Re: Taylor Series in R^n
- To: mathgroup at smc.vnet.net
- Subject: [mg8106] Re: Taylor Series in R^n
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Tue, 12 Aug 1997 00:54:46 -0400
- Organization: University of Western Australia
- Sender: owner-wri-mathgroup at wolfram.com
Andre Deprit wrote:
> The problem is this: Let f[x,y,z] be a numerical function that is
> sufficiently differentiable at the origin. It is proposed to produce the
> Taylor formula for f at the origin to a given order, say 2.
>
> The one-line code below will do the job:
>
> Plus@@Series[f[x,y,z] /. Thread[{x,y,z}->eps {x,y,z}],{eps,0,2}][[3]]
>
> It amounts to multiplying the variables x, y and z by a scale factor
> eps, then initiating a Taylor series in eps at the origin. The
> coefficients of that Taylor series are stored as the third element of
> the structure SeriesData by which Mathematica represents a series
> expansion. Do not try to make the replacement eps-> 1 in the Series
> itself. Mathematica will protest, and rightly so.
I think that using Normal is preferable to using Plus and [[3]] because,
among other things, the internal format of Series could change. After
using Normal you can make the replacement eps-> 1
In[1]:= series[v_List] := Expand[Normal[Series[f @@ v /. Thread[v ->
eps v],
{eps, 0, 2}]] /. eps -> 1]
> Here and now, I am not interested in converting this one-liner into a
> full-fledged code valid for any variables in any (finite!) dimension. I
> just wanted to convey the idea that the one-liner corresponds to what
> mathematicians define as the "Taylor Formula.", save for the remainder
> that the one-liner omits.
The above code works in any (finite!) dimension:
In[2]:= series[{x, y, z}]
Out[2]=
1 (2,0,0) 2 (1,0,0)
- f [0, 0, 0] x + f [0, 0, 0] x +
2
(1,0,1) (1,1,0)
z f [0, 0, 0] x + y f [0, 0, 0] x +
(0,0,1)
f[0, 0, 0] + z f [0, 0, 0] +
1 2 (0,0,2) (0,1,0)
- z f [0, 0, 0] + y f [0, 0, 0] +
2
(0,1,1) 1 2 (0,2,0)
y z f [0, 0, 0] + - y f [0, 0, 0]
2
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA 6907 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
God IS a weakly left-handed dice player
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