System Of Fixed-point Equations Research Articles

RELAXED MODULUS-BASED SYNCHRONOUS MULTISPLITTING MULTI-PARAMETERS TOR (TWO-PARAMETERS OVER-RELAXATION) METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

In 2013, Bai and Zhang [<i>Numerical Linear Algebra with Applications</i>, 20(2013):425-439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problems into a system of fixed-point equations and studied the convergence of them. In 2014, Zhang and Li [<i>Computers and Mathematics with Application</i>, 67(2014):1954-1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In 2008, Zhang et.al. [<i>International Journal of Computer Mathematics</i>, 85(2)(2008):211-224] presented global relaxed non-stationary multisplitting multi-parameter method by introducing some relaxed parameters. In this paper, we generalize Bai and Zhang's methods and study relaxed modulus-based synchronous multisplitting multi-parameters TOR (two-parameters over-relaxation, abbreviated as TOR) methods for linear complementarity problems. Furthermore, the convergence results of our new method in this paper are given when the system matrix is an $H_{+}-$matrix.

GLOBAL RELAXED MODULUS-BASED SYNCHRONOUS BLOCK MULTISPLITTING MULTI-PARAMETERS METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425Ƀ439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307Ƀ312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954Ƀ1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multi-parameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block $ H_{+}- $matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97Ƀ110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.

Relaxed Modulus-Based Synchronous Multisplitting Multi-Parameter Methods for Linear Complementarity Problems

In 2013, Bai and Zhang (Numer Linear Algebra Appl, 20:425–439 2013) constructed modulus-based synchronous multisplitting iteration methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them. In this paper, we generalize Bai and Zhang’s method and study relaxed modulus-based synchronous multisplitting multi-parameter methods for linear complementarity problems. Furthermore, when the system matrix is an H+-matrix, we give convergence results for our new methods under weaker conditions than those in Bai and Zhang’s. Finally, numerical experiments are presented to illustrate the efficiency of the proposed methods.

EIGENVALUE DISTRIBUTIONS OF VARIANCE COMPONENTS ESTIMATORS IN HIGH-DIMENSIONAL RANDOM EFFECTS MODELS.

We study the spectra of MANOVA estimators for variance component covariance matrices in multivariate random effects models. When the dimensionality of the observations is large and comparable to the number of realizations of each random effect, we show that the empirical spectra of such estimators are well-approximated by deterministic laws. The Stieltjes transforms of these laws are characterized by systems of fixed-point equations, which are numerically solvable by a simple iterative procedure. Our proof uses operator-valued free probability theory, and we establish a general asymptotic freeness result for families of rectangular orthogonally-invariant random matrices, which is of independent interest. Our work is motivated in part by the estimation of components of covariance between multiple phenotypic traits in quantitative genetics, and we specialize our results to common experimental designs that arise in this application.

New convergence of modulus-based synchronous block multisplitting multi-parameter methods for linear complementarity problems

In 2013, Bai and Zhang (Numer Linear Algebra Appl 20:425–439, 2013) constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problems into a system of fixed-point equations and studied their convergence. In 2014, Zhang and Li (Comput Math Appl 67:1954–1959, 2014) analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, based on their ideas, we further study modulus-based synchronous block multisplitting multi-parameter methods for linear complementarity problems. Furthermore, the convergence results of our new method in this paper are wider than those in literature when the system matrix is a block \(H_{+}\)-matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters.

The weaker convergence of modulus-based synchronous multisplitting multi-parameters methods for linear complementarity problems

In 2013, Bai and Zhang (2013) constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problems into a system of fixed-point equations and studied their convergence. In this paper, we generalize Bai and Zhang’s methods and study modulus-based synchronous multisplitting multi-parameters methods for linear complementarity problems. Furthermore, the convergence results of our new method in this paper are weaker than those of Bai and Zhang’s when the system matrix is an H+-matrix.

Computational techniques for accurate performance evaluation of multirate, multihop communication networks

Computational techniques are presented for the connection-level performance evaluation of communication networks, with stochastic multirate traffic, state-dependent admission control, alternate routing, and general topology-all characteristics of emerging integrated service networks. The techniques involve solutions of systems of fixed-point equations, which estimate equilibrium network behaviour. Although similar techniques have been applied with success to single-rate fully connected networks, the curse of dimensionality arises when the techniques are extended to multirate, multihop networks, and the cost of solving the fixed point equations exactly is exponential. This exponential barrier is skirted by exploiting, in particular, a close relationship with the network reliability problem, and by borrowing effective heuristics from the reliability domain. A series of experiments are reported on, comparing the estimates from the new techniques to the results of discrete-event simulations.

Self-similar sets 3. Constructions with sofic systems

Using sofic systems, modifications of the self-similar sets of Hutchinson are defined as solutions of systems of fixed-point equations. Their Hausdorff dimension is determined.