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, Comp...

We show how to combine local Shepard operators with Hermite polynomials on the simplex (Chui and Lai, 1990 [3]) so as to raise the algebraic precision of the Shepard-Taylor operators (Farwig, 1986 [8]) that use the same data and contemporaneously maintain the interpolation properties at each sample point (derivative data included) and a good accuracy of approximation. Numerical results are prov...

We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases in a general approach to approximation that also encompasses Taylor sums. Here we give additional details of the construction.

We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating ci...

A method for solving linear boundary value problems is described which consists of approximating the coefficients of the differential operator. Error estimates for the approximate solutions are established and improved results are given for the case of approximation by piecewise polynomial functions. For the latter approximations, the resulting problem can be solved by Taylor series techniques ...

in this letter, the numerical scheme of nonlinear volterra-fredholm integro-differential equations is proposed in a reproducing kernel hilbert space (rkhs). the method is constructed based on the reproducing kernel properties in which the initial condition of the problem is satised. the nonlinear terms are replaced by its taylor series. in this technique, the nonlinear volterra-fredholm integr...

Modern mixed-integer quadratic solvers generally handle binary variables more efficiently than nonlinear solvers. This is relevant to the power system operation models as unit commitment formulations typically contain a large number of variables. paper investigates how achieve accuracy level close one exact models, but by utilising convex and The presented model based on Taylor-series expansion...