Question

consider the function f(x) = cos(2x²) for x ∈ [−3, 1 2 ]. Select n equally spaced nodes (including the interval endpoints). Approximate the function f(x) using the following interpolation methods with n = 10, n = 20, and n = 40. Write a MATLAB code to construct the polynomial interpolant Pn(x). Plot the function versus x and the interpolant versus x for the requested values of n. Also plot the error of the interpolant as a function of x for the various values of n.

Answer 1

The student is asked to use polynomial interpolation to approximate the function f(x) = cos(2x²) at different numbers of nodes using MATLAB, and to plot both the polynomial interpolants and the original function, as well as the errors of approximation.

Step-by-step explanation:

The question involves using polynomial interpolation methods to approximate the function f(x) = cos(2x²) within the interval –[−3, 1/2]. Specifically, the question asks for the construction of polynomial interpolants Pn(x) using different numbers of nodes (n=10, n=20, and n=40), plotting both the interpolants and the original function, and also plotting the error of approximation. To achieve this, one can use MATLAB to generate the required nodes, calculate the coefficients of the interpolating polynomial, and then plot the results along with error calculations.

Using MATLAB, the 'linspace' function can generate equally spaced nodes; 'polyfit' or the Vandermonde matrix can compute the polynomial coefficients; 'polyval' can evaluate the polynomial; and subtracting the polynomial from the original function yields the error.

The code would iteratively perform these operations for each specified value of n, and generate plots to visually represent the interpolation and the resulting errors.