Trefethen & Bau ("numerical linear algebra"; 1997) compute the condition number of polynomial root finding for $f(x) = (x-1)^2$ to be $k=\infty$ because the Jacobian does not exist. I want to know and understand the derivation of that result, because the book only explains it in words with an example perturbation of $x^2-2+0.9999 = (x-0.99)(x-1.01)$. Thank you.
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Condition number has to do with the size of the change in output for a small change in input, right? So if you change the input to $(x-1)^2-\epsilon^2$, you change the output (that is, the root of the polynomial) by $\epsilon$, and the ratio of change of output to change of input is $\epsilon/\epsilon^2$, which is $1/\epsilon$, which goes to infinity as $\epsilon\to0$. – Gerry Myerson Nov 17 '19 at 01:58
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Thanks, Gerry for your very intuitive clarification. That makes sense. – abenol Nov 17 '19 at 04:57