Objectives: Develop and utilize a Matlab algorithm that implements the non$-$linear solution format to solve a system of non$-$linear equations. Plot a $3-$D equation in Matlab using functions meshgrid$(),$ reshape$()$ and contour$().$ Explain each function and describe the output.

Consider the following system of non$-$linear equations.

Write a Matlab script that defines the functional form of each equation using the in$-$line function syntax.

By hand, evaluate the partial derivative for each function. Add to your code the functional form of the first derivative of each equations. Create a Jacobian matrix that utilizes the derivatives.

Use Matlab and the slash command with a loop to solve for the vector. Evaluate each function at an initial guess of x$=1$ and y$=1.$

Loop your code until a maximum tolerance $($in either direction$)$ of $0\mathrm{.}00001$ is reached. Name your solutions $"$xs$"$ and $"$ys$".$

Make a plot of the two original functions and plot the intersection you found using a red circle. Use meshgrid$()$ to set up a matrix of x and y values with $200$ points $....$linspace$(-6,6,200),$ names $"$xn$"$ and $"$yn$".$ Then evaluate each function for the whole matrix of values using the functions you built. Note that meshgrid allows you to do this without a loop. Then use the contour$()$ command to plot the two functions and add a red circle.

Experiment with different initial guesses and see how your solution moves around.

Hint: To plot a function of two variables without solving for one variable use the contour$()$ command with the meshgrid$().$ There are some extra videos in the learning objectives that might help.