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MathGroup Archive 2006

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Re: Points sampled by N[Derivative[]]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71993] Re: Points sampled by N[Derivative[]]
  • From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
  • Date: Thu, 7 Dec 2006 06:25:56 -0500 (EST)
  • References: <ej72s5$ita$1@smc.vnet.net><el68sd$300$1@smc.vnet.net>

Yep I see what's happening now, thanks Paul. I think it's strange for
Mathematica to transparently employ such a strangely non-adaptable
algorithm for this purpose, rather than using e.g. ND from the
NumericalMath`NLimits` package. What do you think?

Cheers,

Andrew

On Dec 6, 9:17 pm, Paul Abbott <p... at physics.uwa.edu.au> wrote:
> In article <ej72s5$it... at smc.vnet.net>,
>  "Andrew Moylan" <andrew.j.moy... at gmail.com> wrote:
>
>
>
> > I am trying to understand the way in which Mathematica automatically
> > computes numerical approximations to derivatives which cannot be
> > differentiated symbolically. Consider the following function:
>
> > f[(x_)?NumericQ] := (Sow[x];
> >    1/(x - 1)^2)
>
> > By supplying a numerical argument to Derivative[f], we can see the
> > points at which f is sampled by Mathematica when approximating the
> > derivative:
>
> > Reap[f'[0.5]]
> >   gives:
> > {15.997945911907477,
> >   {{0.5, 0.5526315789473684,
> >     0.6052631578947368,
> >     0.6578947368421053,
> >     0.7105263157894737,
> >     0.763157894736842,
> >     0.8157894736842105,
> >     0.868421052631579,
> >     0.9210526315789473,
> >     0.4473684210526316,
> >     0.39473684210526316,
> >     0.34210526315789475,
> >     0.2894736842105263,
> >     0.2368421052631579,
> >     0.1842105263157895,
> >     0.13157894736842107,
> >     0.07894736842105265}}}
>
> > It's useful to see the points relative the central point (0.5):
>
> > %[[2]] - 0.5
> >   gives:
> > {{0., 0.05263157894736836,
> >    0.10526315789473684,
> >    0.1578947368421053,
> >    0.21052631578947367,
> >    0.26315789473684204,
> >    0.3157894736842105,
> >    0.368421052631579,
> >    0.42105263157894735,
> >    -0.05263157894736842,
> >    -0.10526315789473684,
> >    -0.15789473684210525,
> >    -0.21052631578947367,
> >    -0.2631578947368421,
> >    -0.3157894736842105,
> >    -0.3684210526315789,
> >    -0.42105263157894735}}
>
> > Why does the number 0.05263157894736836 appear here? A little testing
> > shows that this constant appears frequently when Mathematica
> > automatically computes numerical approximations to derivatives.Rationalize 0.05263157894736836 and you obtain 1/19 -- and so you may be
> able to work out what is happening here. If you multiply your list by 19
> the sampling pattern is clear.
>
> Cheers,
> Paul
>
> _______________________________________________________________________
> Paul Abbott                                      Phone:  61 8 6488 2734
> School of Physics, M013                            Fax: +61 8 6488 1014
> The University of Western Australia         (CRICOS Provider No 00126G)
> AUSTRALIA                              http://physics.uwa.edu.au/~paul


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