Fixed point iteration method

Please refer to description to understand working procedure of the method. Instructions to use the code are provided within the code file.

Consider the equation 𝑓 𝑥 = 0 on [𝑎, 𝑏].

The equation can be written as 𝑥 = 𝜙(𝑥), ( 𝜙(𝑥) is called iterative function)

Let 𝑥0 be initial approximation for the root, then the first approximation is obtained by 𝑥1 = 𝜙(𝑥0), the successive approximation are given by

𝑥2 = 𝜙(𝑥1),

𝑥3 = 𝜙(𝑥2),

and so on, thus 𝑥(𝑘+1) = 𝜙(𝑥(𝑘)) here 𝑘 = 0,1,2,3 …

The above sequence of values 𝑥(𝑘)′𝑠 converges to the solution of given equation; the method depends on the choice of function 𝜙(𝑥) and choice of 𝑥0. The above method is known as fixed point iteration method.

Sufficient condition for the convergence of iterations

(i) If ∝∈ [𝑎, 𝑏] be a root of f 𝑥 = 0 which is equivalent to x = 𝜙 𝑥 ,

(ii) 𝜙(𝑥) and 𝜙′(𝑥) are continuous in [a,b],

(iii) 𝜙′(𝑥) < 1 , for every 𝑥 ∈ [𝑎, 𝑏]

NOTE: If 𝜙′(𝑥0) < 1, the error decreases with each iterations and if 𝜙′(𝑥0) > 1, the error grows.