# Fixed point iteration method

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

### Citar como

Milind Amga (2024). Fixed point iteration method (https://www.mathworks.com/matlabcentral/fileexchange/99594-fixed-point-iteration-method), MATLAB Central File Exchange. Recuperado .

##### Compatibilidad con la versión de MATLAB
Se creó con R2020b
Compatible con cualquier versión