Aitken's Process and Steffensen's Acceleration Method

Theorem (Aitken's Acceleration).Assume that the sequenceconverges linearly to the limit p and thatfor all. If there exists a real number A withsuch that



then the sequencedefined by



converges to p faster than,in the sense that

.

Steffensen's Acceleration

When Aitken's process is combined with the fixed point iteration in Newton�s method, the result is called Steffensen's acceleration.Starting with , two steps of Newton's method are used to computeand,then Aitken's process is used to compute.To continue the iteration set and repeat the previous steps.

Mathematica Subroutine (Newton-Raphson Iteration).

Mathematica Subroutine (Steffensen's Method).

Example.Use Newton's method to construct a linearly convergent sequencewhich converges slowly to the multiple rootof.
Then use the Aitken process to constructwhich converges faster to the root.
Solution.