HighPerformanceMathematics/homotopy-continuation
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finalize report: format with latexindent

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BachoSeven
2024-01-05 17:29:56 +01:00
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commit 05267d6ecc
2 changed files with 65 additions and 58 deletions
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@@ -63,8 +63,8 @@
\begin{large}
\begin{center}
\vbox to 100pt{%
\includegraphics[width=3cm]{cherubino}%
\vfil}
\includegraphics[width=3cm]{cherubino}%
\vfil}
\end{center}
\begin{center}
\begin{Large}
@@ -86,7 +86,7 @@
\end{Huge}
\bigskip\vfill
\begin{tabular}{ll}
\textbf{Studente:} & Francesco Minnocci\\
\textbf{Studente:} & Francesco Minnocci \\
\textbf{Matricola:} & 600455
\end{tabular}
\end{center}
@@ -143,12 +143,12 @@ To do so, we derive the expression \eqref{eq:h2} with respect to $t$, and get th
$$ \frac{\partial H}{\partial z}\frac{\mathrm{d} z}{\mathrm{d} t}+\frac{\partial H}{\partial t}=0 ,$$
where $\frac{\partial H}{\partial z}$ is the Jacobian matrix of $H$ with respect to $z$:
$$
\frac{\partial H}{\partial z}=
\begin{pmatrix}
\frac{\partial H_1}{\partial z_1} & \cdots & \frac{\partial H_1}{\partial z_n}\\
\vdots & \ddots & \vdots\\
\frac{\partial H_n}{\partial z_1} & \cdots & \frac{\partial H_n}{\partial z_n}
\end{pmatrix} .
\frac{\partial H}{\partial z}=
\begin{pmatrix}
\frac{\partial H_1}{\partial z_1} & \cdots & \frac{\partial H_1}{\partial z_n} \\
\vdots & \ddots & \vdots \\
\frac{\partial H_n}{\partial z_1} & \cdots & \frac{\partial H_n}{\partial z_n}
\end{pmatrix} .
$$
This can be rewritten as
\begin{equation}\label{eq:dav} \dot{z}=-\frac{\partial H}{\partial z}^{-1}\frac{\partial H}{\partial t} .\end{equation}
@@ -185,10 +185,10 @@ In the following sections, we go into more detail on each of these steps.
Recall that Euler's method consists in approximating the solution of the initial value problem associated to a system of first-order ordinary differential equations
\begin{equation*}
\left\{
\begin{aligned}
&\dot{z}=f(z,t)\\
&z(t_0)=z_0
\end{aligned}
\begin{aligned}
& \dot{z}=f(z,t) \\
& z(t_0)=z_0
\end{aligned}
\right.
\end{equation*}
by the sequence of points $(z_i)_{i\in\N}$ defined by the recurrence relation
@@ -255,34 +255,34 @@ Below are the plots of four different 2x2 systems for the single- (laptop) and m
\newgeometry{left=.3cm,top=0.1cm}
\begin{figure}[htb]
\begin{tabular}{c c c}
Single-threaded & & Multithreaded \\
Single-threaded & & Multithreaded \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1.png} &
$\left\{\begin{aligned}
&x^3 + 5x^2 - y - 1 \\
&2x^2 - y - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1_6.png} \\
\vspace{0.5cm} \\
& x^3 + 5x^2 - y - 1 \\
& 2x^2 - y - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1_6.png} \\
\vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2.png} &
$\left\{\begin{aligned}
&x^2 + 2y \\
&y - 3x^3 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2_6.png} \\
\vspace{0.5cm} \\
& x^2 + 2y \\
& y - 3x^3 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2_6.png} \\
\vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3.png} &
$\left\{\begin{aligned}
&x^2 + y^2 - 4 \\
&xy - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3_6.png} \\
\vspace{0.5cm} \\
& x^2 + y^2 - 4 \\
& xy - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3_6.png} \\
\vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4.png} &
$\left\{\begin{aligned}
&x^2 + y^2 - 2 \\
&xy - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4_6.png} \\
& x^2 + y^2 - 2 \\
& xy - 1 \\
\end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4_6.png} \\
\end{tabular}
\end{figure}
@@ -298,33 +298,33 @@ on a single node and on 20 nodes (using 1 or 2 threads per node).
\centering
\begin{tikzpicture}
\begin{axis}[
xlabel={\# of tracked roots},
ylabel={Running Times (s)},
legend pos=north west,
grid=major,
]
xlabel={\# of tracked roots},
ylabel={Running Times (s)},
legend pos=north west,
grid=major,
]
\addplot[mark=*,blue] coordinates {
(18, 139.703750)
(24, 171.741583)
(54, 290.947457)
(90, 252.224948)
(108, 266.180392)
(120, 231.164993)
(144, 280.459045)
};
\addlegendentry{Parallel}
\addplot[mark=*,blue] coordinates {
(18, 139.703750)
(24, 171.741583)
(54, 290.947457)
(90, 252.224948)
(108, 266.180392)
(120, 231.164993)
(144, 280.459045)
};
\addlegendentry{Parallel}
\addplot[mark=square,red] coordinates {
(18, 95.067010)
(24, 109.203866)
(54, 251.746024)
(90, 774.436612)
(108, 1098.606851)
(120, 805.911525)
(144, 1908.437483)
};
\addlegendentry{Single Node}
\addplot[mark=square,red] coordinates {
(18, 95.067010)
(24, 109.203866)
(54, 251.746024)
(90, 774.436612)
(108, 1098.606851)
(120, 805.911525)
(144, 1908.437483)
};
\addlegendentry{Single Node}
\end{axis}
\end{tikzpicture}
@@ -336,6 +336,13 @@ systems.
\section{Appendix B: Implementation}
\subsection{Julia code}
The code is also available at the following GitHub repository:
\begin{center}
\fbox{\url{https://github.com/bachoseven/homotopy-continuation}}
\end{center}
\label{sec:listing}\jlinputlisting[caption={solve.jl}]{../solve.jl}
\jlinputlisting[caption={start-system.jl}]{../start-system.jl}
\jlinputlisting[caption={homotopy.jl}]{../homotopy.jl}