[x3d-public] Multiresolution

[John Carlson]

Okay, I may have to look elsewhere for the name of a field series, if there
is such a thing?

I’m actually studying infinitesimal recursion, and stumbled across this
field series idea.

The idea behind infinitesimal recursion is to fractionalize a recursive
step, interpolation is typically done.

The question is, how to do recursive fractionation in a program.   Similar
to rendering NURBS, most likely, but provide interpolation between
resolutions.  Like multiresolution modeling?

Thanks!

John

On Mon, May 15, 2023 at 3:39 PM Vincent Marchetti <vmarchetti at kshell.com>
wrote:

> The Taylor series defined for real numbers, say near 0, is
>
> F(x) = a_0 + a_1 * x + a_2 * x^2 + a_3 * x^3 ....
>
> with the series being absolutely convergent for some region around 0; that
> is |x| < eps for some positive eps > 0
>
> Then, taking a variety of routes, we evenuallt get to a relation between
> the coefficients ant the derivatives of F(x)
>
> There is no bid deal about all of ths a_i themselves being vectors, or
> matrices, perhaps even functions on some domain.
>
> BUT
>
> If the x values are intended to be elements of a vector space, or other
> hgher dimensional space, there's no useful way to define the powers x, x^2,
> x^3 ....
>
> Hence, no Taylor series on vector space domains.
>
> Vince
>
>
>
> The challenge here is that for a domain that's more than one-dimensional,
> there's no obvuj
>
> > On May 15, 2023, at 4:29 PM, John Carlson <yottzumm at gmail.com> wrote:
> >
> > Think of the sum of a taylor series being a vector, matrix, or field.
> This is like
> >
> > f(V) = ...something like a Taylor Series where V is a vector
> >
> > f(M) = ... ditto where M is a matrix
> >
> > f(F) = ... ditto where F is a field (as in physics)
> >
> > I'm looking more for a name of the thing than anything else. Field
> series?
> >
> > On Mon, May 15, 2023 at 2:47 PM GPU Group <gpugroup at gmail.com> wrote:
> > Taylor Series use: linearizing a non-linear equation for iterative
> linear solutions. Take the first 2 terms of the taylor series expansion,
> and that's a straight line equivalent to the tangent line to the non-linear
> curve. Solve for a new guess on each iteration until the error term is
> small.
> > -Doug
> >
> > On Mon, May 15, 2023 at 12:52 PM John Carlson <yottzumm at gmail.com>
> wrote:
> > Is there a kind of Taylor Series that produces a vector or vector field
> instead of a single number?
> >
> > Or maybe just an advanced undergraduate?
> >
> > I guess it's kind of like a matrix sum, where the terms are matrices.
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