Math 1070 HW 11 

1. Consider the nonlinear system

2x^2 + y^2 - 4 = 0
x    + y^2 - 2 = 0

Calculate the first two iterations (x1,y1) and (x2,y2) using Newton's
method with initial guess (x0,y0) = (1,1).

2. For the approximation of the zeros of the function f(x)=(2x^2-3x-2)/(x-1), consider the following fixed-point methods:

a) x^(k+1) = g(x^(k)), where g(x) = (3x^2-4x-2)/(x-1),
b) x^(k+1) = h(x^(k)), where h(x) = x - 2 + x/(x-1).

Analyze the convergence properties of the two methods and determine in particular their order.
Check the behaviour of the two schemes using your fixed-point code.
For the second method, provide an experimental estimate of the interval such that if x^(0) is chosen in the interval, then the method converges to alpha = 2.