🚀 go-pugleaf

In article <1iah0fINNjhj@solar.usc.edu> laydin@solar.usc.edu (Levent Aydin) writes:
>
>	I need to approximate a function given in terms of a product with a
>simple function that I can deal with.The expression is as follows:
>
>               n              /            \  
>              __             |    x-m_i     | 
>        g(x)= || 0.5*{ 1+ erf| -------------| }
>              i=1            |  sqrt(2)*s_i | 
>                              \            /  
>
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I don't think it would be a good idea to approximate by a
single term of the same form , whatever m_0 and s_0.

It would be best to separate the 2 factors with m_i much
larger than the others. Consider the function erf(y). It is
a monotonously increasing func; it is practically 0 from 
- infinity to 0 - 3 standard deviations, and  practically 1 
from 0 +
3 stand. dev. to + infinity; inbetween, it increases 
smoothtly.
Thus, you could approximate one factor  in the above
product by the following
approx.:

f(x) = 0.5 , for x = - infinity to m_i - 3 stand. dev.
     = 1 , for x = + infinity to m_i + 3 stand. dev.
     = a cubic polynomial with inflexion at m_i in-between.
 
Also you could approximate the n-2 factors with the m_i close
together by the function h(x) where

h(x) 	= 0.5^(n-2) , for x= - infinity to min{m_i - a}
     	= 1 , for x = max{m_i + b} to +infinity
	= Ae^(px), in-between 

the parameter a is around 1 to 3 standard dev of  in min m_i
the parameter b is between 1 to 3 standard dev. from max m_i
and A and p are chosen so the straight line px + c1 represent
well enough the log of the product of the (n-2) factors.

--
From:  simardr@iro.umontreal.ca