The fixed-point iteration method is a numerical technique used to approximate the roots of a function. It
is based on the concept of a fixed point, which is a point at which a function f(x) takes on the same value
as x.
The basic idea of the fixed-point iteration method is to start with an initial guess đť‘Ą0, and then use the
function f(x) to generate a sequence of values x1, x2, x3, and so on, by repeatedly applying the function
to the previous value.
Consider a function 𝑓(𝑥) = 0 ………(i) which can be transformed to the form 𝑥 = 𝑔(𝑥) …..….(ii) such that
đť‘Ą is on the left side of the equation. This transformation can be done by simplifying an algebraic expression
or by adding đť‘Ą on both sides of the given equation đť‘“(đť‘Ą) = 0.
Divergence in Fixed-Point Iteration Method
The fixed-point iteration method may not always converge to a root of the function f(x). If the
transformation function g(x) is poorly chosen or the absolute value of the derivative of g(x) at the fixed
point is greater than or equal to 1, the iteration may diverge and not converge to any root. This is called
divergence in the fixed-point iteration method. The following are the types of divergence we may
encounter while following this method:
1. Oscillatory Divergence
Oscillatory divergence occurs when the sequence of iterates oscillates back and forth around the
fixed point, without ever converging to it. This can happen when the absolute value of the
derivative of the function f(x) is greater than 1 in a neighborhood of the fixed point. In this case,
the magnitude of the error in each iteration increases, leading to the oscillation of the sequence
around the fixed point
2. Monotonous Divergence
Monotonous divergence, on the other hand, occurs when the sequence of iterates diverges away
from the fixed point in a monotonous manner, without oscillating. This can happen when the
absolute value of the derivative of the function f(x) is exactly 1 at the fixed point. In this case, the
magnitude of the error in each iteration remains constant, leading to a steady increase in the
distance between the iterates and the fixed point.
