Revisiting Gilbert Strang's "A Chaotic Search for $i$"
For researchers in numerical analysis and dynamical systems, this provides a deeper theoretical understanding of iterative root-finding methods on a simple polynomial, though the results are incremental.
This paper extends Strang's analysis of Newton's method on x^2+1 to higher-order Householder methods and the secant method, providing exact symbolic formulas for the iterations and characterizing their asymptotic behavior.
In the paper "A Chaotic Search for $i$"~(\cite{strang1991chaotic}), Strang completely explained the behaviour of Newton's method when using real initial guesses on $f(x) = x^{2}+1$, which has only a pair of complex roots $\pm i$. He explored an exact symbolic formula for the iteration, namely $x_{n}=\cot{ \left( 2^{n} θ_{0} \right) }$, which is valid in exact arithmetic. In this paper, we extend this to to $k^{th}$ order Householder methods, which include Halley's method, and to the secant method. Two formulae, $x_{n}=\cot{ \left( θ_{n-1}+θ_{n-2} \right) }$ with $θ_{n-1}=\mathrm{arccot}{\left(x_{n-1}\right)}$ and $θ_{n-2}=\mathrm{arccot}{\left(x_{n-2}\right)}$, and $x_{n}=\cot{ \left( (k+1)^{n} θ_{0} \right) }$ with $θ_{0} = \mathrm{arccot}(x_{0})$, are provided. The asymptotic behaviour and periodic character are illustrated by experimental computation. We show that other methods (Schröder iterations of the first kind) are generally not so simple. We also explain an old method that can be used to allow Maple's \textsl{Fractals[Newton]} package to visualize general one-step iterations by disguising them as Newton iterations.