Seminario: Levin-Type Transformations

Mercoledì 2 Marzo 2016, ore 11:00 - Aula 2BC30 - Ernst Joachim Weniger



Mercoledì 2 Marzo 2016 alle ore 11:00 in Aula 2BC30, Ernst Joachim Weniger (University of Regensburg, Germany) terrà un seminario dal titolo "Levin-Type Transformations".

It is generally accepted that the modern theory of non-linear and non-regular sequence transformations starts with two seminal articles by Shanks (1955) and Wynn (1956). These two articles initiated extensive research not only on numerical applications, but also on the derivation of new transformation (see for example the $\theta$-algorithm by Brezinski (1971)). A different approach was pursued by Levin (1973) who introduced a new sequence transformations which was later generalized and extended by Weniger (1989, 2004). It is the characteristic feature of Levin’s transformation that it uses as input data not only a substring of a slowly convergent or divergent sequence $\{s_n\}_{n=0}^\infty$, but also explicit truncation error or remainder estimates $\{\omega_n\}_{n=0}^\infty$. It is the explicit incorporation of the information contained in the remainder estimates which makes Levin-type transformations often remarkably powerful (in particular if factorially divergent power series have to be summed). But Levin-type transformations have also other, purely formal advantages: it is almost trivially simple to construct explicit expressions for Levin-type transformations. In the case of other transformations, this is usually extremely difficult or even practically impossible. These explicit expressions played a major role in the rigorous convergence analysis of the summation of the factorially divergent Euler series $\varepsilon(z) \sim \sum_{n=0}^\infty (-1)^n n! z^n$ by Borghi and Weniger (2015).

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