In this exercise, you will learn what is involved in determining the interpolation polynomial numerically, how this polynomial is stored, and how it is evaluated.

a) Write a function S-divided_differences (nodes,values) in Mat- lab that computes the lower triangular matrix S of divided differences associated with nodes nodes and values values b) Apply the algorithm from task a) to the data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] c) Write a function c-Newton_coefficients (nodes,values) in Mat- lab that computes the vector c of coefficients of the interpolation poly- nomial associated with nodes nodes and values values in Newton form. d) Apply the algorithm from task c) to the data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] e) Write a function y-evaluate_Newton_polynomial (x,nodes,c) in Matlab that evaluates a Newton polynomial with nodes nodes and coefficients c at positions x and returns the corresponding values y, using the Horner-type algorithm from the lecturesotes. Apply the algorithm from task e) with positions x= [05, 15, 2 . 5 3.5] and data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] f) g) Use the algorithm from task e) to plot the interpolation polynomial associated with nodes-[0,1,2,3,4] and values [1,0,2,1,3] on the interval [0, 4]. Indicate the position of the interpolated data with small circles. Give your figure a proper title, axis labels and a legend a) Write a function S-divided_differences (nodes,values) in Mat- lab that computes the lower triangular matrix S of divided differences associated with nodes nodes and values values b) Apply the algorithm from task a) to the data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] c) Write a function c-Newton_coefficients (nodes,values) in Mat- lab that computes the vector c of coefficients of the interpolation poly- nomial associated with nodes nodes and values values in Newton form. d) Apply the algorithm from task c) to the data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] e) Write a function y-evaluate_Newton_polynomial (x,nodes,c) in Matlab that evaluates a Newton polynomial with nodes nodes and coefficients c at positions x and returns the corresponding values y, using the Horner-type algorithm from the lecturesotes. Apply the algorithm from task e) with positions x= [05, 15, 2 . 5 3.5] and data nodes-[0,1,2,3,4] and values-[1,0,2,1,3] f) g) Use the algorithm from task e) to plot the interpolation polynomial associated with nodes-[0,1,2,3,4] and values [1,0,2,1,3] on the interval [0, 4]. Indicate the position of the interpolated data with small circles. Give your figure a proper title, axis labels and a legend