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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