A basin of attraction for a root is defined by a cloud of starting points that converge to that particular

root. You are asked to generate the basins of attraction by colouring each starting point with a unique colour based on the root it converges to$.$

a$.$ You are given the following system of non$-$linear equations:

$3x^3-y=0$

$2y^3-x=0$

Find all $3$ pairs of roots for the equations using an analytical approach, a graphical approach, or MATLAB built$-$in functions.

b$.$ Follow the steps below to find the pairs of roots using the Newton$-$Raphson method for solving a system of two non$-$linear equations until the percent estimated relative error falls below $0\mathrm{.}01\%.$

Step $1$: Define different starting points $($x$0,$ y$0)$ such that they range from $-1$ to $+1$ with an

interval of $0\mathrm{.}01.$

Step $2$: For each of these starting points, determine which root it converges to by measuring the

L$2-$norm between the estimated root and roots in part a$).$ The tolerance for this measurement should be set carefully as this choice can change your plot. Try different tolerances.

Step $3$: Specify a colour for all starting points that converge to the same root.

Step $4$: Plot all starting points with their respective colours on a single figure. Plot all $3$ pairs of roots from part a$)$ in the figure as well.

c$.$ Include your code algorithm in your report.

$(($Note: For the results provide $15$ digits after the decimal point.$))$

please do not forget to include the code and the graph resulted.