FOR FREE YEAR SOLVED

Fixed Point Method

Back to Programming

Description

This method is an iterative method of finding a sequence {xn}, each xn 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 xn 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]
	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]

Code

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.