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Complex-Step Differentiation

A simple way to do a numerical approximations for first derivatives is illustrated by the following:
 In[10]:= exact = Cos[1];
 
 In[13]:= nd[expr_, var_, val_] :=
  10^8 Im[(expr /. var -> val + I 1. 10^-8) ]
 
 In[14]:= nd[Sin[x], x, 1]
 
 Out[14]= 0.540302
 
In[15]:= % - exact

Out[15]= 0.
POSTED BY: Frank Kampas
2 Replies
A paper on it by Martins, Sturdza and Alonso can be found with a Google search.
POSTED BY: Frank Kampas
Posted 10 years ago
This is pretty neat! Here's why it works if anyone is interested:
Im[z] == (z - Conjugate[z])/(2I)
and thus 
Im[f[x + h*I]]/h == (f[x + h*I] - f[x - h*I])/(2h*I)
which is the central difference quotient.

(This only works if we assume x and h are real and f is analytic at x.)
POSTED BY: Greg Hurst
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