@ -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,
]
\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}
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=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}
