Formulation
For a sequence the series
is to be determined. First, the partial sum is defined as:
and forms a new sequence . Provided the series converges, will approach in the limit to as The Shanks transformation of the sequence is defined as
and forms a new sequence. The sequence often converges more rapidly than the sequence Further speed-up may be obtained by repeated use of the Shanks transformation, by computing etc.
Note that the non-linear transformation as used in the Shanks transformation is of similar form as used in Aitken's delta-squared process. But while Aitken's method operates on the coefficients of the original sequence, the Shanks transformation operates on the partial sums
Read more about this topic: Shanks Transformation
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