I really do need help , this has to be done using MATLAB.

Data interpolation and applications

Objective:

This assignment is to empower your ability of using data interpolation to model observed data.

Required tasks:

Prove that the interpolating polynomial is unique. That is, if P(x) and Q(x) are two polynomials that interpolate given n + 1 distinct nodes (x_0, y₀), (x_1, y₁), (x_2, y₂), , (x_n, y_n), then P(x) and Q(x) are the same.

Complete the tasks with the dataset for 1 x 1 provided below:

x	-1.00	-0.96	-0.65	0.10	0.40	1.00
y	-1.0000	-0.1512	0.3860	0.4802	0.8838	1.0000

Interpolate the data by each of the four interpolants discussed in this chapter: piece-wise linear, polynomial interpolation, cubic spline, and piecewise cubic Hermite interpolation. Plot the results.

What are the values of each of the four interpolants at x = 0.3? Which of these values do you prefer? Why?

The data were actually generated from a low-degree polynomial with integer coefficients. What is that polynomial?

The cosine function f(x) = cos x is a transcendental function that cannot be evaluated directly. With arbitrary ten values of cos x for x in the interval (0, 1), one approximates cos x at any point x in (0, 1). Complete the following tasks with MATLAB:

Generate ten points (x_i, cos x_i) for x_i in the interval (0, 1).

Use a polynomial p(x) interpolating the ten points. Compares p(x) and cos x graphically.

For any x in the interval (0, 1), find an upper bound error |p(x) cos x|.

The dataset below is from observing an unknown the function z = f(x, y). The node (0.5, 0.1, 0.165) means 0.165 = f(0.5, 0.1) Apply function polynomial interpolation, visualize the unknown function graphically.

(0.5, 0.1, 0.165), (0.5, 0.2, 0.428), (0.5, 0.3, 0.687)

(1.0, 0.1, 0.271), (1.0, 0.2, 0.640), (1.0, 0.3, 1.003)

(1.5, 0.1, 0.447), (1.5, 0.2, 0.990), (1.5, 0.3, 1.524)

(2.0, 0.1, 0.738), (2.0, 0.2, 1.568), (2.0, 0.3, 2.384)

(2.5, 0.1, 1.216), (2.5, 0.2, 2.520), (2.5, 0.3, 3.800)

Optional: Interpolating your dataset with selected attributes and visualize the approximated function graphically.