Describe the process of the fixed point iteration method for root finding.

The fixed point iteration method is a numerical method used to find the root of a function.

To use the fixed point iteration method, we start with an initial guess for the root, denoted by x0. We then use a fixed formula to generate a sequence of approximations, xn, that hopefully converge to the root. The formula is given by:

xn+1 = g(xn)

where g(x) is a function that is chosen such that the sequence xn converges to the root of the function f(x) that we are trying to find. In other words, we want to find a function g(x) such that:

f(r) = 0 if and only if r = g(r)

Once we have found such a function g(x), we can use the fixed point iteration method to find the root of f(x) as follows:

1. Choose an initial guess x0.
2. Use the formula xn+1 = g(xn) to generate a sequence of approximations xn.
3. Continue this process until the sequence xn converges to the root of f(x).

The convergence of the fixed point iteration method depends on the choice of g(x). If g(x) is chosen such that |g'(r)| < 1, where r is the root of f(x), then the sequence xn will converge to r. If |g'(r)| > 1, then the sequence xn will diverge. If |g'(r)| = 1, then the convergence of the sequence xn is uncertain.

The fixed point iteration method is a simple and intuitive method for finding the root of a function. However, it may converge slowly or not at all for certain functions, and it requires careful choice of the function g(x) to ensure convergence.