Relaxed functional iteration techniques for the numerical solution of M/G/1 type Markov chains

We introduce a new iterative method for the computation of the minimal nonnegative solutionG of the matrix equation , arising in the numerical solution of M/G/1 type Markov chains. The idea consists in applying a relaxation technique to customarily used functional iteration formulas. The proposed method is easy to implement and outperforms, in terms of number of iterations and execution time, the standard functional iteration techniques.     

Local convergence analysis of iterative aggregation-disaggregation methods with polynomial correction

The paper introduces some new results on local convergence analysis of one class of iterative aggregation-disaggregation methods for computing a stationary probability distribution vector of an irreducible stochastic matrix. We focus on methods, where the basic iteration on the fine level corresponds to a multiplication by a polynomial of order one with nonnegative coefficients in the original matrix. We show that this process is locally convergent for matrices with positive diagonals or when the coefficients of the polynomial are positive. On the other hand there are examples for which the process may diverge in a local sense for higher degree polynomials even if it converges for a polynomial of a lower degree for the same matrix.     

The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.     

Comparison theorems for splittings of matrices

Summary. This paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices. On the basis of nonnegative matrix theory, comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived. When the coefficient matrix A is Hermitian positive (semi)definite, comparison theorems about two P-regular splittings and two parallel multisplitting methods are proved. Received April 4, 1998 / Revised version received October 18, 1999 / Published online November 15, 2001     

On the inverse mean first passage matrix problem and the inverse M-matrix problem

The inverse mean first passage time problem is given a positive matrix M∈R^n,n, then when does there exist an n-state discrete-time homogeneous ergodic Markov chain C, whose mean first passage matrix is M? The inverse M-matrix problem is given a nonnegative matrix A, then when is A an inverse of an M-matrix. The main thrust of this paper is to show that the existence of a solution to one of the problems can be characterized by the existence of a solution to the other. In so doing we extend earlier results of Tetali and Fiedler.     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996     

Monotonicity of some perturbations of irreducibly diagonally dominant M-matrices

This paper presents a new result concerning the perturbation theory of M-matrices. We give the proof of a theorem showing that some perturbations of irreducibly diagonally dominant M-matrices are monotone, together with an explicit bound of the norm of the perturbation. One of the assumptions concerning the perturbation matrix is that the sum of the entries of each of its row is nonnegative. The resulting matrix is shown to be monotone, although it may not be diagonally dominant and its off diagonal part may have some positive entries. We give as an application the proof of the second order convergence of an non-centered finite difference scheme applied to an elliptic boundary value problem.     

On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

Summary A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.     

Theorems on Schur complement of block diagonally dominant matrices and their application in reducing the order for the solution of large scale linear systems

We firstly consider the block dominant degree for I-(II-)block strictly diagonally dominant matrix and their Schur complements, showing that the block dominant degree for the Schur complement of an I-(II-)block strictly diagonally dominant matrix is greater than that of the original grand block matrix. Then, as application, we present some disc theorems and some bounds for the eigenvalues of the Schur complement by the elements of the original matrix. Further, by means of matrix partition and the Schur complement of block matrix, based on the derived disc theorems, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and the numerical example illustrates that the iteration can compute out the results faster.     

Nonnegative matrices with prescribed spectrum and elementary divisors

In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors.     

The case of equality in the Dobrushin-Deutsch-Zenger bound

Suppose that A=(a_i,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.     

A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation

In this paper, we propose a structure-preserving doubling algorithm (SDA) for the computation of the minimal nonnegative solution to the nonsymmetric algebraic Riccati equation (NARE), based on the techniques developed for the symmetric cases. This method allows the simultaneous approximation to the minimal nonnegative solutions of the NARE and its dual equation, requiring only the solutions to two linear systems and several matrix multiplications per iteration. Similar to Newton's method and the fixed-point iteration methods for solving NAREs, we also establish global convergence for SDA under suitable conditions, using only elementary matrix theory. We show that sequences of matrices generated by SDA are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the NARE and its dual equation. Numerical experiments show that the SDA algorithm is feasible and effective, and outperforms Newton's iteration and the fixed-point iteration methods. This research was supported in part by RFDP (20030001103) & NSFC (10571007) of China and the National Center for Theoretical Sciences in Taiwan. This author's research was supported by NSFC grant 1057 1007 and RFDP grant 200300001103 of China.     

Some iterative methods for the largest positive definite solution to a class of nonlinear matrix equation

In this paper, we propose some inversion-free iteration methods for finding the largest positive definite solution of a class of nonlinear matrix equation. Then, we consider the properties of the solution for this nonlinear matrix equation. Also, we establish Newton’s iteration method for finding the largest positive definite solution and prove its quadratic convergence. Furthermore, we derive the semi-local convergence of the Newton’s iteration method. Finally, some numerical examples are presented to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.     

一类线性约束矩阵不等式及其最小二乘问题

本文主要考虑一类线性矩阵不等式及其最小二乘问题,它等价于相应的矩阵不等式最小非负偏差问题.之前相关文献提出了求解该类最小非负偏差问题的迭代方法,但该方法在每步迭代过程中需要精确求解一个约束最小二乘子问题,因此对规模较大的问题,整个迭代过程需要耗费巨大的计算量.为了提高计算效率,本文在现有算法的基础上,提出了一类修正迭代方法.该方法在每步迭代过程中利用有限步的矩阵型LSQR方法求解一个低维矩阵Krylov子空间上的约束最小二乘子问题,降低了整个迭代所需的计算量.进一步运用投影定理以及相关的矩阵分析方法证明了该修正算法的收敛性,最后通过数值例子验证了本文的理论结果以及算法的有效性.     

On algebraic Riccati equations associated with M-matrices

We consider the algebraic Riccati equation for which the four coefficient matrices form an M-matrix K. When K is a nonsingular M-matrix or an irreducible singular M-matrix, the Riccati equation is known to have a minimal nonnegative solution and several efficient methods are available to find this solution. In this paper we are mainly interested in the case where K is a reducible singular M-matrix. Under a regularity assumption on the M-matrix K, we show that the Riccati equation still has a minimal nonnegative solution. We also study the properties of this particular solution and explain how the solution can be found by existing methods.     

On the S-matrix conjecture

Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0,1]$[0,1]$. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. His proof is based on the Kiefer–Wolfowitz equivalence theorem from the approximate theory of optimal design. In this note we give a short and simple proof of his result.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

Temperley-Lieb Immanants

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form . The cone generated by these polynomials contains all totally nonnegative polynomials of the form , where, are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative. Received September 30, 2004     

Variational iteration method—Some recent results and new interpretations

This paper is an elementary introduction to the concepts of variational iteration method. First, the main concepts in variational iteration method, such as general Lagrange multiplier, restricted variation, correction functional, are explained heuristically. Subsequently, the solution procedure is systematically addressed, in particular, for nonlinear oscillators. Particular attention is paid throughout the paper to give an intuitive grasp for the method. The main motivation is to put things together in a convenient form for later reference and systematic use.