This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systems Ax+g = 0 where A = A is positive definite. We prove that the Chebyshev method in floating point arithmetic is numerically stable, which means that the computed sequence {x^} approximates the solution a such that lim|k is of order C||a||.||A~ \\.\\y\\ where £ is the relative computer...

Keywords: Set-valued mapping Chebyshev center Uniformly convex Locally uniformly convex Chebyshev fixed point a b s t r a c t The existence of a continuous Chebyshev selection for a Hausdorff continuous set-valued mapping is studied in a Banach space with some uniform convexity. As applications, some existence results of Chebyshev fixed point for condensing set-valued mappings are given, and th...

A Chebyshev-based representation of the state vector is proposed for designing optimal control trajectories of unconstrained, linear, dynamic systems with quadratic performance indices. By approximating each state variable by a finite-term, shifted Chebyshev series, the linear quadratic (LQ) optimal control problem can be cast as a quadratic programming (QP) problem. In solving this QP problem,...

The hypergeometric type differential equations of the second order with polynomial coefficients and their systems are considered. The realization of the Lanczos Tau Method with minimal residue is proposed for the approximate solution of the second order differential equations with polynomial coefficients. The scheme of Tau method is extended for the systems of hypergeometric type differential e...

This paper deals with Vandermonde matrices Vn whose nodes are the Gauss–Lobatto Chebyshev nodes, also called extrema Chebyshev nodes. We give an analytic factorization and explicit formula for the entries of their inverse, and explore its computational issues. We also give asymptotic estimates of the Frobenius norm of both Vn and its inverse and present an explicit formula for the determinant o...

In this paper, a new formula of the spectral differentiation matrices is presented. Therefore, the numerical solutions for higher-order differential equations are presented by expanding the unknown solution in terms of monic Chebyshev polynomials. The resulting systems of linear equations are solved directly for the values of the solution at the extreme points of the Chebyshev polynomial of ord...

A new type ð2; 2Þ-step iterative method related to an optimal Chebyshev method is developed for solving real and nonsymmetric linear systems of the form Ax 1⁄4 b. It is an extension of the ð2; 2Þ-step iterative method introduced in [Numer. Linear Algebra Appl. 7 (2000) 169]. The superiority of the new type ð2; 2Þ-step iterative method over the optimal Chebyshev method is derived in the case whe...

The theoretical and computational aspects of interval methodology based on Chebyshev polynomials for modeling complex nonlinear multi-body dynamic systems in the presence of parametric and external excitation uncertainties is formulated, implemented, and validated. Both the parameters uncertainties and external excitation uncertainties are modeled by uncertain-butbounded interval variables, whe...

The present paper discusses the stability analysis of discrete-time uncertain nonlinear systems with time-varying delay, uncertainties related to parametric uncertainty and unknown nonlinearity. The time-varying delay considered has minimum and maximum bounds. The novelty of Chebyshev Neural Network (CNN) is that it requires much less computation time as compared to multi layer neural network (...