| One of the Hackmem items gives an analytic flow for Newton's method,
which would produce a solution to this problem. My copy isn't
handy or I'd go look it up, I don't remember how the flow was
derived exactly.
An analytic flow is a continuous function of the "iteration number"
for an iterative process - simple examples would be the exponential
expression for the Fibonacci numbers, the Gamma function for the
factorials, etc. but flows have been derived for many other
processes as well, A related subject is Schroeder functions.
- Jim
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| Solution by Stephen C. Lock, Florida Atlantic University, Boca Raton,
Florida.
We will show that x[n] = p[n]/q[n], where p[n] and q[n] are integers
such that p[n]+q[n]sqrt(3) = (2+sqrt(3)^4^n.
Let x[n] = p[n]/q[n], for integers p[n] and q[n]. Then
p[n+1] p[n]^2+18*p[n]^2q[n]^2+9q[n]^4
------ = ------------------------------.
q[n+1] 4p[n]^3*q[n]+12p[n]q[n]^3
It is natural to set p[n+1]=p[n]^4+18p[n]^2*q[n]^2+9q[n]^4, and
q[n+1]=4p[n]^3*q[n]+12p[n]q[n]^3.
Then
p[n+1]+/-q[n+1]sqrt(3) =
p[n]^4+6p[n]^2*(q[n]sqrt(3))^2+(q[n]sqrt(3)^4 +/-
4p[n]^3*(q[n]sqrt(3) +/- 4 p[n](q[n]sqrt(3)^3 =
(p[n]+/-q[n]sqrt(3)^4.
Since we may choose p[0]=2 and q[0]=1, this establishes
p[n]+q[n]sqrt(3)=(2+sqrt(3))^4^n, p[n]-q[n]sqrt(3)=(2-sqrt(3))^4^n.
Solving for p[n] and q[n] yields
p[n] (2+sqrt(3))^4^n + (2-sqrt(3))^4^n
x[n] = ---- = sqrt(3) ---------------------------------.
q[n] (2+sqrt(3))^4^n - (2-sqrt(3))^4^n
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