. A sufficient condition is proved to guarantee the high accuracy of the approximate inversion method for solving the block lower triangular Toeplitz systems, which are easy to verify in practice and have a wide range of applications. The applications of this sufficient condition to several existing finite‐difference schemes are investigated. Numerical experiments are presented to verify the validity of theoretical results. [ABSTRACT FROM AUTHOR]

The uniqueness of multilinear PageRank vectors is discussed, and the new uniqueness condition is given. The new results are better than the one given in the work of Gleich et al. published in SIAM J Matrix Anal Appl. 2015;36;1409-1465. Numerical examples are given to demonstrate the new theoretical results. [ABSTRACT FROM AUTHOR]