Assignment 5 Solution

MECE 390 Assignment 5 Instructions Answer the following questions and prepare a single PDF file including: - Solution for Question 1 - Part I - The MATLAB program for Question 1 - Part II - MATLAB's output of the intermediate estimations for test 1 in Question 1 - Part II - MATLAB's output of the intermediate estimations for test 2 in Question 1 - Part II - Comments on Question 1 - Part III - MATLAB's output of the intermediate estimations for Question 2 - Part I - Discussion for Question 2 - Part II - Example (Not to be added to your report): Note the following equations and information in this example will teach you to use the plotFunc3d only - these are not the equations you will be solving for! Consider the following equations: (1) − ? − ? + ? +1= 0 (2) − ? 2 + ? 2 + ? = 0 (3) − ? 2 − ? 2 − ? 2 +4= 0 These equations respectively describe (1) a plane, (2) a hyperbolic paraboloid, i.e. a saddle function, and (3) a sphere. We are interested in finding the two points, in ( ? , ? , ? ) coordinates, where these three surface plots all intersect. Figure 1 : 3D plot of the plane (Eq. 1), the paraboloid (Eq. 2), and the sphere (Eq. 3) intersecting each other.

You can use plotFun3d (available on eClass) to graphically estimate the position of the two real roots. Roots of the system correspond to common intersections of the three surfaces. In order to use plotFun3d , make ? coordinate the subject of your equations and then use the function handle to insert them as function arguments. A sample script called sampleFunc is provided on eClass to demonstrate how plotFun3d should be executed. Question 1 Consider the following three equations: (1) 𝑓 1( ? , ? , ? ) = − ? − ? + ? +1 = 0 (2) 𝑓 2( ? , ? , ? ) = − ? 2 + ? + ? 2 = 0 (3) 𝑓 3( ? , ? , ? ) = ? 2 + ? 2 + ? 2 −4 = 0 These are the three equations you will use for your answers below. Question 1 – Part I Determine the Jacobian Matrix 𝐽 for the system of Equations (1) to (3). NOTE: - Report your answer as fractions and use the symbols "x", "y" and "z". - For elements of the Jacobian matrix 𝐽 , evaluate the derivatives with respect to ? , ? and ? respectively, e.g. 𝐽 11 =∂ 𝑓 1 / ∂ ? and 𝐽 23 =∂ 𝑓 2 / ∂ ? . [ 1 −1 −1 2? 1 −2? 2? 2? 2? ] Question 1 – Part II Write a MATLAB program to implement the Newton-Raphson Method. You can use the mldivide , backslash operator or your Gaussian Elimination function (Lab 4) to solve the linearized system to obtain Δ ? . You can compare your results (both the roots and the number of iterations) with the built-in MATLAB function fsolve () by using the command [x,fval,exitflag,output] = fsolve(@FUN, X0).