Reference no: EM13721211
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 ;
1. Use Sci/Matlab to plot the behaviour 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).
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 computes solutions for all values of input parameter p.
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.
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