Exercise 3. Conditioning of Polynomial Root-Finding. Let a, be the ith coeficient and r, the jth root of ) of a root z, with respect to a perturbation of the Root-Finding. Let a be the ith coef a polynomial p(z). Consider a small perturbations , and 2y of ai and zj, respectively. (a) Show that the relative condition number Krea single coefficient a, is 18211 /-= laid-il (3 pts.) (b) Consider the Wilkinson's polynomial with exact roots j = j, j = 1, 2, 3, . . . , 19, 20. Use the following code to generate the Wilkinson polynomial function [pl polyWilkinson (n) syms x % Wilkinson polynomial as (x - 1) (x - 2) (x - n) Pn prod(x-(1:n)) % Wilkinson polynomial in monomial basis x^i P expand (Pn); % get coefficients, the polynomial is represented by its coefficients p sym2poly (P) Then, using MATLAB function polyder, compute the following condition numbers Krel (1,a19) Krel 15, a19) Krel (20, a19) rel (15, a15 What can you say about conditioning of the roots of the Wilkinson polynomial? Explain why certain roots are more ill-conditioned than others. Exercise 3. Conditioning of Polynomial Root-Finding. Let a, be the ith coeficient and r, the jth root of ) of a root z, with respect to a perturbation of the Root-Finding. Let a be the ith coef a polynomial p(z). Consider a small perturbations , and 2y of ai and zj, respectively. (a) Show that the relative condition number Krea single coefficient a, is 18211 /-= laid-il (3 pts.) (b) Consider the Wilkinson's polynomial with exact roots j = j, j = 1, 2, 3, . . . , 19, 20. Use the following code to generate the Wilkinson polynomial function [pl polyWilkinson (n) syms x % Wilkinson polynomial as (x - 1) (x - 2) (x - n) Pn prod(x-(1:n)) % Wilkinson polynomial in monomial basis x^i P expand (Pn); % get coefficients, the polynomial is represented by its coefficients p sym2poly (P) Then, using MATLAB function polyder, compute the following condition numbers Krel (1,a19) Krel 15, a19) Krel (20, a19) rel (15, a15 What can you say about conditioning of the roots of the Wilkinson polynomial? Explain why certain roots are more ill-conditioned than others