# CREATE OWN LIBRARY

## Aitken's Method

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### Description

Aitken’s method is a series acceleration method used for accelerating the rate of convergence of a sequence. Alexander Aitken has introduced this method in 1926. The linear convergence of the iterative method can be improved with this method. This method is a first-order process and the convergence is slow generally.

DERIVATION

Let  are the three consecutive iterates which finally converge to α which is the root of the equation.

Therefore, we can write,

Where p and q are constants.

Similarly,

Then we get,

Then

Then if we write

And automatically

Therefore,

The presence of the term  in the process actually explains the name of the process.

From Aitken’s ${∆}^{2}$ process we get the iteration formula as:

When this method is combined with fixed-point iteration, then that result is called Steffensen’s acceleration.

GRAPHICAL REPRESENTATION

### Algorithm

INPUT:

A function f(x)

OUTPUT:

The root of the function f(x)

PROCESS:

``````Step 1: [defining the function f(x)]
Return (1-0.5*x*x)

Step 2: [Aitken’s Method]
Set flag ← 0
Read x0 [the initial value of x]
Read n [the number of iterations]
for i = 1 to n - 1 repeat
Set x1 ← f(x0)
Set x2 ← f(x1)
Set d ← (x2 - x1) - (x1 - x0)
Set x3 ← x2 - (x2 - x1)^2))/d
Print x3
If absolute value of (x3 - x2) < err then
Print x3
Set flag ← 1
break
[End of ‘if’]
Set x0 ← x3
[End of ‘for’ loop]
If flag = 0 then
Print “Not able to find the solution”
Print x3
[End of ‘if’]
[End of Aitken’s method]``````

### Code

1. It accelerates the rate of convergence.

2. It can accelerate the convergence which is converging linearly.

1. The convergence of this method is generally slow.

APPLICATIONS

1. Used to accelerate linearly convergent sequences without depending on the methods used.

2. By using modified Aitken’s method, the root can be found out without using a derivative as the Newton Raphson method.