It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (I.e. see page 133 of Approximation Theory and Approximation Practice by Trefethen.) What is the underlying reason for this? Thank you very much.

It is stated in several sources on numerical analysis that the general problem of polynomial root-finding is ill-conditioned, but that it is well-conditioned if the roots are near the unit circle. (e.g. see page 133 of Approximation Theory and Approximation Practice by Trefethen.) What is the underlying reason for this? Thank you very much.