I am trying to simplify this function : $$ f(x) = \sum_{j=1}^M d_j e^{\frac{a_j}{x^2} + \frac{b_j}{x} + c_j} $$ where $d_j$ and $a_j$ are negative, $c_j$ is positive and the sign of $b_j$ is unknown. Plus, the function is only defined for positive $x$, and $M$ is very large.
I would like to get rid of the sum. I was thinking about approximating it with an integral, but I don't see how I could do so with so many different coefficients depending on $j$.
Any idea? Even a rough approximation would already be interesting.
Thanks a lot.
$$ f(x) = \sum_{j=1}^M d_j e^{\frac{a_j}{x^2} + \frac{b_j}{x} + c_j} $$ I suppose that $\:a_j \,,\,b_j \,,\,c_j\:$ are known values. If not, one can do nothing.
Let $\quad C_j=\ln(-c_j)\quad$ with $c_j<0$.
$$f(x) = -\sum_{j=1}^M e^{\frac{a_j}{x^2} + \frac{b_j}{x} + C_j}$$ Using non-linear least squares fitting method, find a function $\alpha(\xi)$ so that $\alpha(j)\simeq a_j$ for all $j$.
Same method, find a function $\beta(\xi)$ so that $\beta(j)\simeq b_j$ for all $j$.
Same method, find a function $\gamma(\xi)$ so that $\gamma(j)\simeq C_j$ for all $j$.
Then, approximately : $$f(x) = -\int_{\xi=1}^M e^{\frac{\alpha(\xi)}{x^2} + \frac{\beta(\xi}{x} + \gamma(\xi)}d\xi$$
The accuracy will be the best if $M$ is large and if the variations of the functions between one point to the next are not too large, that is if $(a_{j+1}-a_j)$ is not too large compared to $a_j$. The same for the $b_j$ and for the $C_j$ .
If the variations of the functions are too large (functions not smooth enough), one have to split the range of $\xi$ into several sub-ranges, in other words to use piecewise functions.