[PD] better tabread4~
Charles Henry
czhenry at gmail.com
Wed Jun 25 06:50:06 CEST 2008
On Tue, Jun 24, 2008 at 9:24 PM, Mathieu Bouchard <matju at artengine.ca> wrote:
> I don't think that more than one alternative will be necessary. For 4-point
> table lookups that go through all the original points, I don't know why
> anyone would aim lower than a C2 piecewise-polynomial. Unfortunately, it
> would be somewhat too late to just call it [tabread4~]. Or not.
How low is too low? hmmm....
tabread4~ is deficient as Cyrille pointed out, because the resulting
function is not continuously differentiable (thanks for the
correction). So, what characteristics would be best for a "fast"
interpolating function?
When we have an interval between samples, we wish to fit a polynomial
(because it's fast, I guess) that satisfies our constraints.
We could specify the polynomial has the same values at x[-1],x[0],
x[1], x[2] (tabread4~). Four constraints, determines a cubic
polynomial, works out as a linear algebra problem.
or we could set x[0],x[1] and x'[0]=(x[1]-x[-1])/2 and x'[1]=(x[2]-x[0])/2
again, 4 constraints, cubic polynomial, etc...
or another 4 point scheme, with continuous 2nd derivative
setting x[0], x[1], x'[0]=(x[1]-x[-1])/2 and x'[1]=(x[2]-x[0])/2
and x''[0]=x[1]-2*x[0]+x[-1] and x''[1]=x[2]-2*x[1]+x[0]
6 constraints, 5th degree polynomial
and if you additionally wanted it to actually go through x[-1] and
x[2], it would be 7th degree
So, even for 4-point interpolation, there are some options that could
all be called tabread4~.
But not all possibilities are worth analyzing... I'm not even sure
what kind of method to use to narrow the field.
Chuck
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