Sequence transformation

A sequence transformation is in mathematics, a transformation that is used to calculate the limit of a slowly convergent sequence or series, or the Anti limit of a divergent series numerically.

For a given sequence

Is the transformed result

The elements of the transformed sequence are normally calculated as a function of a finite number of elements of the original sequence. So there is a picture of the shape

With a finite. In the simplest case, and the real or complex numbers. In general, there are elements of a vector space or algebra.

It is said that the transformed sequence converges faster than the original sequence if

Where the ( anti-) limit is. If the original sequence converges slowly, we speak in this case of convergence acceleration.

If the mapping linear in each argument, ie, if

Applies, so you call the result of transformation is a linear sequence transformation, otherwise a nonlinear sequence transformation.

A sequence transformation can be employed for convergence acceleration of a convergent series or as a summation method for divergent series: For a number

Simply considered to be the result

Of the partial sums

And applies them to an appropriate follow- transformation.

Important examples of non-linear sequence transformations and Padé approximants Levin -type sequence transformations.

Especially nonlinear sequence transformations often result in highly efficient extrapolation.