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