Suppose we have data x = [k] = and corresponding values y = [YA]=1.... and we want to find a polynomial pn(x) = Σ}}=0 αjx³ which is the "best fit" for the given data. We can find such polynomial by finding coefficients {a}}} 0 that minimizes the sum =0 m Kyk - Pn(k)]². = k=1 The partial derivatives of K with respect to the coefficients of pn must be zero, i.e. for i = 0,...,n, m әк 0 = 2x [Yk - Pn(x)] Jai k=1 Here, we have the convention that x = 1. Let V = [x]i=1,...,m;j=0,...,n, n m m Σα; Σ i+j = Yk (1) j=0 k=1 k=1 and a = = [ai]i-o....n Then system (1) is equivalent to VTVα = V¹y. The solution to this gives the coefficient of the polynomial which is the "best fit" for the given data. Write a python code containing the function vander whose inputs are data vector x and degree of polynomial degree and returns matrix V as above.

Suppose we have data x = [k] = and corresponding values y = [YA]=1.... and we want to find a polynomial pn(x) = Σ}}=0 αjx³ which is the "best fit" for the given data. We can find such polynomial by finding coefficients {a}}} 0 that minimizes the sum =0 m Kyk - Pn(k)]². = k=1 The partial derivatives of K with respect to the coefficients of pn must be zero, i.e. for i = 0,...,n, m әк 0 = 2x [Yk - Pn(x)] Jai k=1 Here, we have the convention that x = 1. Let V = [x]i=1,...,m;j=0,...,n, n m m Σα; Σ i+j = Yk (1) j=0 k=1 k=1 and a = = [ai]i-o....n Then system (1) is equivalent to VTVα = V¹y. The solution to this gives the coefficient of the polynomial which is the "best fit" for the given data. Write a python code containing the function vander whose inputs are data vector x and degree of polynomial degree and returns matrix V as above.