I am aware of many different ways to go about interpolating between the values of known data points. However, whenever I come upon (I work in Quantitative Finance) the need to extrapolate data I find myself with far fewer options. Basically, all I really end up doing is coming up with a model to fit all the internal points and hoping it is sufficiently good to hold for the exterior points. Are there ready made, go to algorithms in Numerical Analysis that I somehow missed along the way, or is the extrapolation problem simply more difficult?
All add in a specific problem that I think there should be some methods for but I don't know of many. Say I have observe values of a function at $x_0, x_1, ..., x_n$ where the points are equally space by $h$. How can I go about extrapolating out to $x_n + h$ or even $x_n + h/2$? I am definitely assuming my function is rather smooth.