Fixed-Point Method || Open Method || Iteration Method

Fixed-point iteration is a method for solving an equation of the form f(x) = 0. The method is carried out by rewriting the equation in the form:
x = g(x)
Obviously, when x is the solution of f(x) = 0, the left side and the right side of the above Eq. are equal. This is illustrated graphically by plotting y = x and y = g( x), as shown in the video. The point of intersection of the two plots, called the fixed point, is the solution. The numerical value of the solution is determined by an iterative process. It starts by taking a value of x near the fixed point as the first guess for the solution and substituting it in g(x). The value of g(x) that is obtained is the new
(second) estimate for the solution. The second value is then substituted back in g(x), which then gives the third estimate of the solution.
The iteration formula is thus given by:
Xi+ I = g(xi)
The function g(x) is called the iteration function.

Sometimes, the form f(x) = 0 does not lend itself to deriving an iteration formula of the form x = g(x) . In such a case, one can always add and subtract x to f ( x) to obtain x + f ( x) - x = O.
The last equation can be rewritten in the form that can be used in the fixed-point iteration method:
x = x+ f(x) = g(x)

Choosing the appropriate iteration function g(x):
For a given equation f(x) = 0, the iteration function is not unique since it is possible to change the equation into the form x = g(x) in different ways. This means that several iteration functions g(x) can be written for the same equation. In cases where there are multiple solutions, one iteration function may yield one root, while a different function yields other roots.

Actually, it is possible to determine ahead of time if the iterations converge or diverge for a specific g( x) as follows:

The fixed-point iteration method converges if, in the neighborhood of the fixed point, the derivative of g(x) has an absolute value that is smaller than 1 (also called Lipschitz continuous):


When should the iterations be stopped?
The true error (the difference between the true solution and the estimated solution) cannot be calculated since the true solution, in general, is not known. As with Newton's method, the iterations can be stopped either when the relative error or the tolerance in f(x) is smaller than some predetermined value.

#Numerical_Methods #Iteration_Methods #Computational_Methods