$2\mathrm{.}10$ A Study of Convergence for Polynomial Interpolation with Uniform and

Chebychev nodes

For a given function f $($x$)=|$x$|,$ we generate interpolating polynomial Pn$($x$)$ on $[-1,1],$ for n $=$

$6,12,18,24,30,36,$ using two types of nodes

$($a$)$ Using uniformly spaced nodes. For each n$,$ the nodes are

hn $=2$

n $,$ xi $=1+$ ih$,$ i $=0,1,2,,$ n$.$

$($b$)$ Using Chebyshev nodes Type I. For each n$,$ the nodes are

xi $=$ cos i$\backslash$pi

n $,$ i $=0,1,2,,$ n$.$

For each of the interpolating problem, do the following:

$($i$)$ Compute the interpolation polynomial Pn using Newton$$s divided differences;

$($ii$)$ Plot Pn and f over points t$=-1$:$0\mathrm{.}001$:$1$;

$($iii$)$ Compute the absolute error, i$.$e$.$ en $=$ max $|$f $($t$)$ Pn$($t$)|$;

$($iv$)$ Plot the error en against hn $=2/$n in loglog$.$

Discuss the convergence of the uniform nodes and Chebyshev nodes. What would you conclude?