This problem studies the rational function: r$($x$)=(4$x$459$x$3+324$x$2751$x $+622)/($x$414$x$3+72$x$2151$x $+112)($a$)$ Apply Horner$$s rule twice $($once for the numerator and once for the denominator$)$

to produce two plots, one for x $=1\mathrm{.}606+252$i and one for x $=2\mathrm{.}400+252$i$.$ Your

results should look like Fig. $2\mathrm{.}3.$

$($b$)$ Create a new Python function that codes up the following expression:

s$($x$)=43($x $2)[($x $5)2+4]/($x $+($x $2)2[($x $5)2+3])$ which is a rewritten version of our rational function. Apply this new function to the

case of the previous two plots and compare the rounding error pattern, size, etc.

$($c$)$ Now introduce two more sets of results, this time for the starting expression for

r$($x$)$ produced using $($not Horner$$s rule but$)$ the naive implementation, using powers

$($i$.$e$.,$ the way you would have coded this up before doing the previous problem$).$

Interpret your findings in python