In an earlier posting, I had written about an approach, for how to find approximations of the roots of polynomials, of an arbitrary degree, but in such away, also to find all the complex roots. (:1)
But with such strategies, there are issues. One concept was, that a home-grown search algorithm would get close to the actual root. Next, polishing would make the result more accurate. And then, an augmented division would be computed, which is also referred to as “Deflation”, resulting in a deflated polynomial, as many times as the original polynomial’s degree, minus one.
Pondering this issue today, I realized that there was still a conceptual weakness in that approach, that being, the fact that some small amount of error is tolerated in the first root found, so that each successive deflated polynomial contains progressively greater degrees of error. What effectively happens next is, that accurate roots are found, of increasingly inaccurate polynomials, and, that there appeared to be few ways to detect and correct the resulting errors, in roots found afterwards. Theoretically, this problem could progress to the point, where doubt is evoked, in whether or not roots found later, were even roots of the original polynomial, since by that time, the object which the roots are being found of, is no longer that original polynomial.
(Update 6/08/2020, 18h35… )