The IBM Power4 processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-stand...

In this paper, by the Chebyshev-type inequalities we define three mappings, investigate their main properties, give some refinements for Chebyshev-type inequalities, obtain some applications.

We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x = T3(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials and b + degC = 3N . If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a ≤ 3. Most results are derived from continued fractions and their matrix represe...

A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree N polynomial in O(N(logN)2/ log logN) operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the di...

The orbital propagation of large sets initial conditions under high accuracy requirements is currently a bottleneck in the development space missions, e.g. for planetary protection compliance analyses. proposed approach can include any force source dynamical model through efficient Picard–Chebyshev (PC) numerical simulations. A two-level augmentation integration scheme proposed, to run an arbit...

in this paper we introduce a type of fractional-order polynomials basedon the classical chebyshev polynomials of the second kind (fcss). also we construct the operationalmatrix of fractional derivative of order $ gamma $ in the caputo for fcss and show that this matrix with the tau method are utilized to reduce the solution of some fractional-order differential equations.

This chapter provides a brief literature review together with detailed descriptions of the authors’ work on the stability and control of systems represented by linear time-periodic delay-differential equations using the Chebyshev and temporal finite element analysis (TFEA) techniques. Here, the theory and examples assume that there is a single fixed discrete delay which is equal to the principa...

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to c...

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

We construct and analyze generalized Gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities. The rules have quadrature points inside the interval of integration and the weights are all strictly positive. Such rules date back to the study of Chebyshev sets, but their use in applications has only recently been appreciated. We provide error estimates an...