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