In the previous question, the data points or nodes were randomly placed. It is therefore natural to wonder whether there is a certain way to place the nodes so that we improve the conditioning of the Vandermonde matrix Vn$.$ We will try $3$ different types of node placements, vary the matrix size and compare results with the expected asymptotic condition number $1,$ i$.$e$.$ the condition number when n $->\backslash$infty $.$

$1.$ Harmonic nodes: xi $=1/$i$,$ i $=1,...,$ n$.$ The expected rate is at least R$\backslash$infty :$=$ limn$->\backslash$infty $\backslash$kappa $($Vn$)>$ nn$+1.$

$2.$ Equidistantnodeson$[0,1]$: xi $=($i$1)/($n$1),$ i$=1,...,$n$.$ Here R$\backslash$infty $428$n$.3.$Chebychevnodeson$(-1,1)$:xi$=$cos$(2$i$1)\backslash$pi $,$i$=1,...,$n$.$Here R$\backslash$infty $33/41+2$n$.$

You must report the first value of n for which the numerically computed condition number satisfies the expected rate or for which numerical errors make the calculation of the condition number unreliable. Support your arguments with plots. In the provided code file part $1$ is shown. Modify this for the other parts. Finally, comment on which node placements of the $3$ would you use in general. Is this a fair

comparison?