Forward stable computation of roots of real polynomials with only real distinct roots

TL;DR

The paper presents a forward stable, efficient algorithm for computing roots of real polynomials with only real, distinct roots, achieving high accuracy and outperforming existing methods in challenging cases.

Contribution

It introduces a novel, stable algorithm leveraging arrowhead matrix representations and eigenvalue computations for accurate polynomial root-finding.

Findings

Achieves almost full accuracy in root computation.

Operates in $O(n^2)$ complexity.

Performs well on difficult problems like Wilkinson's polynomials.

Abstract

As showed in (Fiedler, 1990), any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be chosen real. By using accurate forward stable algorithm for computing eigenvalues of real symmetric arrowhead matrices from (Jakovcevic Stor, Slapnicar, Barlow, 2015), we derive a forward stable algorithm for computation of roots of such polynomials in $O(n^2)$ operations. The algorithm computes each root to almost full accuracy. In some cases, the algorithm invokes extended precision routines, but only in the non-iterative part. Our examples include numerically difficult problems, like the well-known Wilkinson's polynomials. Our algorithm compares favourably to other method for polynomial root-finding, like MPSolve or Newton's method.