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Find zero crossings of interpolant derivatives http://forum.alglib.net/viewtopic.php?f=2&t=589 
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Author:  Testing_Around [ Mon Jun 18, 2012 4:11 pm ] 
Post subject:  Find zero crossings of interpolant derivatives 
Hello, i'm currently trying to interpolate some measured data points via the package 'interpolation'. Target is to find the turning points and the value of first derivate at those points. I wanted to try different types of interpolation, at least spline and polinomal and compare the results with each other and with some other metrics i got my hands on. I started with the spline Interpolation: The proceeded as followed: 1. built the spline and unpack the parameters. 2. then i iterate over all the intervalls: > find solution ("candidate") for the manually setup second derivate of the function = 0. (which i built within the loop by passing the unpacked parameters of the spline to a formula i set up by pen and paper differentiation) > check third derivate < 0 for only left > right turning points (this step is optional and in fact only means to check the intervalls C3 Value for beeing smaller than zero) > check if the "candidate" is within the intervall borders and if yes, add to some output vector. Now what i wondered about was that i had to built up the derivates manually to find that zero crossings. all the ...diff... functions only give me the derivates values at specified points rather than i could let them output some zero crossings. While the thing was quite easy to build manually with the unpacked parameters of cubic spline im now confronted with those parameters of barycentric model which is used in polynomialbuild(). Isn't it quite a common problem to find zero crossings in first to third derivates? Did i oversee the according package to easily find those points in the interpolated models? Or is there an easier way of finding those points in the barycentric model as to build the derivates of those SUM formulas? Thanks for your time! 
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