## Fixed Point Method

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### Description

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

### Algorithm

INPUT:

A function f(x)

OUTPUT:

The root of the function f(x)

PROCESS:

``````Step 1: [Taking the input from the user]

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]``````

### Code

1. They are spaced evenly with the range of values

2. The accuracy of the result is dependable.