Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (2) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Coefficients of ergodicity with respect to vector norms

1983 ◽  
Vol 20 (02) ◽  
pp. 277-287 ◽  
Author(s):  
Choon-Peng Tan
Keyword(s):  
Transition Matrix ◽  
Ergodic Markov Chain ◽  
Numerical Examples ◽  
Localization Property ◽  
New Class ◽  
Geometric Convergence ◽  
Functional Forms ◽  
Ergodicity Coefficients ◽  
Spectrum Localization ◽  
Vector Norms

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞ and l 1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.

Coefficients of ergodicity: structure and applications

1979 ◽  
Vol 11 (03) ◽  
pp. 576-590 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stochastic Matrix ◽  
Numerical Examples ◽  
Localization Property ◽  
Spectrum Localization

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrixP, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property:where λ is any non-unit eigenvalue ofP.The essential purpose is exposition and unification, with the aid of simple informal proofs.

Coefficients of ergodicity: structure and applications

1979 ◽  
Vol 11 (3) ◽  
pp. 576-590 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stochastic Matrix ◽  
Numerical Examples ◽  
Localization Property ◽  
Spectrum Localization

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

scholarly journals A Family of Fourteenth-Order Convergent Iterative Methods for Solving Nonlinear Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fiza Zafar ◽  
Gulshan Bibi
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Efficiency Index ◽  
Numerical Examples ◽  
New Class ◽  
First Derivative ◽  
New Methods ◽  
Specific Step

We present a family of fourteenth-order convergent iterative methods for solving nonlinear equations involving a specific step which when combined with any two-step iterative method raises the convergence order by n+10, if n is the order of convergence of the two-step iterative method. This new class include four evaluations of function and one evaluation of the first derivative per iteration. Therefore, the efficiency index of this family is 141/5 =1.695218203. Several numerical examples are given to show that the new methods of this family are comparable with the existing methods.

scholarly journals The Predictive Power of Transition Matrices

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2096
Author(s):  
André Berchtold
Keyword(s):  
Markov Chain ◽  
Transition Matrix ◽  
Predictive Power ◽  
Distribution Model ◽  
Transition Matrices ◽  
Association Measure ◽  
New Class ◽  
Transition Distribution ◽  
Respective Contribution ◽  
Available Information

When working with Markov chains, especially if they are of order greater than one, it is often necessary to evaluate the respective contribution of each lag of the variable under study on the present. This is particularly true when using the Mixture Transition Distribution model to approximate the true fully parameterized Markov chain. Even if it is possible to evaluate each transition matrix using a standard association measure, these measures do not allow taking into account all the available information. Therefore, in this paper, we introduce a new class of so-called "predictive power" measures for transition matrices. These measures address the shortcomings of traditional association measures, so as to allow better estimation of high-order models.

Ergodicity Coefficients Defined by Vector Norms

2011 ◽  
Vol 32 (1) ◽  
pp. 153-200 ◽  
Author(s):  
Ilse C. F. Ipsen ◽  
Teresa M. Selee
Keyword(s):  
Ergodicity Coefficients ◽  
Vector Norms

scholarly journals Analyzing Electromagnetic Systems on Electrically Large Platform Using a GTM-PO Hybrid Method with Synthetic Basis Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Shang Xiang ◽  
Gaobiao Xiao ◽  
Junfa Mao
Keyword(s):  
Current Distribution ◽  
Hybrid Method ◽  
Computational Efficiency ◽  
Transition Matrix ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Basis Functions ◽  
Physical Optics ◽  
Numerical Examples ◽  
Large Platform

A hybrid method of generalized transition matrix (GTM) and physical optics (PO) with synthetic basis functions (SBF) is proposed to analyze electromagnetic systems on electrically large platforms. Based on domain decomposition method (DDM), the proposed approach is to divide the whole problem into a GTM region and a PO region. The GTM algorithm can simulate antennas and scatterers accurately, and the PO algorithm is applied to obtain current distribution on the electrically large platform. With the characteristics extraction technique using SBFs on the GTM models, the number of unknowns can be greatly reduced and the computational efficiency can be further improved. PO region is regarded as an environment background and the unknowns in the PO region need not to be directly solved. Numerical examples will be shown to demonstrate the feasibility of the hybrid method.

scholarly journals A New Class of “Growth Functions” with Polynomial Variable Transfer Generated by Real Reaction Networks

2020 ◽  
Vol 20 (6) ◽  
pp. 74-81
Author(s):  
Nikolay Kyurkchiev
Keyword(s):  
Growth Models ◽  
Reaction Network ◽  
Initial Time ◽  
Reaction Networks ◽  
Time Interval ◽  
Adequate Model ◽  
Specific Data ◽  
Numerical Examples ◽  
New Class ◽  
Growth Functions

AbstractIn [4, 5], two classes of growth models with “exponentially variable transfer” and “correcting amendments of Bateman-Gompertz-Makeham-type” based on a specific extended reaction network have been studied [1]. In this article we will look at the new scheme with “polynomial variable transfer”. The consideration of such a dynamic model in the present article is dictated by our passionate desire to offer an adequate model with which to well approximate specific data in the field of computer viruses propagation, characterized by rapid growth in the initial time interval. Some numerical examples, using CAS Mathematica illustrating our results are given.

scholarly journals CONSTRUCTION AND ANALYSIS OF BALANCED INCOMPLETE SUDOKU SQUARE DESIGN

2020 ◽  
Vol 4 (2) ◽  
pp. 290-299
Author(s):  
N. S. Dauran ◽  
A. B. Odeyale ◽  
A. Shehu
Keyword(s):  
Linear Model ◽  
Latin Square ◽  
Numerical Examples ◽  
New Class ◽  
Selection Of ◽  
General Method

Sudoku squares have been widely used to design an experiment where each treatment occurs exactly once in each row, column or sub-block.  For some experiments, the size of row (or column or sub-block) may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Sudoku squares design (BISSD) is proposed. A general method for constructing BISSD is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Sudoku designs. The relative efficiencies of a delete-one-transversal balance incomplete Latin Square (BILS) design with respect to Sudoku design are derived. In addition, linear model, numerical examples and procedure for the analysis of data for BISSD are proposed

Design and Analysis of a New Class of Derivative-Free Optimal Order Methods for Nonlinear Equations

2018 ◽  
Vol 15 (03) ◽  
pp. 1850010 ◽  
Author(s):  
Janak Raj Sharma ◽  
Ioannis K. Argyros ◽  
Deepak Kumar
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Numerical Examples ◽  
New Class ◽  
New Methods ◽  
Derivative Free ◽  
Local Convergence Analysis ◽  
Theoretical Results

We develop a general class of derivative free iterative methods with optimal order of convergence in the sense of Kung–Traub hypothesis for solving nonlinear equations. The methods possess very simple design, which makes them easy to remember and hence easy to implement. The Methodology is based on quadratically convergent Traub–Steffensen scheme and further developed by using Padé approximation. Local convergence analysis is provided to show that the iterations are locally well defined and convergent. Numerical examples are provided to confirm the theoretical results and to show the good performance of new methods.

Sign in / Sign up

Export Citation Format

Share Document