On 03/18/2010 03:15 PM, Richard Hennessy wrote:
> I got it this way.
>
>
> bromberg(x^x^x, x, 599975/100000b0, 6b0),brombergit=15,fpprec:2000;
> -> 1.1026649993619407788421378964b36300
> bromberg(x^x^x, x, 599985/100000b0, 6b0),brombergit=15,fpprec:2000;
>> - 1.1026649993619421259432920899b36300
>
> When you are this close to 6 the area under the curve between 5 and
> 5.99985 is irrelevant since it does not affect the answer (at least not
> the first 8-10 digits.)
Yes, this is similar to Raymond Toy's solution. I get a similar answer
with trapezoid rule and an estimate of the truncation error that assumes
similitude between the truncated part and the calculated part.
Is there a solution in terms of an asymptotic series? we assume
integral(f(x),x,5,6) ~ integral(f(x),x,6-eps,6), expand the
approximation in terms of the small parameter eps, and then truncate the
expansion at some small number of terms (perhaps 2 or 3) and get some
kind of decent approximate result?
I tried to do this via integration by parts taking
x^x^x = x * x^(x^x-1)
take x dx = dv and x^(x^x-1) = u, plug into uv - int(v du), iterate
again, and then chop off the remaining integral... but it got me
nowhere, the resulting series explodes.
I still feel like there should be a "clever" way to get good accuracy
using some kind of asymptotic method. This type of extremely fast
growing integral is a very special sort of thing.