Problem $3\mathrm{.}3$

Consider interpolating the following periodic function on interval $-,$ :

$f(x)=\frac{e^{acosx}}{2I_0(a)},$

where $a$ is a given parameter that determines the smoothness of the function, and $I_0(a)$ is a Bessel function

used for normalization purposes, available in MATLAB as besseli $(0,$ a$).$ Fix $a=3.$

We want to see if and how fast the interpolation error converges to zero as the number o interpolation points

increases, for several different types of interpolants.

a$)$ Plot in the same figure function $f(x)$ and polynomial interpolants $P_n(x)$ on $n+1$ evenly spaced points, where

$n=2,4,8,16,32.$ Note that due to periodicity the values of the function at the first and last nodes will be

equal, but that the polynomial itself does not recognize the periodicity.

b$)$ Plot in the same figure function $f(x)$ and piecewise linear interpolants $L_n(x)$ on $n+1$ evenly spaced points,

where $n=2,4,8,16,32.$

c$)$ Plot in the same figure function $f(x)$ and cubic periodic splines $S_n(x)$ on $n+1$ evenly spaced points, where

$n=2,4,8,16,32.$

d$)$ Compare these interpolants. For example, you can estimate the maximum error and present results in a

table. Also, you can plot the errors for, say $n=32.$