> From: Oleg Kobchenko <[EMAIL PROTECTED]> > > > From: Raul Miller > > > To: Programming forum > > Sent: Friday, September 5, 2008 6:29:07 PM > > Subject: Re: [Jprogramming] deriving discontinuous functions from > > polynomials > > > > On Fri, Sep 5, 2008 at 6:16 PM, John Randall > > wrote: > > > Are you looking for the implicit function theorem? This gives a local > > > inverse on the graph of an implicit function whenever the tangent line > > > is not horizontal. > > > > This this theorem looks very relevant. > > > > But I will have to spend some time digesting it. > > There is a simpler "Inverse function theorem" > > http://planetmath.org/encyclopedia/InverseFunctionTheorem.html > > The idea is simply that for y=f(x), on a monotonous stretch > where f(') != 0 and f in C^1 (continuously differentiable), > then there exists x=g(y) on image of f on the stretch. > > It doesn't tell you how to find the stretches or inverses. > > For inverses, the stretches are probably between potential extrema, > ie f(')(x)=0. > > In inverse, where the graph is flipped, the stretches can overlap, > but overlaps can be detected looking at segments from left to right, > since original f was a map. Thus we get non-overlapping stretches. > > Each such stretch can be approximated with a polynomial from a sample > of points obtained with f. > > Does that sound like a recipe for inverse function?
Here's a simple procedure to produce Taylor expansion of implicit functions and inverse Taylor expansion, defined in terms of Derive CAS system: http://www.mathematik.uni-kassel.de/~koepf/Publikationen/Koepf1994.pdf Also a couple of reference from Yacas http://yacas.sourceforge.net/refchapter6.html#InverseTaylor http://yacas.sourceforge.net/refchapter6.html#ReversePoly ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
