Step by step solution to "Numerically find the half-iterate of a quadraticLet's say..."

Zack Wise

Answered question

2022-04-21

Numerically find the half-iterate of a quadratic
Let's say we have a function

f (x) = n + m x + l x^{2}

, and want to find another function

h (x) = a + b x + c x^{2}

such that

h (h (x)) = f (x)

. Here's what you get when composing h on itself:

h (h (x)) = a + a b + b^{2} x + b c x^{2} + a^{2} c + 2 a b c x + 2 a c^{2} x^{2} + b^{2} c x^{2} + 2 b c^{2} x^{3} + c^{3} x^{4}

Elsaidrge

Beginner2022-04-22Added 11 answers

By eliminating a and c, I obtained a polynomial equation in b:

a = \frac{2 n b}{2 b + b^{2} + m},

c = \frac{(m - b^{2}) (m + b^{2} + 2 b)}{4 n b^{2}},

m^{3} + 2 b m^{2} + (m^{2} - 4 \ln - 2 m) b^{3} + m b^{4} - 2 b^{6} - b^{7} = 0 .

You can solve it with a polynomial solver, or even with just Newton. There is at least one real root.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?