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Re: Can't calculate numerical derivative of EllipticTheta

• To: mathgroup at smc.vnet.net
• Subject: [mg82689] Re: [mg82659] Can't calculate numerical derivative of EllipticTheta
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 28 Oct 2007 04:07:56 -0500 (EST)
• References: <200710271006.GAA11228@smc.vnet.net>

On 27 Oct 2007, at 19:06, Scott Hemphill wrote:

> I realize that my version of Mathematica is getting a little old.  Is
> this fixed in later releases?
>
> In[1]:= $Version
>
> Out[1]= 5.1 for Linux (October 25, 2004)
>
> In[2]:= pdf[x_,x0_,sig_] := 1/(Sqrt[2Pi]sig) Exp[-(x-x0)^2/(2sig^2)];
>
> In[3]:= Sum[pdf[x,0,sig],{x,-Infinity,Infinity}]
>
>                                       2
>                              -1/(2 sig )
>         EllipticTheta[3, 0, E           ]
> Out[3]= ---------------------------------
>                  Sqrt[2 Pi] sig
>
> In[4]:= f[sig_]=%;
>
> In[5]:= f'[1.]
>
>                                     (0,0,1)
> Out[5]= -1. + 0.241971 EllipticTheta       [3, 0, 0.606531]
>
> I expected to see a number on this last line.
>
> Scott
> --  
> Scott Hemphill	hemphill at alumni.caltech.edu
> "This isn't flying.  This is falling, with style."  -- Buzz Lightyear
>


You can get correct numerical answer by using instead of f'[1.]


N[f'[1],10]


0.0193719

Note that simply N[f'[1]] will not work. This is reasonable because  
using MachinePrecision would not produce an accurate answer. You can  
see this by defining your function in a somewhat different way.
Replace your definition

f[sig_]=%

with

f[sig_?NumericQ] = %;

Now you can differenitate numerically (but not in Mathematica 6 !)


f'[1.]

-2.115267372136438*^-7

The only problem is that this answer is clearly wrong. This is due to  
numerical instability of this problem which makes it unsuited to  
using MachinePrecision numbers.

Andrzej Kozlowski

• References:
  • Can't calculate numerical derivative of EllipticTheta
    • From: Scott Hemphill <hemphill@hemphills.net>