Problem
Estimate the condition number of this roots-finding problem. Evaluate the relative input error by [ text { relIn }=max left|frac{bar{a}_{j}-a_{j}}{a_{j}}right|, ] 1 and the relative output error by [ text { relOut }=max _{j}left|frac{tilde{r}_{j}-r_{j}}{r_{j}}right|, ] wherer~jandrjrepresents the computed roots ofp~(x)andp(x), respectively. MATLAB function max can compute the maximum. Then the condition number can be evaluated by [ text { Cond }=frac{text { relOut }}{text { relIn }} . ] To be more representative, for a polynomial of a given degree,n, you need compute this estimates for 50 randomly perturbed polynomials and then take the average of these 50 estimates as the final estimate for the condition number. Plot the condition number estimate with respect ton, for1≤n≤30. You may need uselogscale in they-axis to catch the behavior of this plot (see MATLAB function semilogy). Explain what you observe in this plot and the plots you obtained.