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.
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.
Then we get,
Then if we write
The presence of the term in the process actually explains the name of the process.
From Aitken’s process we get the iteration formula as:
When this method is combined with fixed-point iteration, then that result is called Steffensen’s acceleration.
A function f(x)
The root of the function f(x)
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]
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.
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.