I want to make an interpolation with a cubic spline function. And I want to find the coefficients of the polynomials with this function [breaks coefs] = unmkpp(spline(x,y)) . The Matlab function uses natural bouundary or something else?
MATLAB: Cubic spline in Matlab works with natural boundary
cubic spline
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From the documentation:
'spline':
‘Spline interpolation using not-a-knot end conditions. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension.’
It does not appear to be the same as 'cubic'. According to the documentation, 'cubic' is currently the same as 'pchip'.
You were close, in that csape can do it. (My SLM toolbox also has a natural cubic as an option, but there is no need to go that far.) The proper choice in csape is:
pp = csape(x,y,'variational');The flag 'variational' comes from the calculus of variations, I assume. The calculus of variations can also be seen to call those corresponding end conditions the "natural" boundary conditions.
'variational' : set end second derivatives equal to zero (ignoring given end condition values if any)A natural cubic spline is the one that has zero second derivatives at the ends.
As a check, does it produce what I expect? So here a function that has decidedly NON-zero second derivatives at the ends.
x = linspace(0,2*pi,10);y = cos(x);pp = csape(x,y,'variational');pp2 = fnder(pp,2);fnplt(pp2)grid on
Yet as you see, the second derivatives at the ends are forced to zero. This is the classic example that I always used when teaching about splines, as to why a natural cubic spline might often be a poor choice.
So if we look at the fit that arises, compare the natural cubic to that produced by spline.
fnplt(pp,[0 1])hold onpps = spline(x,y);fnplt(pps,[0 1])
In blue is the natural cubic, whereas the green curve is the result of spline, which uses not-a-knot end conditions, generally a safer choice. If we remember these curves are an approximation to cos(x), the green curve is clearly much better.
So while you can find a natural cubic in MATLAB, you need to look deeply enough, and know what you should be looking for. That is in general a good thing in this regard.
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