Numerical Computing :: Project Six

Description

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