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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 ${\mathrm{x}}_{\mathrm{n}-1}, {\mathrm{x}}_{\mathrm{n}}, {\mathrm{x}}_{\mathrm{n}+1}$ 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 ${∆}^2 {\mathrm{x}}_{\mathrm{n}-1}$ 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

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 err [the allowed error]
	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]