Consider the following system of nonlinear equations F : R$2->$ R$2$ given by f$1($x$,$ y$)$ x$2$ y

F$($x$,$y$)=$ f$2($x$,$y$)$ y$2$x $($a$)$ The system has four roots. What are they?

$($b$)$ Recall that the inverse of a $2\backslash$times $2$ matrix is easily computed by a b $11$ d $$b

cd $=$ad$$bc$$ca$.$

What, explicitly, is the Newton iteration function G$($x$,$ y$)$ for the given F $($x$,$ y$)?$ Note$$that it is a vector!

$($c$)$ For $($x$0,$y$0)$T $=(1/4,1/4)$T write a MATLAB code to compute the first $5$ iterations of Newton$$s method and display the iterations in a table with columns n$,$ xn and yn$.$ Submit a copy of your code. To which root do you expect the iteration to converge?