Wondering about Domingo's Rational Mean book"

[Iain Davidson]

> "Iain Davidson" <Sttscitrans at tesco.net> replied:
>
> >I don't see why you choose [3/2, 4/3]
> >as the two initial fractions and why you choose
> >to multiply top and bottom by 2.
> >What you seem to be doing is choosing two fractions
> >a/b and c/d such that ac= 2l and bd = 1.

> The set of two initial fractions used in a rational process
> for approximating the square root of any number P  is:
> [a/b, P*b/a]
> It is clear that  [3/2, 4/3] and consequently all the other new sets
> satisfy the well known condition for generating reduced fractions  |ad-
> bc| = 1.

But then you have "seeded" the process to "get the right answer".
In the above case, a^2 - Pb^2 = 1, which is just Pell's equation.
If P was 61 or 109 , say, then it would be hard to find suitable initial
values.
For most values of a and b you would not get best approximations.
So you have to state a rule that gives the initial conditions explicitly
or finding a and b would be equivalent to solving Pell's equation.

It just seems to be a variant on the method used in schools to find square
roots. You take a 1xn rectangle, then take the average of the sides and
divide n by the average giving two new sides that are "more equal"
This has nothing to do with finding best approximations

If x^2 +ax - b =0
then x(x+a) = b
If you assume x = b and x'+a =1,
then x = b, x' = 1-a, and by taking mean
you can make x and x' converge to same value.
x^2 -n =0 gives special case above.

The same idea can obviously be extended to
higher degree polynomials but conditions for
convergence may be complicated.