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Re: Numerical Differentiation

• To: mathgroup at smc.vnet.net
• Subject: [mg4298] Re: Numerical Differentiation
• From: rubin at msu.edu (Paul A. Rubin)
• Date: Sat, 29 Jun 1996 03:56:47 -0400
• Organization: Michigan State University
• Sender: owner-wri-mathgroup at wolfram.com

In article <4q5k43$bsb at dragonfly.wolfram.com>,
   mrj at cs.su.oz.au (Mark James) wrote:
->Does anyone know of a function that calculates the derivative of
->a function (that can't be differentiated symbolically) at a given
->point by numerical means?  I can't find it as a built-in or in the
->standard packages.  Thanks.
->
A simple solution would be:

In[1]:=  ndiff[ f_, x_, eps_:.0001 ] := 
	  (f[ x + eps ] - f[ x - eps ])/(2 eps) /;
	  Positive[ eps ]
In[2]:=	 ndiff[ Exp, 2 ]
	 ndiff[ Exp, 2, .01 ]  (* less accurate *)
Out[2]=	 7.38906
	 7.38918

The first argument is a function, the second the point at which to 
differentiate, the (optional) third is the step size to use.  I'm doing a 
centered estimate here.  You might find more accurate formulas in a book on 
numerical analysis.  (Unfortunately, mine are buried at home someplace.)

My rather vague recollection from my numerical analysis course (a couple of 
decades ago) is that numerical differentiation is not the most stable of 
processes, due in part to the wonders of rounding error.  In any event, I'm 
not sure how to determine the accuracy of your answer.

Hope this helps.

Paul

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* Paul A. Rubin                                  Phone: (517) 432-3509   *
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