A common practice for computing an elementary transcendental function nowadays has two phases: reductions of input arguments to fall into a tiny interval and polynomial approximations for the function within the interval. Typically the interval is made tiny enough so that one won’t have to go for polynomials of very high degrees for accurate approximations. Often approximating polynomials as su...

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE-eigenvalue problems and typically exclude the use of establis...

We present five theorems concerning the asymptotic convergence rates of Chebyshev interpolation applied to functions transplanted to either a semi-infinite or an infinite interval under exponential or double-exponential transformations. This strategy is useful for approximating and computing with functions that are analytic apart from endpoint singularities. The use of Chebyshev polynomials ins...

We discuss polynomial interpolation and derive sufficient conditions for the uniform convergence of Chebyshev interpolants for different classes of functions. Rigorous results are illustrated with a number of examples which include solitons on an infinite line with algebraic, exponential and Gaussian decay rates. Suitable mappings of the real line to the interval [−1, 1] are considered for each...

In this paper, we describe a novel swarming framework that guides autonomous mobile sensors into a flexible arrangement to interpolate values of a field in an unknown region. The algorithm is devised so that the sensor distribution will behave like a Chebyshev distribution, which can be optimal for certain ideal geometries. The framework is designed to dynamically adjust to changes in the regio...

Let T be a Jordan curve in the complex plane, and let Í) be the compact set bounded by T. Let / denote a function analytic on O. We consider the approximation of / on fî by a polynomial p of degree less than n that interpolates / in n points on T. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation ...

We study the problem of reconstructing a sparse polynomial in a basis of Chebyshev polynomials (Chebyshev basis in short) from given samples on a Chebyshev grid of [−1, 1]. A polynomial is called M -sparse in a Chebyshev basis, if it can be represented by a linear combination of M Chebyshev polynomials. For a polynomial with known and unknown Chebyshev sparsity, respectively, we present efficie...

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 Chebys...

This paper is concerned with a formal linearization problem for a general class of nonlinear time-varying dynamic systems. To a given system, a linearization function is made up of Chebyshev polynomials about its state variables. The nonlinear time-varying system is transformed into a linear time-varying system in terms of the linearization function using Chebyshev interpolation to state variab...

*Correspondence: [email protected] Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran Abstract In this paper, we solve a time-space fractional diffusion equation. Our methods are based on normalized Bernstein polynomials. For the space domain, we use a set of normalized Bernstein polynomials and for the time domain, which is a semi-infinite domain, ...