From b3d979d60752abbf4f705157de3e29e519fc7ec2 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 2 Dec 2020 20:40:33 +0100 Subject: [PATCH] Pade section --- Manuscript/EPAWTFT.bbl | 9 ++++++++- Manuscript/EPAWTFT.bib | 17 +++++++++++++++- Manuscript/EPAWTFT.pdf | Bin 10773864 -> 0 bytes Manuscript/EPAWTFT.tex | 44 ++++++++++++++++++++--------------------- 4 files changed, 45 insertions(+), 25 deletions(-) delete mode 100644 Manuscript/EPAWTFT.pdf diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index ec08f21..03f9f84 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{157}% +\begin{thebibliography}{158}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -1399,6 +1399,13 @@ {journal} {\bibinfo {journal} {Phys. Rev.}\ }\textbf {\bibinfo {volume} {103}},\ \bibinfo {pages} {1116} (\bibinfo {year} {1956})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Shanks}(1955)}]{Shanks_1955}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont + {Shanks}},\ }\href {\doibase https://doi.org/10.1002/sapm19553411} {\bibfield + {journal} {\bibinfo {journal} {Journal of Mathematics and Physics}\ + }\textbf {\bibinfo {volume} {34}},\ \bibinfo {pages} {1} (\bibinfo {year} + {1955})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Surj{\'a}n}\ and\ \citenamefont {Szabados}(2000)}]{Surjan_2000}% \BibitemOpen diff --git a/Manuscript/EPAWTFT.bib b/Manuscript/EPAWTFT.bib index 6bbfd59..e523939 100644 --- a/Manuscript/EPAWTFT.bib +++ b/Manuscript/EPAWTFT.bib @@ -1,13 +1,28 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-12-02 16:02:26 +0100 +%% Created for Pierre-Francois Loos at 2020-12-02 20:06:08 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Shanks_1955, + abstract = {This paper discusses a family of non-linear sequence-to-sequence transformations designated as ek, ekm, {\~e}k, and ed. A brief history of the transforms is related and a simple motivation for the transforms is given. Examples are given of the application of these transformations to divergent and slowly convergent sequences. In particular the examples include numerical series, the power series of rational and meromorphic functions, and a wide variety of sequences drawn from continued fractions, integral equations, geometry, fluid mechanics, and number theory. Theorems are proven which show the effectiveness of the transformations both in accelerating the convergence of (some) slowly convergent sequences and in inducing convergence in (some) divergent sequences. The essential unity of these two motives is stressed. Theorems are proven which show that these transforms often duplicate the results of well-known, but specialized techniques. These special algorithms include Newton's iterative process, Gauss's numerical integration, an identity of Euler, the Pad{\'e} Table, and Thiele's reciprocal differences. Difficulties which sometimes arise in the use of these transforms such as irregularity, non-uniform convergence to the wrong answer, and the ambiguity of multivalued functions are investigated. The concepts of antilimit and of the spectra of sequences are introduced and discussed. The contrast between discrete and continuous spectra and the consequent contrasting response of the corresponding sequences to the e1 transformation is indicated. The characteristic behaviour of a semiconvergent (asymptotic) sequence is elucidated by an analysis of its spectrum into convergent components of large amplitude and divergent components of small amplitude.}, + author = {Shanks, Daniel}, + date-added = {2020-12-02 20:05:53 +0100}, + date-modified = {2020-12-02 20:06:02 +0100}, + doi = {https://doi.org/10.1002/sapm19553411}, + journal = {Journal of Mathematics and Physics}, + number = {1-4}, + pages = {1-42}, + title = {Non-linear Transformations of Divergent and Slowly Convergent Sequences}, + volume = {34}, + year = {1955}, + Bdsk-Url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm19553411}, + Bdsk-Url-2 = {https://doi.org/10.1002/sapm19553411}} + @article{DiSabatino_2015, author = {Di Sabatino,S. and Berger,J. 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