The following is a list of simple polynomials that when iterated converge to the square root of 2.
The higher the degree, the faster it is.
They work best if the starting value is already close enough to the square root of 2, as is typical for all iterate methods.
A starting value between [- 1/3 , 2 ] is usually good.
(x/2) (3 - (1/2) x^2)
(x/2) (3 - (1/2) x^2) (1 + (3/32)(x^2 - 2)^2)
(x/2) (3 - (1/2) x^2) (1 + (3/32)(x^2 - 2)^2) (1 - (1/64)(x^2 - 2)^3)
(x/2) (3 - (1/2) x^2) (1 + (3/32)(x^2 - 2)^2) (1 - (1/64)(x^2 - 2)^3) (1 + (27/2048)(x^2 - 2)^4)
...
We get a mysterious sequence of rational numbers.
They can be computed and we can compute how fast it converges.
But still, it appears a somewhat mysterious sequence.
Is the sequence just as complicated as the digits of the square root of 2 ?
Or is there something useful, practical or interesting about them ?
Can we find asymptotics or properties ?