Why can small changes in polynomial coefficients cause large changes in its roots?

In the field of numerical analysis, the sensitivity of a polynomial’s roots to small changes in its coefficients is a matter of stability. When a problem is highly sensitive to such perturbations, it is described as ill-conditioned.

This phenomenon occurs because the roots of a polynomial are not always robust functions of the coefficients. For example, consider the polynomial $p(x)=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)$. If the coefficient of $x^6$ is altered by a very small amount—such as changing $-28$ to $-28.002$—the roots can shift dramatically, potentially moving from real numbers to complex values.

Numerical analysts emphasize that the stability of a solution is fundamental; a numerical method should ideally be no more sensitive to data changes than the original problem itself. Understanding this sensitivity is crucial because it highlights the inherent limitations when using finite precision arithmetic to solve complex mathematical problems.1