How is this 'root stability' differential equation derived?

[tomizzo]

I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations.

Referring to the Wilkinson polynomial wikipedia page right here,[/PLAIN] you can see that there is a differential equation listed under the 'stability analysis' section. This derivative explains how the rate of change of the roots (with respect to some error scaling parameter 't') equals the error evaluated at the polynomial error function divided by the original polynomials derivative... That was a mouthful..

So my question: How is this differential equation derived? Maybe I'm just rusty on my calculus, but would someone be willing to demonstrate how it was derived given the information from the original problem statement?

Thanks,

 

[pasmith]

I think the notation in the article is confusing; $\alpha_j$ is used both for the root of original polynomial and a function which gives the root of the perturbed polynomial. These really need to be given different symbols.

Define $p(x) = \prod_j (x - \alpha_j)$. We perturb this to $$q(x,t) = p(x) + tc(x) = \prod_j (x - \beta_j(t)).$$ Now by definition $q(\beta_j(t),t) = p(\beta_j(t)) + tc(\beta_j(t)) = 0$ for every $t$, so we can differentiate with respect to $t$ to obtain $$0 = p'(\beta_j)\frac{d\beta_j}{dt} + c(\beta_j) + tc'(\beta_j)\frac{d\beta_j}{dt}$$ which we rearrange to obtain $$\frac{d\beta_j}{dt} = -\frac{c(\beta_j)}{p'(\beta_j) + tc'(\beta_j)}.$$ The right hand side can now be expanded as a Taylor series about $t = 0$, and as $\beta_j(0) = \alpha_j$ this yields to first order $$\frac{d\beta_j}{dt} = -\frac{c(\alpha_j)}{p'(\alpha_j)}$$ as required.