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Manuscript/EPAWTFT.bbl
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%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

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%% 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. A. and Reining,L. and Romaniello,P.},
date-added = {2020-12-02 16:02:21 +0100},

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Manuscript/EPAWTFT.tex
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@ -1339,15 +1339,15 @@ radius of convergence (see Fig.~\ref{fig:RadConv}).
%As frequently claimed by Carl Bender,
\hugh{It is frequently stated that}
It is frequently stated that
\textit{``the most stupid thing that one can do with a series is to sum it.''}
Nonetheless, quantum chemists are basically doing exactly this on a daily basis.
\hugh{As we have seen throughout this review, the MP series can often show erratic,
As we have seen throughout this review, the MP series can often show erratic,
slow, or divergent behaviour.
In these cases, estimating the correlation energy by simply summing successive
low-order terms is almost guaranteed to fail.}
low-order terms is almost guaranteed to fail.
Here, we discuss alternative tools that can be used to sum slowly convergent or divergent series.
\hugh{These so-called ``resummation'' techniques} form a vast field of research and thus we will
These so-called ``resummation'' techniques form a vast field of research and thus we will
provide details for only the most relevant methods.
We refer the interested reader to more specialised reviews for additional information.%
\cite{Goodson_2011,Goodson_2019}
@ -1357,16 +1357,16 @@ We refer the interested reader to more specialised reviews for additional inform
\subsection{Pad\'e Approximant}
%==========================================%
\hugh{The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$
The failure of a Taylor series for correctly modelling the MP energy function $E(\lambda)$
arises because one is trying to model a complicated function containing multiple branches, branch points and
singularities} using a simple polynomial of finite order.
A truncated Taylor series \hugh{can only predict a single sheet and} does not have enough
flexibility to adequately describe the MP energy.
singularities using a simple polynomial of finite order.
A truncated Taylor series can only predict a single sheet and does not have enough
flexibility to adequately describe, for example, the MP energy.
Alternatively, the description of complex energy functions can be significantly improved
by introducing Pad\'e approximants, \cite{Pade_1892} and related techniques. \cite{BakerBook,BenderBook}
\hugh{A Pad\'e approximant can be considered as the best approximation of a function by a
rational function of given order.}
A Pad\'e approximant can be considered as the best approximation of a function by a
rational function of given order.
More specifically, a $[d_A/d_B]$ Pad\'e approximant is defined as
\begin{equation}
\label{eq:PadeApp}
@ -1379,15 +1379,15 @@ chemistry\cite{Loos_2013,Pavlyukh_2017,Tarantino_2019,Gluzman_2020} as they can
which appears at the locations of the roots of $B(\lambda)$.
However, they are unable to model functions with square-root branch points
(which are ubiquitous in the singularity structure of a typical perturbative treatment)
and more complicated functional forms appearing at critical points (
where the nature of the solution undergoes a sudden transition).
\hugh{Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $)
often define a convergent perturbation series in cases where the Taylor series expansion diverges.}
and more complicated functional forms appearing at critical points
(where the nature of the solution undergoes a sudden transition).
Despite this limitation, the successive diagonal Pad\'e approximants (\ie, $d_A = d_B $)
often define a convergent perturbation series in cases where the Taylor series expansion diverges.
\begin{table}[b]
\caption{RMP ground-state energy estimate at $\lambda = 1$ provided by various truncated Taylor
series and Pad\'e approximants at $U/t = 3.5$ and $4.5$.
We also report the \hugh{closest pole to the origin $\lc$} provided by the diagonal Pad\'e approximants.
We also report the distance of the closest pole to the origin $\abs{\lc}$ provided by the diagonal Pad\'e approximants.
\label{tab:PadeRMP}}
\begin{ruledtabular}
\begin{tabular}{lccccc}
@ -1412,7 +1412,7 @@ often define a convergent perturbation series in cases where the Taylor series e
\end{ruledtabular}
\end{table}
Fig.~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e
Figure~\ref{fig:PadeRMP} illustrates the improvement provided by diagonal Pad\'e
approximants compared to the usual Taylor expansion in cases where the RMP series of
the Hubbard dimer converges ($U/t = 3.5$) and diverges ($U/t = 4.5$).
More quantitatively, Table \ref{tab:PadeRMP} gathers estimates of the RMP ground-state
@ -1420,12 +1420,12 @@ energy at $\lambda = 1$ provided by various truncated Taylor series and Pad\'e
approximants for these two values of the ratio $U/t$.
While the truncated Taylor series converges laboriously to the exact energy as the truncation
degree increases at $U/t = 3.5$, the Pad\'e approximants yield much more accurate results.
\hugh{Furthermore, the distance of the closest pole to origin $\abs{\lc}$ in the Pad\'e approximants
Furthermore, the distance of the closest pole to the origin $\abs{\lc}$ in the Pad\'e approximants
indicate that they a relatively good approximation to the position of the
true branch point singularity in the RMP energy.
For $U/t = 4.5$, the Taylor series expansion performs worse and eventually diverges,
while the Pad\'e approximants still offer relaitively accurate energies and recovers
a convergent series.}
while the Pad\'e approximants still offer relatively accurate energies and recovers
a convergent series.
%%%%%%%%%%%%%%%%%
\begin{figure}[t]
@ -1435,22 +1435,20 @@ a convergent series.}
\end{figure}
%%%%%%%%%%%%%%%%%
\hugh{%
We can expect the UMP energy function to be much more challenging
to model properly as it contains three connected branches
(see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}).
Fig.~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case.
Figure~\ref{fig:QuadUMP} and Table~\ref{tab:QuadUMP} indicate that this is indeed the case.
In particular, Fig.~\ref{fig:QuadUMP} illustrates that the Pad\'e approximants are trying to model
the square root branch point that lies close to $\lambda = 1$ by placing a pole on the real axis
(\eg, [3/3]) or with a very small imaginary component (\eg, [4/4]).
The proximity of these poles to the physical point $\lambda = 1$ means that any error in the Pad\'e
functional form becomes magnified in the estimate of exact energy, as seen for the low-order
functional form becomes magnified in the estimate of the exact energy, as seen for the low-order
approximants in Table~\ref{tab:QuadUMP}.
However, with sufficiently high degree polynomials, one obtains
accurate estimates for the position of the closest singularity and the ground-state energy at $\lambda = 1$,
even in cases where the convergence of the UMP series is incredibly slow
(see Fig.~\ref{subfig:UMP_cvg}).
}
%==========================================%
\subsection{Quadratic Approximant}