🚀 go-pugleaf

In article <1991Mar15.063023.6605@dsd.es.com> rthomson@mesa.dsd.es.com
(Rich Thomson) writes:

> . . .
> 
> If you play with Newton's method long enough you come to realize that,
> in general, picking a starting value isn't always obvious. . . .

Some cases are reasonably robust, however.  For example the largest
root of a polynomial with all roots real will be approached by
Newton's method with a starting value larger than that root.

> . . .
>
> A different way to compute square roots (and other rational or
> non-rational numbers) is through the methods of continued fractions.
> If I remember correctly, the continued fraction expansion of Sqrt[2]
> is [1,2,2,2,....], . . .

It is.

> . . .
> 
> This doesn't converge as quickly as the sequence you give, but it
> still interesting.
> 

The rate of convergence is easily worked out, and of the form c^n if n
terms are taken.  This is fine for ordinary machine precision, but
wasteful if you want millions of digits.

> I think it turns out that non-rational numbers have a continued
> fraction expansion that is eventually periodic . . .

It is easy (after Euler) to show that such numbers must be quadratic
irrationals.  Lagrange showed that all quadratic irrationals have
continued fractions which are eventually periodic, and conditions for
pure periodicity have been worked out.  Perron's book gives Galois as
the source for this work, as well as for determining the effect of
reversing the period.  The analytic theory allows the "digits" (called
"partial quotients") of some other numbers (for example e) to be
determined, but the expansion of pi or the cube root of 2 seems to be
random.
-- 

--R. T. Bumby ** Math ** Rutgers ** New Brunswick ** NJ08903 ** USA --
  above postal address abbreviated by internet to bumby@math.rutgers.edu
  voice: 908-932-0277 (but only when I am in my office)