Commit 14cb6d05 authored by Conor McCoid's avatar Conor McCoid
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DD26: updated bib and file names

parent 4c55255e
@inproceedings{davidenko1953new,
title={On a new method of numerical solution of systems of nonlinear equations},
author={Davidenko, DF},
booktitle={Dokl. Akad. Nauk SSSR},
volume={88},
number={4},
pages={601--602},
year={1953}
@article {mccoid_mini_17_davidenko1953new,
AUTHOR = {Davidenko, D. F.},
TITLE = {On a new method of numerical solution of systems of nonlinear
equations},
JOURNAL = {Doklady Akad. Nauk SSSR (N.S.)},
VOLUME = {88},
YEAR = {1953},
PAGES = {601--602},
MRCLASS = {65.0X},
MRNUMBER = {0054339},
MRREVIEWER = {A. S. Householder},
}
@article{branin1972widely,
title={Widely convergent method for finding multiple solutions of simultaneous nonlinear equations},
author={Branin, Franklin H},
journal={IBM Journal of Research and Development},
volume={16},
number={5},
pages={504--522},
year={1972},
publisher={IBM}
@article {mccoid_mini_17_branin1972widely,
AUTHOR = {Branin, Jr., F. H.},
TITLE = {Widely convergent method for finding multiple solutions of
simultaneous nonlinear equations},
JOURNAL = {IBM J. Res. Develop.},
FJOURNAL = {International Business Machines Corporation. Journal of
Research and Development},
VOLUME = {16},
YEAR = {1972},
PAGES = {504--522},
ISSN = {0018-8646},
MRCLASS = {65H10},
MRNUMBER = {418449},
DOI = {10.1147/rd.165.0504},
URL = {https://doi.org/10.1147/rd.165.0504},
}
@article{brent1972davidenko,
title={On the {D}avidenko-{B}ranin method for solving simultaneous nonlinear equations},
author={Brent, RP},
journal={IBM Journal of Research and Development},
volume={16},
number={4},
pages={434--436},
year={1972},
publisher={IBM}
@article {mccoid_mini_17_brent1972davidenko,
AUTHOR = {Brent, R. P.},
TITLE = {On the {D}avidenko-{B}ranin method for solving simultaneous
nonlinear equations},
NOTE = {Mathematics of numerical computation},
JOURNAL = {IBM J. Res. Develop.},
FJOURNAL = {International Business Machines Corporation. Journal of
Research and Development},
VOLUME = {16},
YEAR = {1972},
PAGES = {434--436},
ISSN = {0018-8646},
MRCLASS = {65H10},
MRNUMBER = {334498},
MRREVIEWER = {N. N. Abdelmalek},
DOI = {10.1147/rd.164.0434},
URL = {https://doi.org/10.1147/rd.164.0434},
}
% -- Alternating {S}chwarz --%
@article{lui1999schwarz,
author = {Lui, S.},
title = {On {S}chwarz Alternating Methods for Nonlinear Elliptic {PDE}s},
journal = {SIAM J Sci Comput},
volume = {21},
number = {4},
pages = {1506-1523},
year = {1999},
doi = {10.1137/S1064827597327553},
URL = {https://doi.org/10.1137/S1064827597327553},
eprint = {https://doi.org/10.1137/S1064827597327553}
@article {mccoid_mini_17_lui1999schwarz,
AUTHOR = {Lui, S. H.},
TITLE = {On {S}chwarz alternating methods for nonlinear elliptic
{PDE}s},
JOURNAL = {SIAM J. Sci. Comput.},
FJOURNAL = {SIAM Journal on Scientific Computing},
VOLUME = {21},
YEAR = {1999/00},
NUMBER = {4},
PAGES = {1506--1523},
ISSN = {1064-8275},
MRCLASS = {65N55 (65J15)},
MRNUMBER = {1756041},
MRREVIEWER = {Stephen W. Brady},
DOI = {10.1137/S1064827597327553},
URL = {https://doi.org/10.1137/S1064827597327553},
}
@article{Cai2002,
title={Nonlinearly preconditioned inexact {N}ewton algorithms},
author={Cai, Xiao-Chuan and Keyes, David E},
journal = {SIAM J Sci Comput},
volume={24},
number={1},
pages={183--200},
year={2002},
publisher={SIAM}
@article {mccoid_mini_17_Cai2002,
AUTHOR = {Cai, Xiao-Chuan and Keyes, David E.},
TITLE = {Nonlinearly preconditioned inexact {N}ewton algorithms},
JOURNAL = {SIAM J. Sci. Comput.},
FJOURNAL = {SIAM Journal on Scientific Computing},
VOLUME = {24},
YEAR = {2002},
NUMBER = {1},
PAGES = {183--200},
ISSN = {1064-8275},
MRCLASS = {65H10 (76M25)},
MRNUMBER = {1924420},
MRREVIEWER = {R. P. Tewarson},
DOI = {10.1137/S106482750037620X},
URL = {https://doi.org/10.1137/S106482750037620X},
}
@article{Dolean2016,
author = {V Dolean and M J Gander and W Kheriji and F Kwok and R Masson},
doi = {10.1137/15M102887X},
issue = {6},
journal = {SIAM J Sci Comput},
keywords = {65F10,65N22,nonlinear preconditioning,preconditioning Newton's method AMS subject classifications 65M55,two-level nonlinear Schwarz methods},
pages = {3357-3380},
title = {NONLINEAR PRECONDITIONING: HOW TO USE A NONLINEAR {S}CHWARZ METHOD TO PRECONDITION {N}EWTON'S METHOD},
volume = {38},
url = {http://www.siam.org/journals/sisc/38-6/M102887.html},
year = {2016},
@article {mccoid_mini_17_Dolean2016,
AUTHOR = {Dolean, V. and Gander, M. J. and Kheriji, W. and Kwok, F. and
Masson, R.},
TITLE = {Nonlinear preconditioning: how to use a nonlinear {S}chwarz
method to precondition {N}ewton's method},
JOURNAL = {SIAM J. Sci. Comput.},
FJOURNAL = {SIAM Journal on Scientific Computing},
VOLUME = {38},
YEAR = {2016},
NUMBER = {6},
PAGES = {A3357--A3380},
ISSN = {1064-8275},
MRCLASS = {65F08 (65F10 65N22 65N55)},
MRNUMBER = {3566907},
MRREVIEWER = {Judith M. Ford},
DOI = {10.1137/15M102887X},
URL = {https://doi.org/10.1137/15M102887X},
}
@incollection{Gander2017,
title={On the origins of linear and non-linear preconditioning},
author={Gander, Martin J},
booktitle={Domain Decomposition Methods in Science and Engineering XXIII},
pages={153--161},
year={2017},
publisher={Springer}
@incollection {mccoid_mini_17_Gander2017,
AUTHOR = {Gander, Martin J.},
TITLE = {On the origins of linear and non-linear preconditioning},
BOOKTITLE = {Domain decomposition methods in science and engineering
{XXIII}},
SERIES = {Lect. Notes Comput. Sci. Eng.},
VOLUME = {116},
PAGES = {153--161},
PUBLISHER = {Springer, Cham},
YEAR = {2017},
MRCLASS = {65F08 (01A55 65-03)},
MRNUMBER = {3718350},
DOI = {10.1007/978-3-319-52389-7\_1},
URL = {https://doi.org/10.1007/978-3-319-52389-7_1},
}
@article{liu2015field,
author = {Lulu Liu and David E Keyes},
doi = {10.1137/140970379},
issue = {3},
journal = {SIAM J Sci Comput},
keywords = {65H20,65N22,65N55,Navier-Stokes equations AMS subject classifications 65H10,Newton method,field splitting,nonlinear equations,nonlinear preconditioning},
pages = {A1388-A1409},
title = {FIELD-SPLIT PRECONDITIONED INEXACT {N}EWTON ALGORITHMS},
volume = {37},
url = {http://www.siam.org/journals/sisc/37-3/97037.html},
year = {2015},
@article {mccoid_mini_17_liu2015field,
AUTHOR = {Liu, Lulu and Keyes, David E.},
TITLE = {Field-split preconditioned inexact {N}ewton algorithms},
JOURNAL = {SIAM J. Sci. Comput.},
FJOURNAL = {SIAM Journal on Scientific Computing},
VOLUME = {37},
YEAR = {2015},
NUMBER = {3},
PAGES = {A1388--A1409},
ISSN = {1064-8275},
MRCLASS = {65H10 (65H20 65N22 65N55)},
MRNUMBER = {3352613},
MRREVIEWER = {Luca Gemignani},
DOI = {10.1137/140970379},
URL = {https://doi.org/10.1137/140970379},
}
@book{gander2014scientific,
title={Scientific computing-An introduction using Maple and MATLAB},
author={Gander, Walter and Gander, Martin J and Kwok, Felix},
volume={11},
year={2014},
publisher={Springer Science \& Business}
@book {mccoid_mini_17_gander2014scientific,
AUTHOR = {Gander, Walter and Gander, Martin J. and Kwok, Felix},
TITLE = {Scientific computing},
SERIES = {Texts in Computational Science and Engineering},
VOLUME = {11},
NOTE = {An introduction using Maple and MATLAB},
PUBLISHER = {Springer, Cham},
YEAR = {2014},
PAGES = {xviii+905},
ISBN = {978-3-319-04324-1; 978-3-319-04325-8},
MRCLASS = {65-01},
MRNUMBER = {3287477},
DOI = {10.1007/978-3-319-04325-8},
URL = {https://doi.org/10.1007/978-3-319-04325-8},
}
\ No newline at end of file
......@@ -13,32 +13,7 @@
\smartqed
%--Preamble (some packages in preamble.sty conflict with the svmult class)
% cannot use the subcaption package as it interferes with the svmult class caption settings
%\newcommand{\dxdy}[2]{\frac{d #1}{d #2}}
%\newcommand{\dxdyk}[3]{\frac{d^{#3} #1}{d {#2}^{#3}}}
\newcommand{\pdxdy}[2]{\frac{\partial #1}{\partial #2}}
%\newcommand{\liminfty}[1]{\lim_{#1 \to \infty}}
%\newcommand{\limab}[2]{\lim_{#1 \to #2}}
%
\newcommand{\abs}[1]{\left \vert #1 \right \vert}
%\newcommand{\norm}[1]{\left \Vert #1 \right \Vert}
%\newcommand{\order}[1]{\mathcal{O} \left ( #1 \right )}
%\newcommand{\set}[1]{\left \{ #1 \right \}}
%\newcommand{\Set}[2]{\left \{ #1 \ \middle \vert \ #2 \right \}}
%\newcommand{\vmat}[1]{\begin{vmatrix} #1 \end{vmatrix}}
\DeclareMathOperator{\sign}{sign}
%\newcommand{\sign}{\text{sign}}
%
%\newcommand{\bbn}{\mathbb{N}}
%\newcommand{\bbz}{\mathbb{Z}}
%\newcommand{\bbq}{\mathbb{Q}}
\newcommand{\bbr}{\mathbb{R}}
%\newcommand{\bbc}{\mathbb{C}}
%\newcommand{\bbf}{\mathbb{F}}
%--End Preamble
\definecolor{darkgreen}{rgb}{0,0.6,0.1}
\usepackage{tikz}
......@@ -64,11 +39,11 @@ An algorithm is proposed combining several aspects of this theory and others to
\section{Introduction}
\label{mccoid_mini_17_sec:intro}
ASPIN \cite{Cai2002}, RASPEN \cite{Dolean2016}, and MSPIN \cite{liu2015field} rely on various Schwarz methods to precondition either Newton-Raphson or inexact Newton.
ASPIN \cite{mccoid_mini_17_Cai2002}, RASPEN \cite{mccoid_mini_17_Dolean2016}, and MSPIN \cite{mccoid_mini_17_liu2015field} rely on various Schwarz methods to precondition either Newton-Raphson or inexact Newton.
While \textit{a priori} convergence criteria have been found for the underlying Schwarz methods, so far none exist for their combination with Newton-Raphson.
Like in the linear case when combining a Krylov method and a Schwarz method,
there is an equivalence between preconditioning Newton-Raphson with a Schwarz method and accelerating that same Schwarz method with Newton-Raphson \cite{Gander2017}:
there is an equivalence between preconditioning Newton-Raphson with a Schwarz method and accelerating that same Schwarz method with Newton-Raphson \cite{mccoid_mini_17_Gander2017}:
A domain is first subdivided into subdomains, the problem solved on each subdomain, and the resulting formulation iterated through Krylov and Newton-Raphson, respectively.
We examine cycling behaviour in alternating Schwarz in one dimension that has been accelerated by applying Newton-Raphson.
......@@ -106,12 +81,12 @@ where $v_i(x)=\partial u_i(x) / \partial \gamma$ and $J(u_i)$ is the Jacobian of
\section{Convergence of generic fixed point iterations and Newton-Raphson}
A generic fixed point iteration $x_{n+1}=g(x_n)$ converges when $\abs{g(x_n)-x^*} < \abs{x_n-x^*}$, where $x^*$ is the fixed point.
A generic fixed point iteration $x_{n+1}=g(x_n)$ converges when $\left \vert g(x_n)-x^* \right \vert < \left \vert x_n-x^* \right \vert$, where $x^*$ is the fixed point.
This occurs when $g(x)$ lies between $x$ and $2x^*-x$.
The convergence or divergence of the fixed point iteration is monotonic if $\sign(g(x)-x^*) = \sign(x - x^*)$ and oscillatory otherwise.
This creates four lines, $y=x$, $y=2x^*-x$, $y=x^*$ and $x=x^*$, that divide the plane into octants.
The four pairs of opposite octants form four regions with distinct
behaviour of the fixed point iteration, see left of Figure \ref{mccoid_mini_17_fig:FP} or Figure 5.7 from \cite{gander2014scientific}:
behaviour of the fixed point iteration, see left of Figure \ref{mccoid_mini_17_fig:FP} or Figure 5.7 from \cite{mccoid_mini_17_gander2014scientific}:
\begin{description}
\item[\textcolor{red}{1}, $g(x) < x < x^*$ or $g(x) > x > x^*$:] monotonic divergence;
\item[\textcolor{blue}{2}, $x < g(x) < x^*$ or $x > g(x) > x^*$:] monotonic convergence;
......@@ -206,26 +181,24 @@ Intersections of $g_f(x)$ with $y=2x^*-x$ may be represented as a first order OD
\begin{equation*}
f_C'(x) = -\frac{f_C(x)}{2 (x^*- x)}, \quad f_C(x^*) = 0.
\end{equation*}
The solution to this ODE is $f_C(x) = C \sqrt{\abs{x - x^*}}$ where $C \in \bbr$.
The solution to this ODE is $f_C(x) = C \sqrt{\left \vert x - x^* \right \vert}$ where $C \in \mathbb{R}$.
If a function $f(x)$ with root $x^*$ is tangential to $f_C(x)$ for any value of $C$ then $g_f(x)$ intersects the line $y=2x^*-x$.
The left of Figure \ref{mccoid_mini_17_fig:borders} shows the functions $f_C(x)$.
\newcommand{\faxis}{\clip (-4,-4) rectangle (4,4);
\draw[black,thick,->] (-4,0) -- (4,0);
\draw[black,thick,->] (0,-4) -- (0,4);}
\newcommand{\gaxis}{\clip (-4,-4) rectangle (4,4);
\draw[black,thick,->] (-4,-4) -- (4,4);
\draw[black,thick,->] (0,-4) -- (0,4);}
\begin{figure}
\centering
\begin{tikzpicture}
\matrix[column sep=0.5cm, row sep=0.5cm, every cell/.style={scale=0.4}]
{
\faxis
\clip (-4,-4) rectangle (4,4);
\draw[black,thick,->] (-4,0) -- (4,0);
\draw[black,thick,->] (0,-4) -- (0,4);
\foreach \a in {1,-1,2,-2,0.5,-0.4,1.5,-1.5} {
\draw[domain=-4:4, smooth, samples=200, variable=\x, black, dashed] plot ({\x},{\a * sqrt(abs(\x))});
} &
\gaxis
\clip (-4,-4) rectangle (4,4);
\draw[black,thick,->] (-4,-4) -- (4,4);
\draw[black,thick,->] (0,-4) -- (0,4);
\foreach \a in {1,-1,2,-2,0.5,-0.4,1.5,-1.5} {
\draw[domain=-4:4, smooth, samples=200, variable=\x, black, dashed] plot ({\x},{\a * sqrt(abs(\x)) + \x});
} \\
......@@ -290,16 +263,16 @@ Again by assumption $u^1_1 = u^2_1$ and $\gamma_1 = \gamma_2$.
\end{proof}
We can even prove that $G(\gamma)$ is restricted to region 2 with additional properties.
As an example, we reprove a result from Lui \cite{lui1999schwarz}.
As an example, we reprove a result from Lui \cite{mccoid_mini_17_lui1999schwarz}.
\begin{theorem}[Theorem 2 from \cite{lui1999schwarz}] \label{mccoid_mini_17_thm:lui}
\begin{theorem}[Theorem 2 from \cite{mccoid_mini_17_lui1999schwarz}] \label{mccoid_mini_17_thm:lui}
Consider the equation
$u''(x) + f(x,u,u') = 0$ for $x \in (a,b)$, $u(a) = u(b) = 0$
under the assumptions that
\begin{itemize}
\item $f \in C^1 \left ( [a,b] \times \mathbb{R} \times \mathbb{R} \right )$ ,
\item $\pdxdy{f(x,v,v')}{u} \leq 0$ for all $x \in [a,b]$ and $v \in H_0^1([a,b])$ ,
\item $\abs{f(x,v,v')} \leq C(1 + \abs{v'}^\eta)$ for all $x \in [a,b]$ and $v \in H_0^1([a,b])$ and some $C > 0$, $0 < \eta < 1$ .
\item $\frac{\partial f(x,v,v')}{\partial u} \leq 0$ for all $x \in [a,b]$ and $v \in H_0^1([a,b])$ ,
\item $\left \vert f(x,v,v') \right \vert \leq C(1 + \left \vert v' \right \vert^\eta)$ for all $x \in [a,b]$ and $v \in H_0^1([a,b])$ and some $C > 0$, $0 < \eta < 1$ .
\end{itemize}
The problem is solved using alternating Schwarz with two subdomains and Dirichlet transmission conditions.
Then $G(\gamma)$ for this problem lies within region 2.
......@@ -307,7 +280,7 @@ Then $G(\gamma)$ for this problem lies within region 2.
\begin{proof}
It suffices to prove that the problem is well posed and $0 < G'(\gamma) < 1$ for all $\gamma \in \mathbb{R}$.
The well-posedness of the problem is guaranteed by Proposition 2 from \cite{lui1999schwarz}.
The well-posedness of the problem is guaranteed by Proposition 2 from \cite{mccoid_mini_17_lui1999schwarz}.
As Lui points out, this also means the problem is well posed on any subdomain.
Using Theorem \ref{mccoid_mini_17_thm:mono} this gives monotonicity of $G(\gamma)$.
Moreover, if $u(x)=0$ for any $x \in (a,b)$ then the problem would be well posed on the domains $[a,x]$ and $[x,b]$.
......@@ -315,9 +288,9 @@ As such, $u(x)$ has the same sign as $\gamma$ and $G'(\gamma) > 0$.
Consider the problem in $g_1$:
\begin{equation*}
g_1''(x) + \pdxdy{f}{u} g_1 + \pdxdy{f}{u'} g_1' = 0, \quad x \in [a,\beta], \quad g_1(a) = 0, \quad g_1(\beta) = 1.
g_1''(x) + \frac{\partial f}{\partial u} g_1 + \frac{\partial f}{\partial u'} g_1' = 0, \quad x \in [a,\beta], \quad g_1(a) = 0, \quad g_1(\beta) = 1.
\end{equation*}
From the second assumption on $f$ the operator on $g_1$ satisfies a maximum principle (see, for example, \cite{lui1999schwarz}).
From the second assumption on $f$ the operator on $g_1$ satisfies a maximum principle (see, for example, \cite{mccoid_mini_17_lui1999schwarz}).
Therefore, $g_1(x) < 1$ for all $x \in (a,\beta)$.
By the same reasoning, $g_2(x) < g_1(\alpha) < 1$ for all $x \in (\alpha, b)$ and $G'(\gamma) < 1$.
Incidentally, the same maximum principle applies for the operator on $-g_1$ and $-g_2$, and so $G'(\gamma) > 0$ as we had before.
......@@ -339,9 +312,9 @@ Sadly, its Newton-Raphson acceleration will not do so for all initial conditions
Take $a=3.6$ with an overlap of 0.4 and symmetric regions.
\begin{figure}[t]
\centering
\includegraphics[height=0.3\textwidth]{mccoid_mini_17_NewtonGamma.eps}
\includegraphics[height=0.3\textwidth]{mccoid_mini_17_figure3_left.eps}
\hspace{1em}
\includegraphics[height=0.3\textwidth]{mccoid_mini_17_exp9_02.eps}
\includegraphics[height=0.3\textwidth]{mccoid_mini_17_figure3_right.eps}
\caption{
\textbf{Left:} Results of Newton-Raphson accelerated alternating Schwarz as a function of initial condition in solving equation (\ref{mccoid_mini_17_eq:sin}).
The value of $a$ is 3.6 and the subdomains are $\Omega_1=(-1,0.2)$ and $\Omega_2=(-0.2,1)$.
......@@ -362,7 +335,7 @@ Where stable cycles exist so too must there be period doubling bifurcation.
Changing the value of the parameter $a$ we find that the 2-cycle found in Figure \ref{mccoid_mini_17_fig:ex1} (left) becomes two 2-cycles, then two 4-cycles, and so on until it devolves into chaos, see Figure \ref{mccoid_mini_17_fig:ex2}.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{mccoid_mini_17_bifurcation.eps}
\includegraphics[width=\textwidth]{mccoid_mini_17_figure4.eps}
\caption{Period doubling bifurcation in the example caused by Newton-Raphson acceleration.}
\label{mccoid_mini_17_fig:ex2}
\end{figure}
......@@ -372,8 +345,8 @@ While a change in the parameter $a$ is the most obvious way to alter the dynamic
This has a direct effect on the basin of cycling in the spaces of both initial condition $\gamma$ and the parameter $a$.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{mccoid_mini_17_ParamOverlap.eps}
\includegraphics[width=0.45\textwidth]{mccoid_mini_17_BasinOverlap.eps}
\includegraphics[width=0.45\textwidth]{mccoid_mini_17_figure5_left.eps}
\includegraphics[width=0.45\textwidth]{mccoid_mini_17_figure5_right.eps}
\caption{
\textbf{Left:} value of $a$ at which bifurcation starts.
\textbf{Right:} width of basin of cycling in $\gamma$ and $a$.}
......@@ -389,10 +362,10 @@ Meanwhile, the length of the bifurcation diagram increases, meaning there are mo
\section{Accelerated alternating Schwarz with guaranteed convergence}
Given Theorem \ref{mccoid_mini_17_thm:lui} and the conditions of Table \ref{mccoid_mini_17_tab:conditions} one can construct a series of tests to see if the Newton-Raphson acceleration is suitable for a given iteration.
We present one further useful trick to strengthen convergence, a correction to Newton-Raphson due to Davidenko and Branin \cite{branin1972widely,brent1972davidenko,davidenko1953new}.
We present one further useful trick to strengthen convergence, a correction to Newton-Raphson due to Davidenko and Branin \cite{mccoid_mini_17_branin1972widely,mccoid_mini_17_brent1972davidenko,mccoid_mini_17_davidenko1953new}.
We replace step (3) in the algorithm with
\begin{equation*}
(3*) \quad \tilde{\gamma}_n = \gamma_n - \frac{G(\gamma_n) - \gamma_n}{\abs{G'(\gamma_n) - 1}} .
(3*) \quad \tilde{\gamma}_n = \gamma_n - \frac{G(\gamma_n) - \gamma_n}{\left \vert G'(\gamma_n) - 1 \right \vert} .
\end{equation*}
For $G(\gamma)$ within region 2 the Newton-Raphson acceleration will now always march in the direction of the fixed point.
It may still overshoot and cycle but the direction will always be correct.
......@@ -404,7 +377,7 @@ For a problem satisfying the conditions of Theorem \ref{mccoid_mini_17_thm:lui}
If $G'(\gamma_n) = 1$ then set $\gamma_{n+1} = G(\gamma_n)$, increment $n$ and return to step 2.
If this is not true, proceed to step 3.
\item Perform step (3*), which is the Newton-Raphson acceleration using the Davidenko-Branin trick.
If $\abs{G'(\gamma_n) - 1} \geq 1/2$ then set $\gamma_{n+1} = \tilde{\gamma}_n$, increment $n$ and return to step 2.
If $\left \vert G'(\gamma_n) - 1 \right \vert \geq 1/2$ then set $\gamma_{n+1} = \tilde{\gamma}_n$, increment $n$ and return to step 2.
If this is not true, calculate $\hat{\gamma}_n$, the average of $\gamma_n$ and $\tilde{\gamma}_n$, and proceed to step 4.
\item Calculate $G(\hat{\gamma}_n)$.
If $G(\hat{\gamma}_n)-\hat{\gamma}_n$ has the same sign as $G(\gamma_n) - \gamma_n$ then set $\gamma_{n+1} = \tilde{\gamma}_n$, increment $n$ and return to step 2.
......
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