2020/09/06 by Dominic C. Milioto, Milioto, Dominic C.
Computer Science · Mathematics · #3008 #33B10 (Secondary) #33F05 (Primary) #65E99 #Complex Variables (math.CV) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Numerical Analysis (math.NA) #Numerical Methods and Algorithms
paper · pdf · doi:10.48550/arxiv.2009.02745
openalex publication_date 2020/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper describes a systematic method of numerically computing and indexing fixed points of zzw for fixed z or equivalently, the roots of T2(w;z)=w-zzw. The roots are computed using a modified version of fixed-point iteration and indexed by integer triplets \n,m,p\ which associate a root to a unique branch of T2. This naming convention is proposed sufficient to enumerate all roots of the function with (n,m) enumerated by ℤ2. However, branches near the origin can have multiple roots. These cases are identified by the third parameter p. This work was done with rational or symbolic values of z enabling arbitrary precision arithmetic. A selection of roots up to order \1012,1012,p\ with |z|≤ 1012 was used as test cases. Results were accurate to the precision used in the computations, generally between 30 and 100 digits. Mathematica ver. 12 was used to implement the algorithms.