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How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't change much but is there any theorem around this?

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  • $\begingroup$ Are you asking about roots in the complex plane or real roots? In the complex plane we can prove that perturbations of the coefficients give continuous trajectories of the "existing" roots, but may cause real roots to appear or disappear (when multiplicity is more than one). $\endgroup$ – hardmath Apr 19 '15 at 13:29
  • $\begingroup$ Real roots... in that the polynomial comes from the truncation the Taylor Series for a real (single variable) function about x0. If I know that the Taylor Series has an infinite radius of convergence, and only wish to know a few roots on either side of x0, how does taking more terms in the Taylor series effect the accuracy of these? Is there a theorem? $\endgroup$ – RayR Apr 19 '15 at 13:53
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    $\begingroup$ It sounds like by "accuracy" you have in mind the location of roots of the analytic function represented by the Taylor series. This is more difficult than showing the perturbation of "existing" roots is small by adding a "small" higher order term. Consider the exponential function $y = e^x$ and its truncations. At odd degrees truncations have one real root; at even degrees truncations have no real roots. $\endgroup$ – hardmath Apr 19 '15 at 14:14
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No, a very small coefficient perturbation can affect the roots dramatically, meaning lets say a root changes tens orders of magnitude more than coefficient change. This is actually (for many people) one of the most surprising results in mathematics. Most of the time, this will not be the case, however this still poses a difficult problem in numerical analysis.

Following example is from a Matlab blog post called "Wilkinson’s Polynomials"

Observe this polynomial

$$ w(x) = \prod_{i=1}^{20} (x - i) = (x-1)(x-2) \ldots (x-20)$$

It has roots $1, 2, \dots, 20$.

Now observe this family of polynomials

$$ w(x) - \alpha x^{19}, \alpha = +- 2^{-k}, k=23, 24, \dots, 36 $$

The behavior of roots is illustrated in this picture (red is for negative coefficient perturbation, black for positive)

enter image description here

The root movement is huge, given that coefficient change is less than $2^{-23}$.

A summary and further info on this topic can be found here.

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