Spline Based Series for Sine and Arbitrarily Accurate Bounds for Sine, Cosine and Sine Integral

TL;DR

This paper introduces spline-based series that provide highly accurate bounds for sine, cosine, and sine integral functions, outperforming traditional Taylor series in convergence and precision over the interval [0, Pi/2].

Contribution

It develops a new spline approximation method to generate arbitrarily accurate bounds for sine, cosine, and sine integral functions, with significantly improved convergence properties.

Findings

Maximum relative errors of 3.31 x 10^-4, 2.48 x 10^-8, and 2.02 x 10^-18 for second, fourth, and eighth order approximations.

New series for sine demonstrate better convergence than Taylor series over [0, Pi/2].

Bounded functions can be made arbitrarily accurate.

Abstract

Based on two point spline approximations of arbitrary order, a series of functions that define lower bounds for sin(x) and sin(x)/x, over the interval [0,Pi/2], with increasingly low relative errors and smaller relative errors than published results, are defined. Second, fourth and eighth order approximations have, respectively, maximum relative errors over the interval [0,Pi/2] of 3.31 x 10-4, 2.48 x 10-8, and 2.02 x 10-18. New series for the sine function, which have significantly better convergence that a Taylor series over the interval [0,Pi/2], are proposed. Applications include functions that are upper bounds for the sine function, upper and lower bounds for the cosine function and lower bounds for the sine integral function. These bounded functions can be made arbitrarily accurate.