06/14/2022, 01:22 AM
(06/13/2022, 10:52 PM)tommy1729 Wrote: i advise to use this for the computation around the fixpoint
(a x + k x^2 + ...)^[t] = a^t x + k a^(t-1) ( a^t - 1 )/(a-1) x^2 + ...
This quadration approximation is then plugged in
f^[t](x) = lim f^[n] ( a^t y + k a^(t-1) ( a^t - 1 )/(a-1) y^2 )
where y = f^[-n](x) and a and b can be easily computed from the closed form for x_n and taylors theorem.
Call it the quadratic fixpoint formula or so
regards
tommy1729
Hey, Tommy. Not to let you down or anything, but this is a known procedure. It can be done upto any polynomial, not just quadratic.
It essentially operators off the principle of instead of choosing a fixed point, we choose a fixed polynomial. So we know the first \(3\) coefficients of the Taylor polynomial near the fixed point. Iterate a slightly different Kneser formula, but do it for a fixed point which is polynomial. You'll get a far faster solution. Same Kneser algorithm, you just do it for polynomials.
