Sequence transformation

This is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical).

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To evaluate the limit of a slowly convergent sequence or series, or the antilimit of a divergent series numerically, one may use extrapolation methods or sequence transformations $T$:

For a given series

$S=\{s_{n}\}_{n\in N_{0}}$,

the transformed sequence is

$T(S)=S'=\{s'_{n}\}_{n\in N_{0}}$,

where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, e.g., there is a mapping $F$ according to

$F:(s_{n},s_{n+1},\dots ,s_{n+k})\to s'_{n}$

for some finite $k$.

The transformed sequence is said to converge faster than the original sequence if

$\lim _{n\to \infty }{\frac {s'_{n}-s}{s_{n}-s}}=0$

where $s$ is the (anti)limit of $S$.

If the mapping $F$ is linear in each of its arguments, i.e., for

$s'_{n}=\sum _{m=0}^{k}c_{m}s_{n+m}$ for some constants $c_{0},\dots ,c_{k}$,

the sequence transformation $T$ is called a linear sequence transformation.

Sequence transformations that are not linear are called nonlinear sequence transformations.

Examples of such nonlinear sequence transformations are Pade approximants and Levin-type sequence transformations.

Especially nonlinear sequence transformations often provide highly effective extrapolation methods.

References

Extrapolation Methods. Theory and Practice by C. Brezinski and M. Redivo Zaglia, North-Holland, 1991.

Pade Approximants by G. A. Baker, Jr. and P. Graves-Morris, Cambridge U.P., 1996.