st: How to improve accuracy in numerical integrations using Stata

["Tiago V. Pereira" <tiago.pereira@mbe.bio.br>]

Dear statalisters,

I am using -integ- to numerically integrate a set of functions.

An example of a function to integrate:

function_y  = -2*normal((-`x'-`r'*z)/sqrt(1-`r'^2))*normalden(z)

for variable z.


In my case, the domain ranges from 0 to `x'.



So, what I am doing is  the following:

*/ ------------ start example --------------
local r = 0.1
local x = 6
drop _all
range z 0 `x' 1000
generate y  = -2*normal((-`x'-`r'*z)/sqrt(1-`r'^2))*normalden(z)
dydx     y z, gen(yprime)
integ    y z, gen(Sy)
*/ ------- end example ------------------

dis r(integral)
 -5.288096*07214*e-10


to gain more precision, I have manually edited -integ- to compute values
using the double format (i.e. instead of 'gen float variable = ', it is
using 'gen double variable =').

It seems that some precision is gained:

[using the exactly same code above, but using the 'double' version, one
gets:]

dis r(integral)

-5.288096*31782*e-10

I know that the correct answer would be something like:

-5.28809630924643245856745711e-10

which is obtained from numerical integration using a C program (supposed
to be the most precise approach I am aware of).


Do you have any ideas on how to further increase the precision for
numerical integration in Stata? The problem is that I am working on heavy
tails (alpha levels below 10^-8).

All the best,

Tiago






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