Definitions of extrapolation behavior about interpolations.jl

[tomasaschan]

To state my own opinion, I think what makes the most sense is to exclude the end point from input (that is, to expect that the user does), and to reflect around the edges of input data. In other words, I propose that we expect the red point is not included in the input to correctly interpolate a sine wave from that data, and that we take the magenta approach for reflection.

[timholy]

I agree with that perspective on BCperiodic. As discussed in timholy/Grid.jl#31, I personally find the orange behavior most intuitive/useful for BCreflect, because otherwise sampling a vector of length n at 1:2n will not yield [x, x[end:-1:1]]. But I'm fine with a diversity of options being available.

[tomasaschan]

I guess reflection behavior should also reflect the interpolations we're doing. It's possible to do linear interpolation with data given on half-nodes rather than on nodes (i.e. one line segment has its center on a given data point), in which case reflection on half-nodes give nicer results.

In this image I've cheated, and just evaluated the function in different places for plotting purposes, but it's possible to construct the same linear interpolation from these points as the one up top. With this linear interpolation, there is no "no mans land" between the two end nodes, and sampling at 1:2n will still yield [x, x[end:-1:1]].

However, that's much more complicated than what we're currently doing (although I guess it could probably be expressed in terms of non-interpolating splines) so maybe not what we want to aim for...

[tomasaschan]

This is thoroughly handled in #9, and as soon as that is working well I'll also merge #12 which documents the mathematics, so I'll close this for housekeeping.

tl;dr: All of the use cases above are supported by appropriate choice of OnCell/OnGrid.