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Question $3$: $30$ Marks

An $nn$ Hilbert matrix $H_n$ is defined by

$H_{ij}=\frac{1}{i+j-1},$ for $,1i,jn$

We define $e_n=$ubrace$((1,$dots,$1)ubrace{)}_{n\text{t}\text{i}\text{m}\text{e}\text{s}}.$

$(3\mathrm{.}1)$ Generate the matrices $H_{15},e_{15}$ and $b_{15}=H_{15}{e}_{15}.$

$(3\mathrm{.}2)$ Compute the exact solution for $x_{15}$ from the system $H_{15}{x}_{15}=b_{15}.$

$(3\mathrm{.}3)$ Generate the matrices $F_{15}$ and $c_{15}$ whose entries are the flotation$-$point values of $H_{15}$

and $b_{15}$ respectively. Compute the solution $y_{15}$ of the system $F_{15}{y}_{10}=c_{15}.$ Compare

$y_{15}$ and $x_{15}.$

$(3\mathrm{.}4)$ You noticed that there is a dramatic difference between $x_{15}$ and $y_{15}.$ Solve numerically

the system $F_{15}{z}_{15}=c_{15}$ such that for $1i15.$