a neat idea, easy to do in Maxima, perhaps. Numerical derivatives

This is an increasingly popular method for doing sensitivity analysis of
complex computational physics codes.  I was playing around with it and did a
little 'grid convergence' study showing that in fact it is second order
accurate and avoids subtractive cancellation:
http://j-stults.blogspot.com/2008/12/complex-step-subtractive-cancellation.html

You can extend the idea to higher derivatives / orders. Those don't enjoy
the same benefit of arbitrarily high accuracy by avoiding subtractive
cancellation, but they do increase the accuracy with fewer additional
function evaluations.

Another good short write-up is Squire and Trapp 1998:
http://www.math.uu.se/~warwick/summer04/material/reading/week2/complexDerivative.pdf

I even wrote a little Maxima batch file that calculates the expressions (for
use in fortran) for arbitrary derivatives using a mixture of real and
complex steps:
http://j-stults.blogspot.com/2008/12/asymmetric-complex-step.html

Neat stuff.

On Fri, Dec 4, 2009 at 11:54 PM, Richard Fateman <fateman at cs.berkeley.edu>wrote:

>   *Accurate numerical derivatives in MATLAB*
>
>    *Source*
>   ACM Transactions on Mathematical Software (TOMS) archive<http://portal.acm.org/toc.cfm?id=J782&type=periodical&coll=portal&dl=ACM&CFID=525354554&CFTOKEN=525354554>;
>  Volume 33 ,  Issue 4  (August 2007) table of contents<http://portal.acm.org/toc.cfm?id=1268776&type=issue&coll=portal&dl=ACM&CFID=525354554&CFTOKEN=525354554>;
>    Article No. 26
>  Year of Publication: 2007
>  ISSN:0098-3500
>   *Author *
>   L. F. Shampine<http://portal.acm.org/author_page.cfm?id=81100197580&coll=portal&dl=ACM&trk=0&CFID=525354554&CFTOKEN=525354554>;   Southern
> Methodist University, Dallas, TX
>   *Publisher*
>  ACM <http://www.acm.org/publications>;  New York, NY, USA
>
> The basic observation is that if you can evaluate a function accurately at
> a complex point, you have a way of getting very very accurate numerical
> values of its derivative.
>
> Observe that  Im(  f(x+i*h)) /h   = f'(x)- h^3/6*f'''(x) + ....
>
> solve for f'(x).
>
> Much better than finite differences.  Read the paper if you want more
> interesting observations.
>
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> Maxima at math.utexas.edu
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>
>


-- 
Joshua Stults
Website: http://j-stults.blogspot.com
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