**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**80

# Search results for: Chebyshev polynomials

##### 80 Generalized Chebyshev Collocation Method

**Authors:**
Junghan Kim,
Wonkyu Chung,
Sunyoung Bu,
Philsu Kim

**Abstract:**

In this paper, we introduce a generalized Chebyshev collocation method (GCCM) based on the generalized Chebyshev polynomials for solving stiff systems. For employing a technique of the embedded Runge-Kutta method used in explicit schemes, the property of the generalized Chebyshev polynomials is used, in which the nodes for the higher degree polynomial are overlapped with those for the lower degree polynomial. The constructed algorithm controls both the error and the time step size simultaneously and further the errors at each integration step are embedded in the algorithm itself, which provides the efficiency of the computational cost. For the assessment of the effectiveness, numerical results obtained by the proposed method and the Radau IIA are presented and compared.

**Keywords:**
Generalized Chebyshev Collocation method,
Generalized Chebyshev Polynomial,
Initial value problem.

##### 79 System Overflow/Blocking Transients For Queues with Batch Arrivals Using a Family of Polynomials Resembling Chebyshev Polynomials

**Authors:**
Vitalice K. Oduol,
C. Ardil

**Abstract:**

The paper shows that in the analysis of a queuing system with fixed-size batch arrivals, there emerges a set of polynomials which are a generalization of Chebyshev polynomials of the second kind. The paper uses these polynomials in assessing the transient behaviour of the overflow (equivalently call blocking) probability in the system. A key figure to note is the proportion of the overflow (or blocking) probability resident in the transient component, which is shown in the results to be more significant at the beginning of the transient and naturally decays to zero in the limit of large t. The results also show that the significance of transients is more pronounced in cases of lighter loads, but lasts longer for heavier loads.

**Keywords:**
batch arrivals,
blocking probability,
generalizedChebyshev polynomials,
overflow probability,
queue transientanalysis

##### 78 Computable Function Representations Using Effective Chebyshev Polynomial

**Authors:**
Mohammed A. Abutheraa,
David Lester

**Abstract:**

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.

**Keywords:**
Approximation Theory,
Chebyshev Polynomial,
Computable Functions,
Computable Real Arithmetic,
Integration,
Numerical Analysis.

##### 77 A Note on the Numerical Solution of Singular Integral Equations of Cauchy Type

**Authors:**
M. Abdulkawi,
Z. K. Eshkuvatov,
N. M. A. Nik Long

**Abstract:**

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels.

**Keywords:**
Singular integral equations,
Cauchy kernel,
Chebyshev polynomials,
interpolation.

##### 76 Numerical Solution of Riccati Differential Equations by Using Hybrid Functions and Tau Method

**Authors:**
Changqing Yang,
Jianhua Hou,
Beibo Qin

**Abstract:**

A numerical method for Riccati equation is presented in this work. The method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The operational matrices of derivative and product of hybrid functions are presented. These matrices together with the tau method are then utilized to transform the differential equation into a system of algebraic equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

**Keywords:**
Hybrid functions,
Riccati differential equation,
Blockpulse,
Chebyshev polynomials,
Tau method,
operational matrix.

##### 75 Hybrid Function Method for Solving Nonlinear Fredholm Integral Equations of the Second Kind

**Authors:**
jianhua Hou,
Changqing Yang,
and Beibo Qin

**Abstract:**

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm type equations which have many applications in mathematical physics are then considered. The method is based on hybrid function approximations. The properties of hybrid of block-pulse functions and Chebyshev polynomials are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

**Keywords:**
Hybrid functions,
Fredholm integral equation,
Blockpulse,
Chebyshev polynomials,
product operational matrix.

##### 74 On Bounds For The Zeros of Univariate Polynomial

**Authors:**
Matthias Dehmer1 Jürgen Kilian

**Abstract:**

Problems on algebraical polynomials appear in many fields of mathematics and computer science. Especially the task of determining the roots of polynomials has been frequently investigated.Nonetheless, the task of locating the zeros of complex polynomials is still challenging. In this paper we deal with the location of zeros of univariate complex polynomials. We prove some novel upper bounds for the moduli of the zeros of complex polynomials. That means, we provide disks in the complex plane where all zeros of a complex polynomial are situated. Such bounds are extremely useful for obtaining a priori assertations regarding the location of zeros of polynomials. Based on the proven bounds and a test set of polynomials, we present an experimental study to examine which bound is optimal.

**Keywords:**
complex polynomials,
zeros,
inequalities

##### 73 Edge Detection in Low Contrast Images

**Authors:**
Koushlendra Kumar Singh,
Manish Kumar Bajpai,
Rajesh K. Pandey

**Abstract:**

The edges of low contrast images are not clearly distinguishable to human eye. It is difficult to find the edges and boundaries in it. The present work encompasses a new approach for low contrast images. The Chebyshev polynomial based fractional order filter has been used for filtering operation on an image. The preprocessing has been performed by this filter on the input image. Laplacian of Gaussian method has been applied on preprocessed image for edge detection. The algorithm has been tested on two test images.

**Keywords:**
Chebyshev polynomials,
Fractional order
differentiator,
Laplacian of Gaussian (LoG) method,
Low contrast
image.

##### 72 Bilinear and Bilateral Generating Functions for the Gauss’ Hypergeometric Polynomials

**Authors:**
Manoj Singh,
Mumtaz Ahmad Khan,
Abdul Hakim Khan

**Abstract:**

The object of the present paper is to investigate several general families of bilinear and bilateral generating functions with different argument for the Gauss’ hypergeometric polynomials.

**Keywords:**
Appell’s functions,
Gauss hypergeometric functions,
Heat polynomials,
Kampe’ de Fe’riet function,
Laguerre polynomials,
Lauricella’s function,
Saran’s functions.

##### 71 Rigid Registration of Reduced Dimension Images using 1D Binary Projections

**Authors:**
Panos D. Kotsas,
Tony Dodd

**Abstract:**

**Keywords:**
binary projections,
image registration,
reduceddimension images.

##### 70 Optimal Image Representation for Linear Canonical Transform Multiplexing

**Authors:**
Navdeep Goel,
Salvador Gabarda

**Abstract:**

**Keywords:**
Chirp signals,
Image multiplexing,
Image
transformation,
Linear canonical transform,
Polynomial
approximation.

##### 69 Design of Nonlinear Observer by Using Chebyshev Interpolation based on Formal Linearization

**Authors:**
Kazuo Komatsu,
Hitoshi Takata

**Abstract:**

**Keywords:**
nonlinear system,
nonlinear observer,
formal linearization,
Chebyshev interpolation.

##### 68 Near-Lossless Image Coding based on Orthogonal Polynomials

**Authors:**
Krishnamoorthy R,
Rajavijayalakshmi K,
Punidha R

**Abstract:**

**Keywords:**
Near-lossless Coding,
Orthogonal Polynomials
Transform,
Embedded Zerotree Coding

##### 67 Numerical Inverse Laplace Transform Using Chebyshev Polynomial

**Authors:**
Vinod Mishra,
Dimple Rani

**Abstract:**

In this paper, numerical approximate Laplace transform inversion algorithm based on Chebyshev polynomial of second kind is developed using odd cosine series. The technique has been tested for three different functions to work efficiently. The illustrations show that the new developed numerical inverse Laplace transform is very much close to the classical analytic inverse Laplace transform.

**Keywords:**
Chebyshev polynomial,
Numerical inverse Laplace transform,
Odd cosine series.

##### 66 Multi-objective Optimization of Vehicle Passive Suspension with a Two-Terminal Mass Using Chebyshev Goal Programming

**Authors:**
Chuan Li,
Ming Liang,
Qibing Yu

**Abstract:**

**Keywords:**
Vehicle,
passive suspension,
two-terminal mass,
optimization,
Chebyshev goal programming

##### 65 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves

**Authors:**
Taweechai Nuntawisuttiwong,
Natasha Dejdumrong

**Abstract:**

**Keywords:**
Newton interpolation,
Lagrange interpolation,
linear
complexity.

##### 64 On the Construction of m-Sequences via Primitive Polynomials with a Fast Identification Method

**Authors:**
Abhijit Mitra

**Abstract:**

**Keywords:**
Finite field,
irreducible polynomial,
primitive polynomial,
maximal length sequence,
additive shift register,
multiplicative
shift register.

##### 63 Some Results on the Generalized Higher Rank Numerical Ranges

**Authors:**
Mohsen Zahraei

**Abstract:**

**Keywords:**
Rank−k numerical range,
isometry,
numerical range,
rectangular matrix polynomials.

##### 62 Best Coapproximation in Fuzzy Anti-n-Normed Spaces

**Authors:**
J. Kavikumar,
N. S. Manian,
M. B. K. Moorthy

**Abstract:**

The main purpose of this paper is to consider the new kind of approximation which is called as t-best coapproximation in fuzzy n-normed spaces. The set of all t-best coapproximation define the t-coproximinal, t-co-Chebyshev and F-best coapproximation and then prove several theorems pertaining to this sets.

**Keywords:**
Fuzzy-n-normed space,
best coapproximation,
co-proximinal,
co-Chebyshev,
F-best coapproximation,
orthogonality

##### 61 4D Flight Trajectory Optimization Based on Pseudospectral Methods

**Authors:**
Kouamana Bousson,
Paulo Machado

**Abstract:**

**Keywords:**
Pseudospectral Methods,
Trajectory Optimization,
4DTrajectories

##### 60 Holistic Face Recognition using Multivariate Approximation, Genetic Algorithms and AdaBoost Classifier: Preliminary Results

**Authors:**
C. Villegas-Quezada,
J. Climent

**Abstract:**

Several works regarding facial recognition have dealt with methods which identify isolated characteristics of the face or with templates which encompass several regions of it. In this paper a new technique which approaches the problem holistically dispensing with the need to identify geometrical characteristics or regions of the face is introduced. The characterization of a face is achieved by randomly sampling selected attributes of the pixels of its image. From this information we construct a set of data, which correspond to the values of low frequencies, gradient, entropy and another several characteristics of pixel of the image. Generating a set of “p" variables. The multivariate data set with different polynomials minimizing the data fitness error in the minimax sense (L∞ - Norm) is approximated. With the use of a Genetic Algorithm (GA) it is able to circumvent the problem of dimensionality inherent to higher degree polynomial approximations. The GA yields the degree and values of a set of coefficients of the polynomials approximating of the image of a face. By finding a family of characteristic polynomials from several variables (pixel characteristics) for each face (say Fi ) in the data base through a resampling process the system in use, is trained. A face (say F ) is recognized by finding its characteristic polynomials and using an AdaBoost Classifier from F -s polynomials to each of the Fi -s polynomials. The winner is the polynomial family closer to F -s corresponding to target face in data base.

**Keywords:**
AdaBoost Classifier,
Holistic Face Recognition,
Minimax Multivariate Approximation,
Genetic Algorithm.

##### 59 Numerical Solution of Linear Ordinary Differential Equations in Quantum Chemistry by Clenshaw Method

**Authors:**
M. Saravi,
F. Ashrafi,
S.R. Mirrajei

**Abstract:**

**Keywords:**
Chebyshev polynomials,
Clenshaw method,
ODEs,
Spectral methods

##### 58 Rational Chebyshev Tau Method for Solving Natural Convection of Darcian Fluid About a Vertical Full Cone Embedded in Porous Media Whit a Prescribed Wall Temperature

**Authors:**
Kourosh Parand,
Zahra Delafkar,
Fatemeh Baharifard

**Abstract:**

The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy-s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.

**Keywords:**
Tau method,
semi-infinite,
nonlinear ODE,
rational Chebyshev,
porous media.

##### 57 A Numerical Solution Based On Operational Matrix of Differentiation of Shifted Second Kind Chebyshev Wavelets for a Stefan Problem

**Authors:**
Rajeev,
N. K. Raigar

**Abstract:**

**Keywords:**
Operational matrix of differentiation,
Similarity
transformation,
Shifted second kind Chebyshev wavelets,
Stefan
problem.

##### 56 Fuzzy Fingerprint Vault using Multiple Polynomials

**Authors:**
Daesung Moon,
Woo-Yong Choi,
Kiyoung Moon

**Abstract:**

Fuzzy fingerprint vault is a recently developed cryptographic construct based on the polynomial reconstruction problem to secure critical data with the fingerprint data. However, the previous researches are not applicable to the fingerprint having a few minutiae since they use a fixed degree of the polynomial without considering the number of fingerprint minutiae. To solve this problem, we use an adaptive degree of the polynomial considering the number of minutiae extracted from each user. Also, we apply multiple polynomials to avoid the possible degradation of the security of a simple solution(i.e., using a low-degree polynomial). Based on the experimental results, our method can make the possible attack difficult 2192 times more than using a low-degree polynomial as well as verify the users having a few minutiae.

**Keywords:**
Fuzzy vault,
fingerprint recognition multiple polynomials.

##### 55 Image Adaptive Watermarking with Visual Model in Orthogonal Polynomials based Transformation Domain

**Authors:**
Krishnamoorthi R.,
Sheba Kezia Malarchelvi P. D.

**Abstract:**

**Keywords:**
Orthogonal Polynomials based Transformation,
Digital Watermarking,
Copyright Protection,
Visual model.

##### 54 Optimal Control of a Linear Distributed Parameter System via Shifted Legendre Polynomials

**Authors:**
Sanjeeb Kumar Kar

**Abstract:**

**Keywords:**
Optimal control,
linear systems,
distributed parametersystems,
Legendre polynomials.

##### 53 Codebook Generation for Vector Quantization on Orthogonal Polynomials based Transform Coding

**Authors:**
R. Krishnamoorthi,
N. Kannan

**Abstract:**

In this paper, a new algorithm for generating codebook is proposed for vector quantization (VQ) in image coding. The significant features of the training image vectors are extracted by using the proposed Orthogonal Polynomials based transformation. We propose to generate the codebook by partitioning these feature vectors into a binary tree. Each feature vector at a non-terminal node of the binary tree is directed to one of the two descendants by comparing a single feature associated with that node to a threshold. The binary tree codebook is used for encoding and decoding the feature vectors. In the decoding process the feature vectors are subjected to inverse transformation with the help of basis functions of the proposed Orthogonal Polynomials based transformation to get back the approximated input image training vectors. The results of the proposed coding are compared with the VQ using Discrete Cosine Transform (DCT) and Pairwise Nearest Neighbor (PNN) algorithm. The new algorithm results in a considerable reduction in computation time and provides better reconstructed picture quality.

**Keywords:**
Orthogonal Polynomials,
Image Coding,
Vector Quantization,
TSVQ,
Binary Tree Classifier

##### 52 Factoring a Polynomial with Multiple-Roots

**Authors:**
Feng Cheng Chang

**Abstract:**

**Keywords:**
Polynomial roots,
greatest common divisor,
Longhand polynomial division,
Euclidean GCD Algorithm.

##### 51 A Comparison of Recent Methods for Solving a Model 1D Convection Diffusion Equation

**Authors:**
Ashvin Gopaul,
Jayrani Cheeneebash,
Kamleshsing Baurhoo

**Abstract:**

In this paper we study some numerical methods to solve a model one-dimensional convection–diffusion equation. The semi-discretisation of the space variable results into a system of ordinary differential equations and the solution of the latter involves the evaluation of a matrix exponent. Since the calculation of this term is computationally expensive, we study some methods based on Krylov subspace and on Restrictive Taylor series approximation respectively. We also consider the Chebyshev Pseudospectral collocation method to do the spatial discretisation and we present the numerical solution obtained by these methods.

**Keywords:**
Chebyshev Pseudospectral collocation method,
convection-diffusion equation,
restrictive Taylor approximation.