New Progress in Classic Area: Polynomial Root-squaring and Root-finding

TL;DR

This paper introduces novel polynomial root-finding algorithms that improve numerical stability and efficiency by leveraging reduction techniques and black box evaluations, enabling fast zero and eigenvalue approximations.

Contribution

The authors develop new root-finding methods based on polynomial reduction and black box evaluations, avoiding classical DLG iterations and enhancing computational efficiency.

Findings

Algorithms can be applied to black box polynomials

Enables efficient concurrent zero approximation

Applicable to eigenvalue problems of matrices

Abstract

The DLG root-squaring iterations, due to Dandelin 1826 and rediscovered by Lobachevsky 1834 and Graeffe 1837, have been the main approach to root-finding for a univariate polynomial p(x) in the 19th century and beyond, but not so nowadays because these iterations are prone to severe numerical stability problems. Trying to avoid these problems we have found simple but novel reduction of the iterations applied for Newton's inverse ratio -p'(x)/p(x) to approximation of the power sums of the zeros of p(x) and its reverse polynomial. The resulting polynomial root-finders can be devised and performed independently of DLG iterations, based on Newton's identities or Cauchy integrals. In the former case the computation involve a set of leading or tailing coefficients of an input polynomial. In the latter case we must scale the variable and increase the arithmetic computational cost to ensure numerical stability. Nevertheless the cost is still manageable, at least for fast root-refinement, and the algorithms can be applied to a black box polynomial p(x)--given by a black box for the evaluation of the ratio p'(x)/p(x) rather than by its coefficients. This enables important computational benefits, including efficient recursive as well as concurrent approximation of a set of zeros of p(x) or even all of its zeros, acceleration where an input polynomial can be evaluated fast, and extension to approximation of the eigenvalues of a matrix or a polynomial matrix, being efficient if the matrix can be inverted fast, e.g., is data sparse. We also recall our recent fast algorithms for approximation of the root radii, that is, the distances to the roots from the origin or any complex value to the zeros of p(x), and apply it for fast black box initialization of polynomial root-finding by means of functional iterations, e.g., Newton's, Ehrlich's, and Weierstrass's.