[EM] Taylor or McLaurin polynomial for the complicated functions would reduce the numerical work.

[Kristofer Munsterhjelm]

On 07/21/2012 08:01 AM, Michael Ossipoff wrote:
> I spoke of using a polynomial approximation of G(q), the cumulative
> state number,and differentiating it to get F(q), the
> probability-density.
>
> I'd like to add that a Taylor or McLaurin polynomial approximation of
> a complicated function could be used.  ...after you've determined, by
> whatever method, exactly what the complicated function is to be, and
> what its constants should be.

For log-normal distributions, both the pdf and the cdf is defined and 
relatively simple. As would be expected of something related to the 
Gaussian, the cumulative distribution function involves the error 
function; but as long as you can calculate the error function, it's 
entirely possible to calculate the integral of the pdf (i.e. the cdf) 
directly without having to resort to numerical integration.

However, I don't think a divisor method making use of a log-normal 
approximation would retain the generality of Warren's exponential 
solution. You'd still have to find the distribution parameters (mu and 
sigma), and I imagine these would differ for different countries, 
depending, if not on more factors, on the rate and variance of 
population growth. In the cdf calculation, those parameters would be 
inside the error function call.

So I guess we would have something between Warren's exact solution and 
your numerical integration/root-finding situation: the divisor formula 
would be something like f(x) = floor(x + g(x, a_1..a_n)) where a_1...a_n 
have to be found in an empirical manner, but where g(...) itself can be 
directly calculated instead of having to be found by numerical 
integration for each x.

I could be wrong, though. It's been a while and I've been busy 
elsewhere, so perhaps I have missed something that would indicate that g 
would have to be numerically integrated every time, not just fitted.