scholarly journals q-Difference Systems for the Jackson Integral of Symmetric Selberg Type

2020 ◽  
Author(s):  
Masahiko Ito ◽  
Keyword(s):  
Explicit Expression ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Gauss Hypergeometric Function ◽  
Symmetric Polynomials ◽  
Difference System ◽  
Difference Systems ◽  
Repeated Use ◽  
Jackson Integral ◽  
Interpolation Polynomials

We provide an explicit expression for the first order q-difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of q-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the q-KZ equation. Our main result is an explicit expression for the coefficient matrix of the q-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.

A Kind of Preconditioned IGMRES(m) Algorithm

2012 ◽  
Vol 482-484 ◽  
pp. 413-416
Author(s):  
Chun Xiao Yu
Keyword(s):  
Krylov Subspace ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Residual Method ◽  
Minimal Residual Method ◽  
Algorithm Convergence ◽  
Minimal Residual

Fundamental theories are studied for an Incomplete Generalized Minimal Residual Method(IGMRES(m)) in Krylov subspace. An algebraic equations generated from the IGMRES(m) algorithm is presented. The relationships are deeply researched for the algorithm convergence and the coefficient matrix of the equations. A kind of preconditioned method is proposed to improve the convergence of the IGMRES(m) algorithm. It is proved that the best convergence can be obtained through appropriate matrix decomposition.

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

A note on the algebra of S-functions

1949 ◽  
Vol 45 (3) ◽  
pp. 328-334 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Explicit Expression ◽  
Symmetric Function ◽  
Arbitrary Number ◽  
Symmetric Polynomials ◽  
Considerable Part ◽  
Algebraic Form ◽  
General Validity ◽  
Linearly Independent ◽  
The One ◽  
Alternative Solutions

1. The particular set of symmetric polynomials known as S-functions has recently been shown to be of importance in a variety of algebraic problems. Many of the applications of these functions depend upon the properties of the operation which Littlewood terms ‘new multiplication’, by which, from two given S-functions {μ}, {ν} of respective degrees m and n, is constructed a symmetric function {μ} ⊗ {ν} of degree mn. Littlewood has devoted a considerable part of his paper (2) to explaining various methods by which this function can be expressed in terms of S-functions of degree mn. None of these methods is really simple (in the sense that the rule for expressing the ordinary product {μ} {ν} in terms of S-functions of degree m + n is simple); and, indeed, it would be unreasonable to expect any simple rule of general validity for writing the resulting expression down since, for instance, a knowledge of the explicit expression for {μ} ⊗ {n} would yield immediately a knowledge of all the linearly independent concomitants, of degree n, of an algebraic form of type {μ} in an arbitrary number of variables. Nevertheless, the evaluation of {μ} ⊗ {ν} in particular cases is often necessary. Of the methods suggested by Littlewood for performing this evaluation the one which seems to be normally the simplest (his ‘third method’) involves a process which is not shown to be free from ambiguity, and which can actually be shown by examples to give, in certain cases, alternative solutions, the choice between which must be made by other considerations. It is therefore perhaps worth while putting on record a quite different method of procedure, which, apart from any intrinsic interest which it may possess, seems to be quite practicable for most of the actual evaluations performed by Littlewood.

scholarly journals Coefficient Matrix Decomposition Method and BIBO Stabilization of Stochastic Systems with Time Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xia Zhou ◽  
Shouming Zhong
Keyword(s):  
Control Systems ◽  
Time Delays ◽  
Model Transformation ◽  
Stochastic Systems ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Linear Matrix ◽  
Mean Square ◽  
The Mean ◽  
Delay Dependent

The mean square BIBO stabilization is investigated for the stochastic control systems with time delays and nonlinear perturbations. A class of suitable Lyapunov functional is constructed, combined with the descriptor model transformation and the decomposition technique of coefficient matrix; thus some novel delay-dependent mean square BIBO stabilization conditions are derived. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Finally, three numerical examples are given to demonstrate that the derived conditions are effective and much less conservative than those given in the literature.

scholarly journals Incremental Nonnegative Matrix Factorization for Face Recognition

2008 ◽  
Vol 2008 ◽  
pp. 1-17 ◽  
Author(s):  
Wen-Sheng Chen ◽  
Binbin Pan ◽  
Bin Fang ◽  
Ming Li ◽  
Jianliang Tang
Keyword(s):  
Face Recognition ◽  
Matrix Factorization ◽  
Nonnegative Matrix Factorization ◽  
Computational Cost ◽  
Nonnegative Matrix ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Large Matrix ◽  
Training Samples ◽  
Almost All

Nonnegative matrix factorization (NMF) is a promising approach for local feature extraction in face recognition tasks. However, there are two major drawbacks in almost all existing NMF-based methods. One shortcoming is that the computational cost is expensive for large matrix decomposition. The other is that it must conduct repetitive learning, when the training samples or classes are updated. To overcome these two limitations, this paper proposes a novel incremental nonnegative matrix factorization (INMF) for face representation and recognition. The proposed INMF approach is based on a novel constraint criterion and our previous block strategy. It thus has some good properties, such as low computational complexity, sparse coefficient matrix. Also, the coefficient column vectors between different classes are orthogonal. In particular, it can be applied to incremental learning. Two face databases, namely FERET and CMU PIE face databases, are selected for evaluation. Compared with PCA and some state-of-the-art NMF-based methods, our INMF approach gives the best performance.

An Effective Improved Algorithm for Finite Particle Method

2016 ◽  
Vol 13 (04) ◽  
pp. 1641009 ◽  
Author(s):  
Yang Yang ◽  
Fei Xu ◽  
Meng Zhang ◽  
Lu Wang
Keyword(s):  
Computing Time ◽  
Matrix Decomposition ◽  
Particle Method ◽  
Coefficient Matrix ◽  
Computational Accuracy ◽  
Sph Method ◽  
Modified Method ◽  
Transient Problems ◽  
Basic Equations ◽  
Finite Particle

The low accuracy near the boundary or the interface in SPH method has been paid extensive attention. The Finite Particle Method (FPM) is a significant improvement to the traditional SPH method, which can greatly improve the computational accuracy for boundary particles. However, there are still some inherent defects for FPM, such as the long computing time and the potential numerical instability. By conducting matrix decomposition on the basic equations of FPM, an improved FPM method (IFPM) is proposed, which can not only maintain the high accuracy of FPM for boundary particles, but also keep the invertibility of the coefficient matrix in FPM. The numerical results show that the IFPM is really an effective improvement to traditional FPM, which could greatly reduce the computing time. Finally, the modified method is applied to two transient problems.

Sign in / Sign up

Export Citation Format

Share Document