Discuss the concept of fixed-point iteration for solving systems of nonlinear equations. Provide an example.

Fixed-point iteration is a numerical method used to solve systems of nonlinear equations. It involves transforming the original system of equations into an equivalent form where each equation is expressed in terms of a single variable. The method then iteratively updates the values of these variables until a solution is reached.

To illustrate the concept, let's consider the following system of equations:

f(x, y) = x^2 + y^2 - 4 = 0
g(x, y) = x^2 - y - 1 = 0

To apply fixed-point iteration, we need to rewrite each equation in terms of a single variable. Let's solve the first equation for x:

x^2 = 4 - y^2
x = √(4 - y^2)

Now, substitute this expression for x into the second equation:

(√(4 - y^2))^2 - y - 1 = 0
4 - y^2 - y - 1 = 0
-y^2 - y + 3 = 0

We can rewrite this equation as:

y = -y^2 + 3

Now, we have transformed the original system of equations into an equivalent form where each equation is expressed in terms of a single variable. The next step is to choose an initial guess for y and then iteratively update the value of y using the equation y = -y^2 + 3 until convergence is achieved.

For example, let's start with an initial guess of y = 1. Plugging this value into the equation, we get:

y = -(1)^2 + 3
y = 2

Now, we update the value of y using the equation:

y = -(2)^2 + 3
y = -1

Continuing this process, we update the value of y again:

y = -(-1)^2 + 3
y = 2

We repeat these steps until the value of y converges to a specific value. In this case, the value of y oscillates between 2 and -1, indicating that the iteration does not converge to a solution.

In summary, fixed-point iteration is a method used to solve systems of nonlinear equations by transforming the original equations into an equivalent form where each equation is expressed in terms of a single variable. The method then iteratively updates the values of these variables until convergence is achieved. However, it is important to note that fixed-point iteration may not always converge to a solution, as demonstrated in the example above.