  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.
