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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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@ -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]
@ -1435,32 +1435,30 @@ 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}
%==========================================%
Quadratic approximants \hugh{are designed} to model the singularity structure of the energy
Quadratic approximants are designed to model the singularity structure of the energy
function $E(\lambda)$ via a generalised version of the square-root singularity
expression \cite{Mayer_1985,Goodson_2011,Goodson_2019}
\begin{equation}
\label{eq:QuadApp}
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ]
E(\lambda) = \frac{1}{2 Q(\lambda)} \qty[ P(\lambda) \pm \sqrt{P^2(\lambda) - 4 Q(\lambda) R(\lambda)} ],
\end{equation}
with the polynomials
\begin{align}
@ -1469,27 +1467,28 @@ with the polynomials
&
Q(\lambda) & = \sum_{k=0}^{d_Q} q_k \lambda^k,
&
R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k
R(\lambda) & = \sum_{k=0}^{d_R} r_k \lambda^k,
\end{align}
defined such that $d_P + d_Q + d_R = n - 1$, and $n$ is the truncation order of the Taylor series of $E(\lambda)$.
Recasting Eq.~\eqref{eq:QuadApp} as a second-order expression in $E(\lambda)$, \ie,
\begin{equation}
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}}
Q(\lambda) E^2(\lambda) - P(\lambda) E(\lambda) + R(\lambda) \sim \order*{\lambda^{n+1}},
\end{equation}
and substituting $E(\lambda$) by its $n$th-order expansion and the polynomials by
their respective expressions \eqref{eq:PQR} yields $n+1$ linear equations for the coefficients
$p_k$, $q_k$, and $r_k$ (where we are free to assume that $q_0 = 1$).
A quadratic approximant, characterised by the label $[d_P/d_Q,d_R]$, generates, by construction,
$n_\text{bp} = \max(2d_p,d_q+d_r)$ branch points at the roots of the polynomial $P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ \hugh{and $d_q$ poles at the roots of $Q(\lambda)$}.
$n_\text{bp} = \max(2d_p,d_q+d_r)$ branch points at the roots of the polynomial
$P^2(\lambda) - 4 Q(\lambda) R(\lambda)$ and $d_q$ poles at the roots of $Q(\lambda)$.
Generally, the diagonal sequence of quadratic approximant,
\ie, $[0/0,0]$, $[1/0,0]$, $[1/0,1]$, $[1/1,1]$, $[2/1,1]$,
is of particular interest \hugh{as the order of the corresponding Taylor series increases on each step.
is of particular interest as the order of the corresponding Taylor series increases on each step.
However, while a quadratic approximant can reproduce multiple branch points, it can only describe
a total of two branches.
Since every branch point must therefore correspond to a degeneracy of the same two branches, this constraint
\titou{Since every branch points must therefore correspond to a degeneracy of the same two branches,} this constraint
can hamper the faithful description of more complicated singularity structures such as the MP energy surface.
Despite this limitiation,} Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants
Despite this limitation, Ref.~\onlinecite{Goodson_2000a} demonstrates that quadratic approximants
provide convergent results in the most divergent cases considered by Olsen and
collaborators\cite{Christiansen_1996,Olsen_1996}
and Leininger \etal \cite{Leininger_2000}
@ -1501,9 +1500,8 @@ In such a scenario, the quadratic approximant will tend to place its branch poin
The remedy for this problem involves applying a suitable transformation of the complex plane (such as a bilinear conformal mapping) which leaves the points at $\lambda = 0$ and $\lambda = 1$ unchanged. \cite{Feenberg_1956}
\begin{table}[b]
\caption{Estimate \hugh{for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$}
in the UMP energy function provided
by various resummation techniques at $U/t = 3$ and $7$.
\caption{Estimate for the distance of the closest singularity (pole or branch point) to the origin $\abs{\lc}$
in the UMP energy function provided by various resummation techniques at $U/t = 3$ and $7$.
The truncation degree of the Taylor expansion $n$ of $E(\lambda)$ and the number of branch
points $n_\text{bp} = \max(2d_p,d_q+d_r)$ generated by the quadratic approximants are also reported.
\label{tab:QuadUMP}}
@ -1557,14 +1555,13 @@ The remedy for this problem involves applying a suitable transformation of the c
\subcaption{\label{subfig:ump_cp} [3/0,4] Quadratic}
\end{subfigure}
\caption{%
\hugh{%
Comparison of the [3/2,2] and [3/0,4] quadratic approximants with the exact UMP energy surface in the complex $\lambda$
plane with $U/t = 3$.
Both quadratic approximants correspond to the same truncation degree of the Taylor series and model the branch points
using a radicand polynomial of the same order.
However, the [3/2,2] approximant introduces poles into the surface that limits it accuracy, while the [3/0,4] approximant
is free of poles.}
\label{fig:nopole_quad}}
\label{fig:nopole_quad}
\end{figure*}
For the RMP series of the Hubbard dimer, the $[0/0,0]$ and $[1/0,0]$ quadratic approximant
@ -1572,24 +1569,22 @@ are quite poor approximations, but the $[1/0,1]$ version perfectly models the RM
function by predicting a single pair of EPs at $\lambda_\text{EP} = \pm \i 4t/U$.
This is expected from the form of the RMP energy [see Eq.~\eqref{eq:E0MP}], which matches
the ideal target for quadratic approximants.
\hugh{%
Furthermore, the greater flexibility of the diagonal quadratic approximants provides a significantly
improved model of the UMP energy in comparison to the Pad\'e approximants or Taylor series.
In particular, these quadratic approximants provide an effective model for the avoided crossings
(Fig.~\ref{fig:QuadUMP}) and an improved estimate for the distance of the
closest branch point to the origin.
Table~\ref{tab:QuadUMP} shows that they provide remarkably accurate
estimates of the ground-state energy at $\lambda = 1$.}
estimates of the ground-state energy at $\lambda = 1$.
\hugh{%
While the diagonal quadratic approximants provide significanty improved estimates of the
ground-state energy, we can use our knowledge of the UMP singularity structure to develop
even more accurate results.
We have seen in previous sections that the UMP energy surface
contains only square-root branch cuts that approach the real axis in the limit $U/t \rightarrow \infty$.
contains only \titou{square-root branch cuts} that approach the real axis in the limit $U/t \to \infty$.
Since there are no true poles on this surface, we can obtain more accurate quadratic approximants by
taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term.
Fig.~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic
taking $d_q = 0$ and increasing $d_r$ to retain equivalent accuracy in the square-root term [see Eq.\eqref{eq:QuadApp}].
Figure~\ref{fig:nopole_quad} illustrates this improvement for the pole-free [3/0,4] quadratic
approximant compared to the [3/2,2] approximant with the same truncation degree in the Taylor
expansion.
Clearly modelling the square-root branch point using $d_q = 2$ has the negative effect of
@ -1597,23 +1592,19 @@ introducing spurious poles in the energy, while focussing purely on the branch p
leads to a significantly improved model.
Table~\ref{tab:QuadUMP} shows that these pole-free quadratic approximants
provide a rapidly convergent series with essentially exact energies at low order.
}
\hugh{Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic,
Finally, to emphasise the improvement that can be gained by using either Pad\'e, diagonal quadratic,
or pole-free quadratic approximants, we consider the energy and error obtained using only the first 10 terms of the UMP
Taylor series in Table~\ref{tab:UMP_order10}.
The accuracy of these approximants reinforces how our understanding of the MP
energy surface in the complex plane can be leveraged to significantly improve estimates of the exact
energy using low-order perturbation expansions.
}
\begin{table}[h]
\caption{
\hugh{%
Estimate and associated error of the exact UMP energy at $U/t = 7$ for
various approximants using up to ten terms in the Taylor expansion.
}
\label{tab:UMP_order10}}
\begin{ruledtabular}
\begin{tabular}{lccc}
@ -1640,7 +1631,7 @@ will be fast enough for low-order approximations to be useful.
However, these low-order partial sums or approximants often contain a remarkable amount of information
that can be used to extract further information about the exact result.
The Shanks transformation presents one approach for extracting this information
and accelerating the rate of convergence of a sequence.\cite{Shanks_1955}
and accelerating the rate of convergence of a sequence.\cite{Shanks_1955,BenderBook}
\hugh{Consider the partial sums
$S_N = \sum_{k=0}^{N} a_k$