Math 451: Consider the Polynomial Interpolation - Mathematics Assignment Help

Assignment Task:

Task:

Problem 1
Consider the polynomial interpolation for the following data points

x -2 0 1 2
y -5 1 1 7

(a). Write down the linear system in matrix form for solving the coefficients ai (i = 0, · · · , n) of the polynomial pn(x).
(b). Use the Lagrange interpolation process to obtain a polynomial to approximate these data points (simplify your answer).

Problem 5: Newton’s Divided Difference in Matlab.
(a). Write two functions in Matlab. The first function, called divdiff, should read in two vectors x and y, and return a table (a matrix) of the divided difference values. This means, the first few lines of your divdiff.m file should be:
function a=divdiff(x,y)
% input: x,y: the data set to be interpolated
% output: a: table for Newton’s divided differences.
The second function, called polyvalue, should read in the table of divided difference values generated by the function divdiff, the x-vector, and a vector t. The output of the function are the values of Newton’s polynomial computed at points given in the vector t. This means, your file polyvalue.m should start with the following few lines:

function v=polyvalue(a,x,t)
% input: a= Newton’s divided differences
% x= the points for the data set to interpolate,
% same as in divdiff.
% t= the points where the polynomial should be evaluated
% output: v= value of polynomial at the points in t
What to hand in: Hand in the files divdiff.m and polyvalue.m.
(b). Here you can test your Matlab functions in (a). Using 21 equally spaced nodes on the interval [−5, 5], find the interpolating polynomial p of degree 20 for the function f(x) = (x 2+1)−1
.
Plot the functions f(x) and p(x) together at at least 201 equally spaced points (you may use more points if you like), including the nodes. Plot also the error e(x) = |f(x) − p(x)|. Why does this work so poorly? Write your comments.
What to hand in: Hand in your Matlab script file, and the plots of f(x) with p(x), and the error, and your comments.
(c). With the same problems in (b), now we use the Chebyshev nodes of Type I and Type II:

Type I: xi = 5 cos(iπ/20), 0 ≤ i ≤ 20 ,
Type II: xi = 5 cos[(2i + 1)π/42] , 0 ≤ i ≤ 20 .

Find the interpolating polynomial p(x) for each Type, and plot f(x) with p(x) for both Types, and the errors e(x) = |f(x) − p(x)| as well. Do these nodes work better than uniform nodes?
Give your comments.
What to hand in: Hand in your Matlab script file, and the plots of f(x) with p(x), and the error, and your comments.

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