# Linear approximation higher order terms

by bacon
Tags: approximation, linear, order, terms
 Sci Advisor HW Helper PF Gold P: 3,171 Regarding the higher-order terms for approximating e^x, use however many terms of the Taylor series you need: $$e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$$ Thus for example a quadratic approximation you would use $$e^x \approx 1 + x + \frac{x^2}{2}$$ and for a cubic you would use $$e^x \approx 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$ etc. I'm not sure specifically about the GPS relativistic correction, but in general algorithmic complexity is not much of an issue, as the calculations are not performed on board the satellite. Ground stations upload the (time-varying) orbital information needed to compute the correction, and the satellite simply echoes this information to receivers. The receivers then calculate the correction based on the orbital parameters and the current time and user position. The bottleneck is the craptacular 50 bits per second data communication speed from the satellite to the receiver, so there's only so much orbital information that can be sent, and this is further limited by the fact that this information must be repeated frequently (every 30 seconds) because a receiver may "tune in" at any time. This limitation means that only first and second order coefficients are used for most parameters, each represented by as few data bits as possible. This in turn means that the ground stations have to broadcast updated orbital parameters fairly often, every few hours.