# Wondering about Domingo's Rational Mean book

Iain Davidson Sttscitrans at tesco.net
Wed May 31 19:35:01 CEST 2000

```Kirby Urner <urner at alumni.princeton.edu> wrote in message
news:aae9js85444llpfnao0n9nnhfqcq7p546t at 4ax.com...
> "Iain Davidson" <Sttscitrans at tesco.net> wrote:

> >The advantage of standard CFs is that they produce all
> >best approximations. The mean method for cbrt(2)-1 misses
> >131/ 504, for example.
> >
>
> I presume by "best" you mean most accuracy for the least
> digits.  Like, the mean method gets me closer to cbrt(2)-1
> than 131/504 with fractions like 4159/16001 or 236845/911219
> or 168286661033/647452990441 -- but you're saying standard
> CFs will converge more quickly, yes?
>
> In other words the standard CFs will give a better approximation
> when pushed to the same total number of digits -- something
> along those lines?

You can always find better and better approximations
to say, cbrt(2)-1 , but there is no fraction with a smaller denominator
than 504 which will be closer. So in this sense 131/504 is optimal.

> Do you know of an URL where the standard algorithms for
> approximating the nth root of k as p/q are spelled out,
> something suitable for computerizing?

Do you mean an algorithm that finds convergents directly
from x^n -k or just from a decimal approximation of the nth root ?

Cohen's book "A Course in Computational Algebraic Number
Theory" covers the latter I think.

```