Hi, I have a function f(t), where f is numerically sampled at discrete points in time t that are NOT evenly sampled in time. (This is unavoidable.) I need to take the first and second derivatives of f(t), compute a complicated function (call it D(t)) of f' and f'' over that time interval, and integ…

You could use Roots.jl to find zeros of the first derivative of the spline. Here is a quick example using Newton’s method: using BSplineKit using Roots f(x) = cos(4π * x) * exp(-4x) xs = 0:0.05:1 S = spline(interpolate(xs, f.(xs), BSplineOrder(6))) S′ = diff(S, Derivative(1)) S″ = diff(S, Deriv…

If you know the true answer just take the norm of difference on a dense grid If you can increase the density of samples then you can investigate convergence rate (here AAA + ApproxFun should have spectral convergence so will beat algebraic convergence of splines ) If all you have is the data of a …

@dlfivefifty , you recommend computing the norm (or root mean square errors) for f, df/dt and d2f/dt2? It might well happen that some methods work best for some type of functions and not for others. How do mathematicians bring order to such a wide range of possibilities?

Yea. This would be a Sobolev norm.

[image] rafael.guerra: It might well happen that some methods work best for some type of functions and not for others Generally speaking if you know the regularity (ie differentiability) of the function there’s precise results on the convergence rates in Sobolev norms of various methods. Apar…

Just to add here that there is a mistake in my example above in that I used a function defined on [0,1] to fit to a ApproxFun Fun defined on [-1, 1] (oops). I fixed that mistake and also added a comparison the the “exact” integral (starting with a Fun that had adaptively decided how many points to …

When fitting curves piecewise with splines of degree d, expect the fit at the initial d xs and the final d xs to be unsatisfactory. The closer to the first x or the final x, the less likely it is that the y will be fit robustly. Additionally, larger d asks to use higher order polynomials, which ma…

So, I realized that I was anchoring the initial and final limits of my integration (of my key function RCurv) to the randomly selected time points, which is why each time I ran the program the range of numerical integration was slightly different, so the result was slightly different. If I keep the…

The function RCurv(t) has a division by 0 around t0 = 1.20608 and it complicates matters to integrate it starting at t = 1. In the revised code above, the analyses started at t = 1.21. Used splines of degree 4 (thanks @jipolanco for the correction) for both BSplineKit and Dierckx, and ApproxFun.j…

That looks great! Just a note for fairer comparison between spline packages. There are two different conventions for the definition of the spline order. The one used in BSplineKit is the same as in classical textbooks (e.g. de Boor) and is also the one used in Wikipedia . Namely, splines of order k …

While I have you guys here, once I have a spline representation of a fit to a data set, what’s the easiest way to find the locations & values of any maxima & minima of the function? Example partial code: Sfn = spline(BSplineKit.interpolate(timeDatArr, FunctionDatArr, BSplineOrder(5))) Sd1 = diff(S…

Best way to take derivatives of unevenly spaced data (with interpolations? discrete derivatives?)

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Best way to take derivatives of unevenly spaced data (with interpolations? discrete derivatives?)

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