Higher-order root distillers
For numerical analysts, this provides a preconditioning technique for root-finding on intervals, though it is incremental in nature.
The paper introduces a root distiller method using high-order recursive maps to globally separate and compute real roots of functions, demonstrating accurate root computation for high-degree inexact Chebyshev polynomials.
Recursive maps of high order of convergence $m$ (say $m=2^{10}$ or $m=2^{20}$) induce certain monotone step functions from which one can filter relevant information needed to globally separate and compute the real roots of a function on a given interval $[a,b]$. The process is here called a root distiller. A suitable root distiller has a powerful preconditioning effect enabling the computation, on the whole interval, of accurate roots of an high degree polynomial. Taking as model high-degree inexact Chebyshev polynomials and using the {\sl Mathematica} system, worked numerical examples are given detailing our distiller algorithm.