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MathGroup Archive 2006

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Re: New Analytical Functions - Mathematica Verified

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66976] Re: New Analytical Functions - Mathematica Verified
  • From: "Mohamed Al-Dabbagh" <mohamed_al_dabbagh at hotmail.com>
  • Date: Tue, 6 Jun 2006 06:27:01 -0400 (EDT)
  • References: <200605280104.VAA23436@smc.vnet.net> <200606011055.GAA20733@smc.vnet.net> <e5osg1$hvp$1@smc.vnet.net> <200606030726.DAA17310@smc.vnet.net> <e60our$gbv$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Daniel Lichtblau wrote:
>
> This is not exactly new. The function FractionalPart[f[x]] is
> differentiable wherever f[x] is differentiable except where it takes on
> integer (or, in the complex plane, Gaussian integer) values.
>
Of course it is not new to say that fractional part is differentiable!
The new thing is that to say derivative of fractional part is the same
as the derivative in continuity intervals. I don't know what harm in
confessing that! However, if it is not new and you have a solution for
that then you should consider that instead of the very costly Dirac
Delta involvement.

> At such points one might regard the derivative as undefined, or as a
> generalized function in terms of derivatives of delta functions,

Delta function is VERY SLOW in high derivatives.

>> To the extent that calculating 5th derivative of
> > the FractionalPart(x^5) to 1000 places of decimal would take about ONE
> > HOUR!!!!

Try this at your side and post your results with Timing function,
please.

> At the bottom of your note you say: "Now you understand how powerful my
> formulas are compared to the existing ones you use in Mathematica."
> While I am at a loss to understand why the computations are so slow when
> you run them, I think you are vastly overstating the case here.

I don't!

> Differentiation in Mathematica by default does but little with
> discontinuous functions such as FractionalPart. But as seen above a user
> can readily add derivatives to suit ones needs, and evaluations
> involving those derivatives seem (to me) to be reasonably fast.

They are not fast in higher order derivatives.

Last word to say that it is all up to you to use my results or not.
Mathematica is a great software and I am very proud of using it. It
deserves all improvement. I wouldn't spend all this time unless I like
to see myself doing some improvement on this great software. When you
use it symbolically, it still returns no result for the derivative of
fractional part, while it is great to return the correct answer which
is the derivative of the function itself. 

Mohamed Al-Dabbagh


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