Rounding errors $10+10pt$ Computers use finite precision to represent real numbers, which

leads to rounding. You can see the size of the rounding error for real numbers around $1$ using the

MATLAB command eps $(1)$ or the numpy command np$.$spacing$(1).$ This number, also called

the machine epsilon, is $lon=2\mathrm{.}22{10}^{-16}$dots for the standard $($double precision$)$ representation of

numbers in a computer. Try the following experiments in MATLAB $($or Python$/$Octave$/$Julia$).{?}^1$

$($a$)$ The $n-$th Hilbert matrix $H_{n}in{R}^{nn}$ has the entries $h_{ij}=(i+j-1{)}^{-1}$ for $i,j=1,$dots,$n.{?}^2$

It is known that solving systems with the Hilbert matrix increases rounding errors since the

matrix is poorly conditioned. Let $e_n$ be the column vector of length $n$ that contains all $1\text{'}$s$.$

Report the exact and the numerically computed values for

$||H_n(H_n^{-1}{e}_n)-e_n|{|}_2,$ for $n=5,10,20.$

Here, $||*|{|}_2$ is the $2-$norm. Also report the $1-$norm, $2-$norm and $-$norm condition numbers

of $H_n$ for $n=5,10,20.{?}^3$