Showing papers on "M-matrix published in 2011"
TL;DR: In this article, the existence and characterization of matrix pth roots of stochastic matrices is studied. But the problem of determining the pth root of a matrix is not addressed.
77 citations
TL;DR: It is proved that, under relatively flexible conditions on the computational parameters of the method, the technique yields bounded numerical approximations for every set of bounded initial estimates, and it is successfully tested by means of computational experiments in some particular instances.
39 citations
TL;DR: In this paper, a class of functionals containing the Schatten q-norms for q ∈ (0, 1) and q < 0 were investigated, and some subadditivity results involving symmetric (unitarily invariant) norms were obtained.
Abstract: Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.
33 citations
01 May 2011
TL;DR: This paper investigates the existence and uniqueness of a fuzzy solution to the fuzzy Sylvester matrix equation and uses the accelerated over-relaxation method to compute an approximate solution to this system.
Abstract: In this paper, we consider the fuzzy Sylvester matrix equation $$AX+XB=C,$$ where $$A\in {\mathbb{R}}^{n \times n}$$ and $$B\in {\mathbb{R}}^{m \times m}$$ are crisp M-matrices, C is an $$n\times m$$ fuzzy matrix and X is unknown. We first transform this system to an $$(mn)\times (mn)$$ fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.
20 citations
TL;DR: By applying the generalized accelerated overrelaxation (GAOR) and the symmetric successive overrelAXation (SSOR) techniques, two class of synchronous matrix multisplitting methods to solve LCP (M,q) are introduced.
12 citations
TL;DR: This paper generalizes Arnal’s algorithms and study the non-stationary matrix multisplitting multi-parameters methods for almost linear systems, finding that the parameters can be adjusted suitably so that the convergence property of methods can be substantially improved.
Abstract: In 1999, Arnal et al. [ Numerical linear algebra and its applications, 6(1999): 79-92] introduced the non-stationary matrix multisplitting algorithms for almost linear systems and studied the convergence of them. In this paper, we generalize Arnal’s algorithms and study the non-stationary matrix multisplitting multi-parameters methods for almost linear systems. The parameters can be adjusted suitably so that the convergence property of methods can be substantially improved. Furthermore, the convergence results of our new method in this paper are weaker than those of Arnal’s. Finally, numerical examples show that our new convergence results are better and more efficient than Arnal’s
11 citations
TL;DR: In this article, a large class of exact solutions of the non-autonomous chiral model equation for an m m matrix function on a two-dimensional space appeared in particular in general relativity.
Abstract: The non-autonomous chiral model equation for an m m matrix function on a two-dimensional space appears in particular in general relativity, where for m = 2 a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for m = 3 solutions of the Einstein{Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Demia nski{
9 citations
TL;DR: In this article, it was shown that the estimation quality of sparse recovery algorithms degrades with the number of sparsest submatrices, and that to obtain a predetermined guaranty on the maximum of ||s_hat-s0||_2, s_hat is needed to be sparse with a better approximation.
Abstract: Let A be an n by m matrix with m>n, and suppose that the underdetermined linear system As=x admits a sparse solution s0 for which ||s0||_0 < 1/2 spark(A). Such a sparse solution is unique due to a well-known uniqueness theorem. Suppose now that we have somehow a solution s_hat as an estimation of s0, and suppose that s_hat is only `approximately sparse', that is, many of its components are very small and nearly zero, but not mathematically equal to zero. Is such a solution necessarily close to the true sparsest solution? More generally, is it possible to construct an upper bound on the estimation error ||s_hat-s0||_2 without knowing s0? The answer is positive, and in this paper we construct such a bound based on minimal singular values of submatrices of A. We will also state a tight bound, which is more complicated, but besides being tight, enables us to study the case of random dictionaries and obtain probabilistic upper bounds. We will also study the noisy case, that is, where x=As+n. Moreover, we will see that where ||s0||_0 grows, to obtain a predetermined guaranty on the maximum of ||s_hat-s0||_2, s_hat is needed to be sparse with a better approximation. This can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where ||s0||_0 grows.
9 citations
TL;DR: The answer is positive, and a tight bound is constructed based on minimal singular values of submatrices of A, which enables us to study the case of random dictionaries and obtain probabilistic upper bounds.
Abstract: Let A be an n × m matrix with m >; n, and suppose that the underdetermined linear system As = x admits a sparse solution S0 for which ||S0||0 <; 1/2 spark( A). Such a sparse solution is unique due to a well-known uniqueness theorem. Suppose now that we have somehow a solution s as an estimation of s0, and suppose that s is only "approximately sparse," that is, many of its components are very small and nearly zero, but not mathematically equal to zero. Is such a solution necessarily close to the true sparsest solution? More generally, is it possible to construct an upper bound on the estimation error ||s - s0||2 without knowing S0? The answer is positive, and in this paper, we construct such a bound based on minimal singular values of submatrices of A. We will also state a tight bound, which is more complicated, but besides being tight, enables us to study the case of random dictionaries and obtain probabilistic upper bounds. We will also study the noisy case, that is, where x = As + n. Moreover, we will see that where ||s0 ||0 grows, to obtain a predetermined guaranty on the maximum of ||s - s0 ||2, s is needed to be sparse with a better approximation. This can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where ||s0||0 grows.
8 citations
Journal Article•
TL;DR: In this article, the authors gave sufficient conditions for generalized diagonally dominant matrices, improved and generalized some related results, and illustrated the advantage of results obtained by a numerical example.
Abstract: In this paper,we gave some sufficient conditions for generalized diagonally dominant matrices,improved and generalized some related resultsThe advantage of results obtained was illustrated by a numerical example
8 citations
Dissertation•
01 Dec 2011
TL;DR: In this article, Ashford et al. proposed a time-varying feedback approach to reach control for affine systems defined on simplices, which is shown to solve RCP for all cases in the literature where continuous state feedback fails.
Abstract: A Time-Varying Feedback Approach to Reach Control on a Simplex Graeme Ashford Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2011 This thesis studies the Reach Control Problem (RCP) for affine systems defined on simplices. The thesis focuses on cases when it is known that the problem is not solvable by continuous state feedback. Previous work has proposed (discontinuous) piecewise affine feedback to resolve the gap between solvability by open-loop controls and solvability by feedbacks. The first results on solvability by time-varying feedback are presented. Timevarying feedback has the advantage to be more robust to measurement errors circumventing problems of discontinuous controllers. The results are theoretically appealing in light of the strong analogies with the theory of stabilization for linear control systems. The method is shown to solve RCP for all cases in the literature where continuous state feedback fails, provided it is solvable by open loop control. Textbook examples are provided. The motivation for studying RCP and its relevance to complex control specifications is illustrated using a material transfer system.
TL;DR: A direct algorithm for the solution to the affine two-sided obstacle problem with an M-matrix has the polynomial bounded computational complexity O(n3) and is more efficient than those in (Numer. Algebra Appl. 2006).
Abstract: A direct algorithm for the solution to the affine two-sided obstacle problem with an M-matrix is presented. The algorithm has the polynomial bounded computational complexity O(n3) and is more efficient than those in (Numer. Linear Algebra Appl. 2006; 13:543–551). Copyright © 2010 John Wiley & Sons, Ltd.
TL;DR: In this paper, the authors characterized all central linear generalized polynomial (GP) for a division algebra and determined the lengths of all central GP for A, where A = M m (D ), the m × m matrix ring over D.
TL;DR: In this paper, the equivalence between two sets of m + 1 bipartite quantum states under local unitary transformations was studied and a randomized polynomial-time algorithm was proposed to solve the problem with an arbitrarily high success probability.
Abstract: In this brief report, we consider the equivalence between two sets of m + 1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree m matrix polynomials are unitarily equivalent; i.e. UAiV† = Bi for 0 ≤ i ≤ m where U and V are unitary and (Ai, Bi) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices U and V.
TL;DR: In this paper, the existence and uniqueness theorem of global solutions to a class of neutral stochastic differential equations with unbounded delay was established without the linear growth condition, by the use of Lyapunov function, and examined the pathwise stability of this solution with general decay rate.
Abstract: Without the linear growth condition, by the use of Lyapunov function, this paper establishes the existence-and-uniqueness theorem of global solutions to a class of neutral stochastic differential equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.
TL;DR: In particular, when perturbing the second diagonals (elements ( l, l + 2 ) and ( l, l - 2 ) of M, these sufficient bounds are shown to be the actual maximum allowable perturbations.
Journal Article•
TL;DR: In this article, it was shown that a real matrix has positive real eigenvalues and some sufficient and necessary conditions and sufficient conditions were obtained, and a method was also pointed to count the number of different real negative eigen values of a general matrix by using Sturm theorem.
Abstract: It was studied in this paper that a real matrix had positive real eigenvalues.Some sufficient and necessary conditions and sufficient conditions were obtained.Some methods were given to assert a special matrix which has positive real eigenvalues.A method was also pointed to count the number of different real negative eigenvalues of a general matrix by using Sturm theorem.
TL;DR: In this article, the question of whether a 4× 4 SPP-matrix is a P-mixture was investigated and it was shown that it is not true for matrices of order greater than 3.
Abstract: It is known that inverse M-matrices are strict path product (SPP) matrices, and that the converse is not true for matrices of order greater than 3. In this paper, given a normalized SPP-matrix A, some new values sfor which A+sI is an inverse M-matrix are obtained. Our values sextend the values s given by Johnson and Smith (C.R. Johnson and R.L. Smith. Positive, path product, and inverse M-matrices. Linear Algebra Appl., 421:328-337, 2007.). The question whether or not a 4× 4 SPP-matrix is a P-matrix is settled.
27 May 2011
TL;DR: This paper first discusses the method of diagonal transformation, an algorithm for the spectral radius of irreducible nonnegative matrices, and an algorithms for the minimal eigenvalue of an M-matrix is constructed afterwards.
Abstract: The M-matrix is an important kind of structured matrix with many applications in various physics and engineering problems. In this paper, we focus on the minimal eigenvalue of an M-matrix. We first discuss the method of diagonal transformation, an algorithm for the spectral radius of irreducible nonnegative matrices. Based on it, an algorithm for the minimal eigenvalue of an M-matrix is constructed afterwards. A numerical example is given at last, showing the feasibility and validity of the algorithm presented.
29 May 2011
TL;DR: By applying vector Lyapunov function method and M matrix theory which are different from all the existing study methods, some sufficient conditions ensuring stochastic exponential stability of the equilibrium point of a class of Cohen-Grossberg neural networks with Markovian jumping parameters and mixed delays are derived.
Abstract: In this paper by applying vector Lyapunov function method and M matrix theory which are different from all the existing study methods (LMI technique), some sufficient conditions ensuring stochastic exponential stability of the equilibrium point of a class of Cohen-Grossberg neural networks with Markovian jumping parameters and mixed delays are derived.
TL;DR: It is shown that the homogeneous linear system Ax=0 has Ω(|GN|m−k) monochromatic solutions for each r-coloring of GN\{0} and sufficiently large N.
TL;DR: A preconditioned AOR iterative method for solving the systems of linear equations with M-matrix coefficient and some numerical results are given to compare the proposed preconditionser with an available precONDitioner.
Abstract: In this paper, we propose a preconditioned AOR iterative method for solving the systems of linear equations with M-matrix coefficient. Some numerical results are given to compare the proposed preconditioner with an available preconditioner.
15 Apr 2011
TL;DR: In this article, the global exponential stability of a class of stochastic linear interconnected large-scale systems with time delay was analyzed based on M-matrix theory and box theory, by constructing a vector Lyapunov function.
Abstract: The global exponential stability of a class of stochastic linear interconnected large-scale systems with time delay was analyzed based on M-matrix theory and box theory, by constructing a vector Lyapunov function. A criterion was concluded for global exponential stability of the systems by analyzing the stability of differential inequalities. It is obtained that a stochastic large-scale system with time delay is described as global exponential stable if the test matrix is an M-matrix, where the test matrix is constructed by employing the coefficient matrices of the system and the solutions of the Lyapunov equations which are interconnected with the system. The computation is convenient, so it is easy for application.