Enhancing the Power of Cardinal's Algorithm

TL;DR

This paper improves Cardinal's factorization algorithm by reducing its computational complexity and extending its applicability, making it more practical for polynomial root-splitting tasks.

Contribution

It amends and extends Cardinal's algorithm, reducing the final computational stage to manageable matrix operations and broadening its scope to more general polynomial factors.

Findings

Reduced the computational cost of the final stage

Extended the algorithm to more general polynomial factors

Introduced techniques of independent interest for matrix computations

Abstract

Cardinal's factorization algorithm of 1996 splits a univariate polynomial into two factors with root sets separated by the imaginary axis, which is an important goal itself and a basic step toward root-finding. The novelty of the algorithm and its potential power have been well recognized by experts immediately, but by 2016, that is, two decades later, its practical value still remains nil, particularly because of the high computational cost of performing its final stage by means of computing approximate greatest common divisor of two polynomials. We briefly recall Cardinal's algorithm and its difficulties, amend it based on some works performed since 1996, extend its power to splitting out factors of a more general class, and reduce the final stage of the algorithm to quite manageable computations with structured matrices. Some of our techniques can be of independent interest for matrix computations.