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

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