Perturbation to the coefficients of a polynomial

Question

I am reading 'Trefethen and Bau: Numerical Linear Algebra' book and I came across the following problem- Assume that the polynomial $p(x) = \sum_{k=0}^n a_k x^k$ with real coefficients has $n$ distinct real roots. Show that if the $i^{th}$ coefficient $a_i$ is perturbed by an infinitesimal quantity $\delta a_i$ , then the perturbation of the $j^{th}$ root $x_j$ is $$ \delta x_j = \frac{(\delta a_i)x_j^i}{p'(x_j)} $$ I am not sure how to approach the above problem.

Answer 1

The claim as stated is false (there is a missing minus sign).

To see this, consider $n=2$ so $p(x)=a_0+a_1 x$ and $p'(x)=a_1$. The root $x$ satisfies $x=-\frac{a_0}{a_1}$. Thus $\frac{dx}{da_0}=-\frac{1}{a_1}$ and $\frac{dx}{da_1}=\frac{a_0}{a_1^2}$. These expression have the opposite sign to the formula you gave.

To derive the correct formula, suppose: $$0=p(x)=\sum_{k=0}^n{a_k x^k}$$ This holds for all $a_i$. So we can differentiate both sides with respect to $a_i$ and the equality will still hold. This gives: $$0=\sum_{k=0}^n{k a_k x^{k-1}\frac{dx}{da_i}}+x^i=p'(x)\frac{dx}{da_i}+x^i$$ Thus: $$\frac{dx}{da_i}=-\frac{x^i}{p'(x)}$$