Fluctuationless Univariate Integration Through Taylor Expansion with Remainder by Using Oscillatory Function Basis Sets

Abstract

Taylor series expansion with the fluctuation freely approximated remainder over gauss wave type basis functions AIP Conf. Proc. 1504, 816 (2012); 10.1063/1.4771819Taylor series based integration with the fluctuation freely approximated remainder over gausswave type basis functions AIP Conf.Abstract. This work uses a recently developed fluctuation free matrix representation method in approximating the integral of the Taylor expansion remainder term. The basis set used for the matrix representation contains common factors of sine and cosine functions with the same frequencies and the same origin. This provides a new numerical univariate integration method to us such that the approximation quality can be controlled by the number of the expansion terms in the Taylor expansion of the integrand and by the dimension of the subspace over which the matrix representations are built. The number of oscillations in the basis set is also another quality control agent and may help to get better approximants in the case of high oscillations. Due to the limitation of space in this extended abstract results of the implementations are left to the presentation.

Influence of a simple pole on the convergence of separate node ascending derivatives expansion (SNADE) on a sequence of nodes alternating between 2 values

“…One very widely used method is to expand the considered function to the nonnegative integer powers of a first‐degree polynomial (a monomial taking zero value at a specific independent variable value, which may be called the expansion point). This is the backbone of very well known‐Taylor series, and the obtained series converges in the complex‐plane analyticity domain of the target function. As long as that domain is not empty, the target function can be exactly represented by that power series even though its finite truncations are preferred to be used for practical purposes.…”