Consider a mountain that is perfectly described by the Gumbel$$s function $()=$

$$

$$

$.$ The

mountain is reconstructed by sampling $($surveying$)$ points from it and interpolating all other points

from the interpolant. To put it in context, assume you are a surveyor, and you want to minimize

the number of points to sample and from which you should reconstruct the Runge Mountain as

accurately as possible. Define the reconstruction error as

$=|()()|$

where $()$ is the interpolation at $$ in $[5,3]$ at intervals of $0\mathrm{.}1.$ Compare and discuss your

results for the following methods of sampling and interpolation:

a$)$ Polynomial interpolation $($any method$)$ using $9$ randomly sampled points,

$($in Matlab, $=$a$+($b$-$a$)*$rand is a random number on $[$a$,$ b$])$

b$)$ Polynomial interpolation using $9$ equally spaced points

c$)$ Polynomial interpolation using $9$ Chebyshev points,

$=$a$+($b$-$a$)*(-$ cos

$()$

$$

$+1)/2$ on $[$a$,$b$]$ for i $=0$ to n$-1$

d$)$ RBF interpolation $($any$)$ using $9$ equally spaced points $($use appropriate value for parameter $)$

e$)$ RBF interpolation using $9$ Chebyshev points