Aitken Interpolation
After reading this topic, you should be able to:
1. derive the Aitken interpolation method,
2. apply the Aitken iteration formula
Overview
The Aitken iteration formula is similar to Newtonโs formula. This formula is also successively
generating the higher degree interpolation polynomials. But, the most advantage of this formula is,
it can be easily programmed for a computer.
Assume that the points ๐ฅ๐ฅ0,๐ฅ๐ฅ1,๐ฅ๐ฅ2,โฆ,๐ฅ๐ฅ๐๐ and values ๐ฆ๐ฆ0=๐๐(๐ฅ๐ฅ0),๐ฆ๐ฆ1=๐๐(๐ฅ๐ฅ1),๐ฆ๐ฆ2=๐๐(๐ฅ๐ฅ2),โฆ , ๐ฆ๐ฆ๐๐=
๐๐(๐ฅ๐ฅ๐๐), of the function ๐๐(๐ฅ๐ฅ) being interpolated are given. The interpolating polynomial ๐๐๐๐(๐ฅ๐ฅ) can be
evaluated by using the following ๐๐-iterative algorithm.
First Iteration
At the beginning, ๐๐ polynomials ๐๐0๐๐(๐ฅ๐ฅ) of degree 1 are created in the following manner:
๐๐0๐๐(๐ฅ๐ฅ)=1
๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ0๏ฟฝ๐ฆ๐ฆ0๐ฅ๐ฅ0โ๐ฅ๐ฅ
๐ฆ๐ฆ๐๐๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ๏ฟฝ,๐๐= 1, 2, 3, โฆ , ๐๐.
That is, each of these polynomials passes through the points (๐ฅ๐ฅ0,๐ฆ๐ฆ0) and ๏ฟฝ๐ฅ๐ฅ๐๐,๐ฆ๐ฆ๐๐๏ฟฝ, ๐๐= 1, 2, 3, โฆ , ๐๐.
Second Iteration
Polynomials ๐๐0๐๐(๐ฅ๐ฅ) evaluated above allow us to create the set of ๐๐โ1 polynomials of degree 2:
๐๐01๐๐(๐ฅ๐ฅ)=1
๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ1๏ฟฝ๐๐01 ๐ฅ๐ฅ1โ๐ฅ๐ฅ
๐๐0๐๐ ๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ๏ฟฝ,๐๐= 2, 3, 4, โฆ , ๐๐.
The quadratic polynomial passing through the points (๐ฅ๐ฅ0,๐ฆ๐ฆ0),(๐ฅ๐ฅ1,๐ฆ๐ฆ1) and ๏ฟฝ๐ฅ๐ฅ๐๐,๐ฆ๐ฆ๐๐๏ฟฝ is determined.
Third Iteration
At the third stage of iteration:
๐๐012๐๐(๐ฅ๐ฅ)=1
๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ๐๐โ1๏ฟฝ๐๐012 ๐ฅ๐ฅ2โ๐ฅ๐ฅ
๐๐01๐๐๐ฅ๐ฅ๐๐โ๐ฅ๐ฅ๏ฟฝ,๐๐= 3, 4, โฆ , ๐๐.
Successive Iterations
In successive iterations the polynomials of higher degrees are evaluated analogously, as shown in
Table 1.
Last Iteration
Only one polynomial ๐๐0123โฆ๐๐=๐๐๐๐(๐ฅ๐ฅ) of degree ๐๐ is evaluated in the last (๐๐) iteration.
