Numerical Computation Homework 6

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) 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?

4. 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).

4. 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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