# Evaluate accuracy of polynomial root finding

Given the coefficients of a polynomial of high degree, i.e. n > 10.000. The polynomials have some nice structure such that I was able to find a way to compute its roots. Now I would like to evaluate how accurate these computed roots actually are. What is the best way to do this?

Initially, I tried to compute the coefficients from the roots, but this turns out to be numerically unstable. For more details here.

• How did you manage the roots? Which method? (Newton, bisection...) Sep 25, 2017 at 15:46
• @Raffaele: The strongest property of the polynomials $p$ are that the roots are on the unit circle. So, having $p$ as the denominator of a transfer function, its impulse response consists of sinusoids with angles corresponding to the root location. (Hope this makes sense) In the end I refine with Newton.
– Jiro
Sep 25, 2017 at 15:56
• I find it difficult to believe that the structure which allowed you to find the roots would not be helpful when trying to evaluate their quality. Consider writing a few words about the structure. Also add a few words about the desired properties of a "more collective evaluation of the roots". Error estimates can be derived from successive iterates of Newton's method. at little extra cost. Why would a SIMD/OpenMP implementation of this not be good enough? Sep 25, 2017 at 17:38
• @CarlChristian Sorry, my idea for a "more collective evaluation of the roots" was pointless.
– Jiro
Sep 25, 2017 at 17:56