Aitken's delta-squared process is a method for accelerating the convergence of numerical sequences. It works by extrapolating the partial sums of a series whose convergence is approximately geometric. The method eliminates the largest part of the absolute error, improving the rate of convergence. Aitken's method can be applied to root-finding algorithms and iterative processes to achieve faster linear or quadratic convergence.

1. THE AITKEN’S METHOD

2. Aitken’s Method
In numerical analysis, Aitken's delta-squared
process or Aitken Extrapolation is a series
acceleration method, used for accelerating the rate
of convergence of a sequence. It is named
after Alexander Aitken, who introduced this method
in 1926. Its early form was known to Seki Kōwa (end
of 17th century) and was found for rectification of
the circle, i.e. the calculation of π. It is most useful
for accelerating the convergence of a sequence that
is converging linearly.

3. Aitken’sAlgorithm

5. Aitken's Delta Squared Process
Also called Aitken Extrapolation. An Algorithm which
extrapolates the partial sums of a series whose
Convergence is approximately geometric and accelerates its
rate of Convergence. A simple nonlinear sequence
transformation is the Aitken extrapolation or delta-squared
method
This transformation is commonly used to improve the rate of
convergence of a slowly converging sequence; heuristically, it
eliminates the largest part of the absolute error.

6. FORMULA:

8. Derivation:

9. Adding and Subtracting and In numerator
then group terms

10. Finally

11. ACCELERATING CONVERGENCE:
Aitken’s ∆2 process
Used to accelerate linearly
convergent sequences,
regardless of the method used.
this acceleration method is not
only applicable to root-finding
algorithms.

12. Aitken’s process is an extrapolation
r delta r delta^2 r
rk-2
rk-1 rk-1-rk-2
rk rk-rk-1 (rk-rk-1)-(rk-
1-rk-2)

13. Steffensen’s Method:
a modified Aitken’s delta-squared process
applied to fixed point iteration.
In numerical analysis, Steffensen's method is
a root-finding technique similar to Newton's
method, named after Johan Frederik
Steffensen. Steffensen's method also
achieves quadratic convergence, but without
using derivatives as Newton's method does.

14. Derivation usingAitken's delta-squared process
Implemented in the MATLAB can be found using the Aitken's
delta-squared process for accelerating convergence of a
sequence. This method assumes starting with a linearly
convergent sequence and increases the rate of convergence of
that sequence. If the signs of agree and is sufficiently close to
the desired limit of the sequence , we can assume the following:

16. Example : Find a root of cos[x] - x * exp[x] = 0 with
x0 = 0.0
Let the linear iterative process be
xi+1 = xi + 1/2(cos[xi]- xi * exp[xi] ) i = 0, 1, 2 . . .

17. Aitken extrapolation can greatly accelerate the convergence
of a linearly convergent iteration
xn+1 = g(xn)
This shows the power of understanding the behaviour
of the error in a numerical process. From that understanding,
we can often improve the accuracy, thru
extrapolation or some other procedure.
This is a justification for using mathematical analyses
to understand numerical methods. We will see this
repeated at later points in the course, and it holds
with many different types of problems and numerical
methods for their solution.
General Comment