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

[robert bristow-johnson]

On 7/22/12 6:23 AM, Kristofer Munsterhjelm wrote:
> 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.

ya know, i didn't expect this crossover to happen with EM and DSP (the 
latter i'm supposed to have some competence).  anyway, these were really 
written for embedded systems, to implement good (but not perfect) and 
fast elementary and transcendentals, but i have an old C file that i 
wrote a decade ago, if someone is doing some big, massive simulation.  
can't guarantee that it would work as fast as the intel processor   
stuff like exp(), log(), sin(), sqrt(), etc.

all using finite order polynomials to cover a portion of the function.  
anyone lemme know, and i'll send you the files.

if your computer is blazingly fast, might not make any difference.

-- 

r b-j                  rbj at audioimagination.com

"Imagination is more important than knowledge."