Free(OnGrid()) boundary conditions produce surprising results

[cchapmanbird]

I'm interpolating a 3-dimensional regular surface with scale(interpolate(Espldata, BSpline(Cubic(<bc>))) where is either Line(OnGrid()) or Free(OnGrid()). I am confused by the below result, which is a 1D slice through the grid with the other two parameters being grid points. Why do the Free boundary conditions behave so poorly? My impression is that Free BCs are the same as not-a-knot (continuity in third derivative for second-last spline knot) but this is clearly not what is happening here.

[mkitti]

You've encountered https://en.wikipedia.org/wiki/Runge%27s_phenomenon

[cchapmanbird]

I have never seen such a bad case of it for not-a-knot cubic splines - especially for such a simple function. The true curve appears almost linear at the boundary but the cubic polynomial is totally different...

I suppose more nodes is the only answer?

[mkitti]

Perhaps. The reality is that you probably only need more nodes near the boundaries.

true curve appears almost linear at the boundary

Have you tried Natural?

Also consider https://github.com/jipolanco/BSplineKit.jl https://github.com/JuliaApproximation/ApproxFun.jl

[cchapmanbird]

Thanks for the recommendations - I did not know about BSplineKit.jl.

I thought that Line is an alias for Natural? In the docs....

"The following boundary conditions are implemented: Flat, Line (alternatively, Natural), Free, Periodic and Reflect."