Beginning with this assignment, homework tasks are described at a higher level. Apply your experience from previous assignments to cogently formulate a solution.

1Track 1 & 2

Study the convergence of polynomial interpolation, $p_n\left(x_i\right)=f\left(x_i\right)$, $i=1,2,\dots ,n$ with increasing number of sample points $n$ for the following functions:

1. $f:[-1,1]\to ℝ$, $f\left(t\right)=\cos (\pi t/2)$;

2. $f:[-1,1]\to ℝ$, $f\left(t\right)=1/(1+25t^2)$;

3. $f:[-1,1]\to ℝ$, $f\left(t\right)=\exp (-25t^2)$.

For each case consider both equidistant and Chebyshev sample points, present plots of the function and interpolant, plots of the error as a function of $n$, and compare the observed error with that predicted by the formula

Comment on what you observe.

2Track 1

For $n=5$ equidistant sample points, explicitly write the Lagrange and Newton forms of the interpolating polynomial.

3Track 2

Repeat the above convergence study for Hermite interpolation $p_n\left(x_i\right)=f\left(x_i\right)$, $p_n^{\text{'}}\left(x_i\right)=f^{\text{'}}\left(x_i\right)$, $i=1,2,\dots ,n$.