Problem $5$: Error analysis in the Riemann Sum Algorithm

When using the left Riemann sum algorithm, one does not know a priori what step size or number of intervals to choose so that the result is accurate to within a desired error tolerance $($the maximum error that can be tolerated$).$ This choice depends on the given function to integrate. For example, if one integrates the constant function, an algorithm using $N=1$ produces the exact result, i$.$e$.,$ no error.

a$)$ Modify the rmlsum file to include a while$-$loop so that the program code terminates when an accuracy based on relative approximate error is achieved. In each iteration of the while$1$oop, double the number of the intervals $($i$.$e$.,$ the step size $h$ is reduced by one half$)$ in the numerical integration algorithm. The function's input$/$output structure should be:

function fun, $a,b,$ maxtol