HighPerformanceMathematics
/
homotopy-continuation
mirror of https://github.com/BachoSeven/homotopy-continuation/
2
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity

finalize report: format with latexindent

Browse Source
main
Francesco Minnocci 1 year ago
parent 191adf7a60
commit 05267d6ecc
No known key found for this signature in database
GPG Key ID: 76DA3AF9BAED1A32
2 changed files with 67 additions and 60 deletions
Show Stats Download Patch File Download Diff File

BIN
report/report.pdf
View File

Binary file not shown.

39
report/report.tex
Unescape Escape View File

@ -86,7 +86,7 @@
\end{Huge} \end{Huge}
\bigskip\vfill \bigskip\vfill
\begin{tabular}{ll} \begin{tabular}{ll}
\textbf{Studente:} & Francesco Minnocci\\ \textbf{Studente:} & Francesco Minnocci \\
\textbf{Matricola:} & 600455 \textbf{Matricola:} & 600455
\end{tabular} \end{tabular}
\end{center} \end{center}
@ -143,12 +143,12 @@ To do so, we derive the expression \eqref{eq:h2} with respect to $t$, and get th
$$ \frac{\partial H}{\partial z}\frac{\mathrm{d} z}{\mathrm{d} t}+\frac{\partial H}{\partial t}=0 ,$$ $$ \frac{\partial H}{\partial z}\frac{\mathrm{d} z}{\mathrm{d} t}+\frac{\partial H}{\partial t}=0 ,$$
where $\frac{\partial H}{\partial z}$ is the Jacobian matrix of $H$ with respect to $z$: where $\frac{\partial H}{\partial z}$ is the Jacobian matrix of $H$ with respect to $z$:
$$ $$
\frac{\partial H}{\partial z}= \frac{\partial H}{\partial z}=
\begin{pmatrix} \begin{pmatrix}
\frac{\partial H_1}{\partial z_1} & \cdots & \frac{\partial H_1}{\partial z_n}\\ \frac{\partial H_1}{\partial z_1} & \cdots & \frac{\partial H_1}{\partial z_n} \\
\vdots & \ddots & \vdots\\ \vdots & \ddots & \vdots \\
\frac{\partial H_n}{\partial z_1} & \cdots & \frac{\partial H_n}{\partial z_n} \frac{\partial H_n}{\partial z_1} & \cdots & \frac{\partial H_n}{\partial z_n}
\end{pmatrix} . \end{pmatrix} .
$$ $$
This can be rewritten as 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} \begin{equation}\label{eq:dav} \dot{z}=-\frac{\partial H}{\partial z}^{-1}\frac{\partial H}{\partial t} .\end{equation}
@ -186,8 +186,8 @@ Recall that Euler's method consists in approximating the solution of the initial
\begin{equation*} \begin{equation*}
\left\{ \left\{
\begin{aligned} \begin{aligned}
&\dot{z}=f(z,t)\\ & \dot{z}=f(z,t) \\
&z(t_0)=z_0 & z(t_0)=z_0
\end{aligned} \end{aligned}
\right. \right.
\end{equation*} \end{equation*}
@ -258,29 +258,29 @@ Below are the plots of four different 2x2 systems for the single- (laptop) and m
Single-threaded & & Multithreaded \\ Single-threaded & & Multithreaded \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1.png} & \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1.png} &
$\left\{\begin{aligned} $\left\{\begin{aligned}
&x^3 + 5x^2 - y - 1 \\ & x^3 + 5x^2 - y - 1 \\
&2x^2 - y - 1 \\ & 2x^2 - y - 1 \\
\end{aligned}\right.$ & \end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1_6.png} \\ \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions1_6.png} \\
\vspace{0.5cm} \\ \vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2.png} & \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2.png} &
$\left\{\begin{aligned} $\left\{\begin{aligned}
&x^2 + 2y \\ & x^2 + 2y \\
&y - 3x^3 \\ & y - 3x^3 \\
\end{aligned}\right.$ & \end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2_6.png} \\ \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions2_6.png} \\
\vspace{0.5cm} \\ \vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3.png} & \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3.png} &
$\left\{\begin{aligned} $\left\{\begin{aligned}
&x^2 + y^2 - 4 \\ & x^2 + y^2 - 4 \\
&xy - 1 \\ & xy - 1 \\
\end{aligned}\right.$ & \end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3_6.png} \\ \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions3_6.png} \\
\vspace{0.5cm} \\ \vspace{0.5cm} \\
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4.png} & \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4.png} &
$\left\{\begin{aligned} $\left\{\begin{aligned}
&x^2 + y^2 - 2 \\ & x^2 + y^2 - 2 \\
&xy - 1 \\ & xy - 1 \\
\end{aligned}\right.$ & \end{aligned}\right.$ &
\includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4_6.png} \\ \includegraphics[width=0.45\textwidth,valign=c]{../plots/solutions4_6.png} \\
\end{tabular} \end{tabular}
@ -336,6 +336,13 @@ systems.
\section{Appendix B: Implementation} \section{Appendix B: Implementation}
\subsection{Julia code} \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} \label{sec:listing}\jlinputlisting[caption={solve.jl}]{../solve.jl}
\jlinputlisting[caption={start-system.jl}]{../start-system.jl} \jlinputlisting[caption={start-system.jl}]{../start-system.jl}
\jlinputlisting[caption={homotopy.jl}]{../homotopy.jl} \jlinputlisting[caption={homotopy.jl}]{../homotopy.jl}

Write Preview
Loading…
Cancel
Save