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) defined as
f1
(x1
, x2) = x13
− x23
+ x1,
(1)
f2(x1, x2) = x12 + x22 − 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 define 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 find the two solutions to the sys-tem of equations (f1 = 0, f2 = 0). Try several (10 or so) different 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?
1
