On Fri, 27 Jun 2008, Charles Henry wrote:
I get what you're saying now. I had to read it a couple times through
to see :) You're referring to piecewise cubic polynomials, right?
Yes, I'm always assuming that piecewise-cubics is all that we'll need.
We would wind up with an overdetermined system of equations if we didn't
float the 1st *and* 2nd derivatives, which would come out as a linear
algebra problem of the size of the table.
Yes, which is why we don't want to do that.
but I think it gets even worse. There could be a non-zero null space to
the problem. There are infinite solutions to interpolate a table full
of zeros, with these conditions. What a mess :)
In that case (which is when the algebra problem has the size of the
table), there are two missing conditions, and then when you set them to
x''[0]=0 and x''[last]=0, it is called «natural cubic spline».
By expanding it out to more points, we could use a more accurate
calculation of the derivative.
Yes, but we don't want to get into that for this particular application,
because the point is to be fast, and 4-point is the first N-point that
makes sense (enough interpolation). Well, there is also 3-point, as used
in Tk's Bézier splines (-smooth 1), but... hmm... what is possible with
3-point ?
There's always a frequency dependent effect on the accuracy of 1st
derivative approximations.
And on the accuracy of all Nth derivative approximations.
So there might be some value in expanding the number of points to
include better approximations.
Perhaps, but there is still a need for something fairly expedient to be
used as the main interpolation method in pd.
Yeah, a cubic polynomial makes the most sense for small changes. I
haven't ever heard of people interpolating 4 points with a 5th degree
polynomial.... but I think I could make it work....
It could work, but I don't know how much it's worth it. Higher derivatives
of anything quite discrete will be rather jumpy. The Nth derivative of a
white noise sample doubles its RMS at every derivation, for example, but
when looking just at the near-Nyquist hiss, it's much worse than that. The
more you use derivatives, the trickier it gets.
_ _ __ ___ _____ ________ _____________ _____________________ ...
| Mathieu Bouchard - tél:+1.514.383.3801, Montréal, Québec
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