Function (gif)
An animated plot representing the convergence of iterative refinement.
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%%% Full, compilable sources are on Github:
%%% https://github.com/bvieuble/TeXFantasy/tree/main/illustrations/fig2
%%% Appears in my lecture ``Mixed precision iterative refinement'' (2023).
% Compiled with XeLaTeX
% Generate the GIF with convert -delay 100 -loop 0 -density 200 -alpha on
% file.pdf file.gif (package imagemagick is required)
\documentclass[tikz,border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepackage{bm}
\input{color_theme.tex}
\newcommand{\alertgm}[1]{\color{mygreen} \bm{#1}}
\newcommand{\alertrm}[1]{\color{myred} \bm{#1}}
\begin{document}
\newcommand{\plota}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
}
\newcommand{\plotb}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\draw [dashed,thick] (axis cs:{8,0}) -- (axis cs:{8,3.2});
\addplot [only marks,mygreen,mark=*] coordinates { (8,3.25) };
\node at (axis cs:8,3.7) {$\alertgm{f(x_0)}$};
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
}
\newcommand{\plotc}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\draw [dashed,thick] (axis cs:{8,0}) -- (axis cs:{8,2.9});
\addplot [only marks,myred,mark=*] coordinates { (8,2.9) };
\node at (axis cs:8.6,2.9) {$\alertrm{\widehat{f}(x_0)}$};
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
}
\newcommand{\plotd}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[mygreen,no marks,domain=-0.5:9,thick,restrict y to domain={-1:5}]
{0.5*x - 1.1};
\addplot [only marks,mark=*] coordinates { (8,2.9) };
\node[rotate=25] at (axis cs:7,2) {$\alertgm{\nabla f(x_0)}$};
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
}
\newcommand{\plote}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[myred,no marks,domain=-0.5:9,thick,restrict y to domain={-1:5}]
{0.65*x - 2.3};
\addplot [only marks,mark=*] coordinates { (8,2.9) };
\node[rotate=30] at (axis cs:7.1,1.9) {$\alertrm{\widehat{\nabla}
f(x_0)}$};
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
}
\newcommand{\plotf}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[no marks,domain=-0.5:9,thick,restrict y to domain={-1:5}]
{0.65*x - 2.3};
\addplot [only marks,mygreen,mark=*] coordinates { (3.55,0) };
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
\node at (axis cs:3.55,-0.4) {\large \alertgm{$x_{1}$}};
}
\newcommand{\plotg}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[no marks,domain=-0.5:9,thick,restrict y to domain={-1:5}]
{0.65*x - 2.3};
\addplot [only marks,myred,mark=*] coordinates { (3.9,0) };
\node[rotate=20] at (axis cs:6,2.6) {$f(x) = Ax-b$};
\node at (axis cs:3.9,-0.4) {\large \alertrm{$\widehat{x}_{1}$}};
}
\newcommand{\ploth}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\draw [dashed,thick] (axis cs:{3.9,0}) -- (axis cs:{3.9,1.6});
\addplot [only marks,mark=*] coordinates { (3.9,1.6) };
\node at (axis cs:3.9,2.1) {$\widehat{f}(x_1)$};
\addplot [only marks,mark=|] coordinates { (3.9,0) };
\node at (axis cs:3.9,-0.4) {\large $\widehat{x}_{1}$};
}
\newcommand{\ploti}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[no marks,domain=-0.5:9,thick] {0.45*x - 0.155};
\addplot [only marks,mark=*] coordinates { (3.9,1.6) };
\node[rotate=20] at (axis cs:2.7,1.5) {$\widehat{\nabla}f(x_1)$};
\addplot [only marks,mark=|] coordinates { (3.9,0) };
\node at (axis cs:3.9,-0.4) {\large $\widehat{x}_{1}$};
}
\newcommand{\plotj}{
\addplot[myblue,no marks,domain=-0.5:9,thick] {0.5*x - 0.75};
\addplot[no marks,domain=-0.5:9,thick] {0.45*x - 0.155};
\addplot [only marks,mark=*] coordinates { (0.7,0) };
\node at (axis cs:0.7,-0.4) {\large $\widehat{x}_{2}$};
\addplot [only marks,mark=|] coordinates { (3.9,0) };
\node at (axis cs:3.9,-0.4) {\large $\widehat{x}_{1}$};
}
\foreach \whichplot in
{\plota,\plotb,\plotc,\plotd,\plote,\plotf,\plotg,\ploth,\ploti,\plotj}
{\begin{tikzpicture}
\begin{axis}
[
ymajorticks=false,
xtick={8,1.5},
xticklabels = {\large $x_0$, \large $x$},
width=1.1\textwidth,
height=0.6\textwidth,
ymin=-1,
ymax=5,
xmax=9,
xmin=-0.5,
axis lines=middle,
tick style={black, very thick},
]
\whichplot
\end{axis}
\end{tikzpicture}}
\end{document}