**This method is an iterative method of finding a sequence **$\mathbf{\left\{}{\mathbf{x}}_{\mathbf{n}}\mathbf{\right\}}$**, each **${\mathbf{x}}_{\mathbf{n}}$** being the successive approximation of the real root α of the equation f(x) = 0**. It is a method by which a fixed point of an iterated function is computed.

__DERIVATION__

**The first stage of this method is to locate the interval [a, b] by any method (tabular method or graphical method) and then reset f(x) = 0 in the form of x = ∅(x)**

**Thus,**

**Therefore, α = ∅(α)… gives the point α fixed under the mapping ∅ and a root of the equation is fixed under the mapping ∅. So it is called fixed-point iteration.**

**Now, **

**Then the successive approximations are calculated as**

**This above iteration is generated by the formula **

**This formula is called the fixed point iteration formula** and ${\mathbf{x}}_{\mathbf{n}}$ is the nth approximation of **α**, the root of **f(x) = 0**.

__GRAPHICAL REPRESENTATION__

__INPUT:__

A function f(x)

__OUTPUT:__

The root of the function f(x)

__PROCESS:__

```
Step 1: [Taking the input from the user]
Read arr[0] [the initial guess]
Step 2: [defining a function f(x)]
Return (cos x + 2)/3
[End of the function f(x)]
Step 3: [Fixed point Iteration]
Set e ← 100.0
Set j ← 0
While e > 0.00001 repeat
Set arr[j+1] ← f(arr[j])
Set e ← arr[j + 1] - arr[j]
Set e ← absolute value of e
Print arr[j]
[End of ‘while’ loop]
Print arr[j]
[End of fixed point iteration]
```

__ADVANTAGES__

**1.** They are spaced evenly with the range of values

**2.** The accuracy of the result is dependable.

__DISADVANTAGES__

**1.** It has a fixed rate of convergence, so it is slower than the other methods.

**2.** It requires a starting interval that contains a change of sign, so it can not find repeated roots.

__APPLICATIONS__

**1.** It is used to design an optimal Cassegrain antenna structure.

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