The output shall be generated by the following output function.

function outputErrorResults$($method$,$dt$,$err,maxIt$)$

$\%...$

Here, err is the array with the error values mentioned above, and maxIt is the maximum number of

time step size iterations $($in this case $10).$ Note also that in this function, $dt$ is an array that includes all

time step sizes considered in this problem whereas it is a scalar value in the computeRungeKutta$-$

Solution function $($consequently$,$ the computeRungeKuttaSolution function needs to be called

with only one element of the dt array$).$

By examining the error ratios shown above, explain how you can verify through numerical means

that the orders of approximation of the four different Runge$-$Kutta methods are given by $O(t),$

$O((t{)}^2),O((t{)}^3),$ and $O((t{)}^4)$Start your calculations with the time step size $t=1s,$ and halve $t$ successively nine times to gen$-$

erate ten calculations for each Runge$-$Kutta method. For each time step $($in each calculation associated

with one $($constant$)$ time step size $t),$ call the computeRungeKuttaSolution function repeatedly

to solve the given IVP up to $t_{\text{f}\text{i}\text{n}\text{a}\text{l}}=6s$ using each of the four numerical methods. Generate three plots

associated with the time step sizes $tin\{1,0\mathrm{.}5,0\mathrm{.}25\}.$ Each plot should consist of the exact solution

and piecewise linear interpolants $($MATLAB default$)$ of the four Runge$-$Kutta approximations.

Determine the order of approximations of the various Runge$-$Kutta methods in terms of the ratios of

the $($absolute values of the$)$ exact errors, as obtained from each of the four Runge$-$Kutta approxima$-$

tions. More precisely, determine the ratios of the exact errors from the previous time step size to the

current time step size at the times tin$\{1,3,5\}.$ Use a $1-$D $($or $2-$D$)$ array to store the necessary exact

error values for each of the four Runge$-$Kutta approximations at the given times, and output the results

to the command window in the following format $($the values shown are for illustration purposes only$)$: