The challenge is to find the intersection points of two curves as a parameter p varies. Consider the following two curves;

x² + 9y² = 16,

y - x2 + 2x = p,

where 0 ≤p ≤2.5

1. Use Sci/Matlab to plot the behavior of the intersection points of the two curves as the parameter p is systematically changed (i.e. plot the intersection for p = 0, 1 and 2).

Hint: Plotting an ellipse can be problematic. The simplest way if possible, is to recast the equation of the ellipse in canonical form-

x²/a² + y²/b² = 1

and then calculate x; y using the canonical form's parametric solution-

x(t) = a cos(t)

y(t) = b sin(t)

where t = [0,2π].

When attempting to plot relations always use the parametric form wherever possible.

2. Write a Sci/Matlab function that returns the function values and the Jacobian needed for Newton's method.

3. Write a Sci/Matlab function to solve the above equations for a given value of p using Newton's method.

4. Solve the above pair of equations for p = 0 , p = 1 and p = 2 .

Hint: Use the estimates from Task 2 as your initial guess when p = 0 . As p is changed use the solution of the system from the previous value of p as your initial guess of the solution for the new value of p.

5. Now do something that you may find difficult.

First, recognise that the computation method in Task 3 is not well vectorised: it does a lot of computation on scalars and on vectors with a pathetically short length only two. Write a new Newton's method function that is invoked just once and simultaneously com- putes solutions for all values of input parameter p.

Note: instead of computing and storing the Jacobian as a 2 2 array, you will need to compute and store as a Jacobian for each value of p, say store the four elements as a 4 n array where you are solving for n values of p. Then, instead of using Sci/Matlab's linear equation solver x=J\f, explicitly solve all the 2 2 equations using Cramer's Rule. Assuming:

we get

x1 = (f1J4 - J2f2)/ det J ; and x2 = (J1f2 - f1J3)/ det J ;

where det J = J1J4 - J2J3 :

You will need the vector .* and ./ operators.

6. Plot the solutions x and y as a function of parameter p. That is, show graphically how the intersection point of the two equations varies with p.

If you did not get Task 4 working, then proceed with this last question by utilising your algorithm of Task 3 to compute solutions for many values of parameter p in the range 0 ≤ p ≤2.5 , and then plot these.