Forward stable computation of roots of real polynomials with only real distinct roots
Provides a provably forward stable and efficient algorithm for a specific class of polynomials, offering high accuracy for numerically challenging problems.
The paper presents an O(n^2) forward stable algorithm for computing all roots of real polynomials with only real distinct roots, achieving almost full accuracy per root. It outperforms methods like MPSolve and Newton's method on difficult problems such as Wilkinson's polynomials.
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.