Turn in your own writeup that includes your code. List any resources you used including collaborating with others. Submit a PDF on Canvas by Friday, Oct. 15 at 5pm.
Consider the following nonlinear system of equations with two equations and two unknowns. The math problem can be stated as follows. Given f1(x1; x2) and f2(x1; x2) de ned as
f1(x1; x2) = x13
x23 + x1
;
f2(x1; x2) = x12
+ x22 1:
(1)
Find r1 and r2 such that f1(r1; r2) = 0 and f2(r1; r2) = 0.
1. Note that all the points such that f2 = 0 de ne a circle of radius 1 centered at the origin. Make a plot that shows (i) all the points that satisfy f1 = 0 and (ii) all the points that satisfy f2 = 0. Identify the points on the plot that satisfy both f1 = 0 and f2 = 0.
2. By hand, calculate the 2 2 Jacobian matrix of the system (f1; f2).
3. Use Newton’s method for systems to nd the two solutions to the system of equations (f1 = 0; f2 = 0). Try several (10 or so) di erent initial guesses. Make a table of the answer that Newton’s method gives|something like:
Initial guess (x1(0); x2(0))
Newton’s answer (r1; r2)
####, ####
####, ####
The superscript in the column heading indicates the iteration number, i.e., 0 means the initial guess. Check the plot you made in problem 1 to see whether the answers you’re getting make sense.
4. Find a starting point where Newton’s method fails. Why did it fail?
5. BONUS (10%): Plot the iterates from Newton’s method in (x1; x2) space. Interpret what you see (related to the solution that Newton’s method gives).
6. BONUS (10%): Set up a grid on the (x1; x2) space; you can use meshgrid in Matlab/Numpy. For each point in the grid, compute the Newton direction
p = J(x1; x2) 1 f(x1; x2);
(2)
where J is the Jacobian matrix and f = (f1; f2). Make a quiver plot of the Newton directions on the grid. How does this relate to the solution that Newton’s method gives?
7. BONUS (20%): Derive a xed point iteration for nonlinear systems. (That is: show how to transform the nonlinear system problem to a xed point problem and then write the iteration). Implement your method. Solve the same two-equation system with your xed point method. Anything interesting?
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