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Re: Points sampled by N[Derivative[]]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71968] Re: Points sampled by N[Derivative[]]
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Wed, 6 Dec 2006 06:04:01 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <ej72s5$ita$1@smc.vnet.net>
In article <ej72s5$ita$1 at smc.vnet.net>,
"Andrew Moylan" <andrew.j.moylan 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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