Koenigs Function Expansion Implementation

The Koenigs function, $h$, offers a method to linearize the dynamics around a fixed point for holomorphic functions. This transformation has broad applications, including enhancing the convergence properties of iterative methods like Newton's method and beyond.

Overview

The Koenigs function $h$ for a given function $f$ that maps a domain into itself and fixes a point (typically the origin) is defined by Schröder's equation:

$$ h(f(z)) = \lambda h(z) $$

where $\lambda = f'(0)$.

The recursive formula for the coefficients of the Koenigs function expansion is given by:

$$ an = \frac{1}{n} f^{(n-1)}(0) - \sum{k=2}^{n-1} k a_k f^{(n-k)}(0) $$

To-Do

[ ] Implement the $k$-next function expansion in Arb4j to test its capabilities.
[ ] Explore applications to iterative methods such as Newton's method.

Koenigs Function Expansion Proof

The Koenigs function, denoted as $h$, linearizes the dynamics around a fixed point for a holomorphic function $f$. This document presents the recursive formula for the coefficients of the Koenigs function using Schröder’s equation.

Overview

Given a function $f(z)$ that maps a domain into itself, fixes a point (typically the origin), and is not an automorphism, the Koenigs function $h$ satisfies Schröder's equation:

$$ h(f(z)) = \lambda h(z), $$

where $\lambda = f'(0)$ and $|\lambda| < 1$.

Function Expansions

The function $f(z)$ and the Koenigs function $h(z)$ can be expanded as:

$$ f(z) = \lambda z + \sum_{n=2}^\infty b_n z^n, $$

$$ h(z) = z + \sum_{n=2}^\infty a_n z^n. $$

where ( b_n ) is the coefficient of ( z^n ) in the Taylor series expansion of ( f ), defined as:

$$ b_n = \frac{f^{(n)}(0)}{n!} $$

Schröder’s Equation and Recursive Formula

Applying Schröder’s equation and equating the coefficients of $z^n$ on both sides, we obtain:

$$ \lambda z + \sum_{n=2}^\infty an \lambda^n z^n = \lambda z + \sum{n=2}^\infty \lambda a_n z^n. $$

For the coefficients of $z^n$, the detailed recursive relation is:

$$ a_n \lambda^n = \lambda an + \sum{k=2}^{n-1} k ak b{n-k} $$

Isolating $a_n$:

$$ an (\lambda^n - \lambda) = \sum{k=2}^{n-1} k ak b{n-k} $$

Solving for $a_n$:

$$ an = \frac{1}{\lambda^n - \lambda} \left( \sum{k=2}^{n-1} k ak b{n-k} \right) $$

Conclusion

This recursive method provides an efficient way to compute the coefficients of the Koenigs function, particularly useful in applications requiring fast and accurate computation of iterative dynamics near a fixed point.

crowlogic / arb4j