# Approximation for the logarithm of a summatory

I would like to find an approximation for:

$$\log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right)$$

with

$$a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + e_i^2)} \\ c_i = \frac{c-f_i}{2(g^2 + g_i^2)}$$

where $b, e, c, g$ are floats independent of $i$ and $d_i, e_i, f_i, g_i$ are also floats which depend on $i$. These floats can safely be considered to take random values between $(0,100]$.

Finally $N>1$ and $a_i>0 \,\forall\, i$.

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It depends an incredible amount on the nature of the sequence $b_i^2+c_i^2$. – Greg Martin Mar 14 '14 at 22:17
@GregMartin ok, I'll expand the question being a bit more precise about those factors. – Gabriel Mar 14 '14 at 22:24
There are more names now, but the nature of the sequence is still a mystery. How are the terms weighted? Are they all large/small? Are they larger/smaller when $i$ is larger? etc. These kinds of qualitative details are what's important. Could you give a typical example of some $a_i$, $b_i$, and $c_i$? – Antonio Vargas Mar 15 '14 at 12:02