We will only consider \textit{square} systems of polynomial equations, i.e. systems of $n$ polynomial equations in $n$ variables, although or over- or under-determined systems can
often be solved by reducing them to square systems, by respectively choosing a suitable square subsystem or squaring it by adding equations. Morever, we will restrict ourselves to
There are many ways to choose the "simpler" system, from now on called a \textit{start system}, but in general we can observe that, by Bezout's theorem, a system
where $x:=(x_1,\ldots,x_n)$ and $t\in[0,1].$ This is such that the roots of $H(x,0)=G(x)$ are known, and the roots of $H(x,1)=F(x)$ are the solutions of the original system (the
reason why we place the start system at $t=0$ and the original system at $t=1$ is that we need higher numerical precision for the solutions of the original system, and there are more
floating point numbers near to $t=0$; see \cite{BertiniBook}, p. 33).
define a curve $z(t)$ in $\C^n$ by the equation \begin{equation}\label{eq:h2} H(z(t),t)=0,\end{equation} so that in order to approximate the roots of $F$ it is enough to numerically track $z(t)$.
To do so, we derive the expression \eqref{eq:h2} with respect to $t$, and get the \textit{Davidenko Differential Equation}
This is a system of $n$ first-order differential equations, which can be solved numerically for $z(t)$ as an initial value problem, which is called \textit{path tracking}.
While \eqref{eq:h1} is a fine choice of a homotopy, it's not what it's called a \textit{good homotopy}: in order to ensure that the solution paths $z(t)$ for different roots
a root $z_0$ of the start system by solving the initial value problem associated to the Davidenko differential equation \eqref{eq:dav} with starting value $z_0$ and
This will be done numerically, by using a first-order predictor-corrector tracking method, whose typical iteration goes like this:
\begin{itemize}
\item\textbf{Predictor:} we first apply Euler's method to get an approximation $\widetilde{z}_i$ of the next value of the solution path;
\item\textbf{Corrector:} we then use Newton's method to correct $\widetilde{z}_i$ using equation \eqref{eq:h2}, so that it becomes a good approximation $z_i$ of the next value of the solution path.
\end{itemize}
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
In the case of the Davidenko equation \eqref{eq:dav}, we have
$$f(z,t)=-\left(\frac{\partial H}{\partial z}(z,t)\right)^{-1}\frac{\partial H}{\partial t}(z,t)$$ and $t_0=1$, since we are tracking from $1$ to $0$. For the same
To test the method's scalability, we first launched it on a single-threaded machine, then one a multi-threaded one, and finally parallelized it on a Cluster.
The latter was done by using the Julia package \textit{Distributed.jl} to parallelize the tracking of the roots on separate nodes, and the \texttt{SlurmClusterManager} package, which allows
to run Julia code using the \texttt{Slurm} workload manager.
In order to scale the method to larger systems, we also implemented a random polynomial generator which can be found in \hyperref[sec:random]{random-poly.jl}; this was used to
evaluate the performance of the parallel implementation, by generating square systems of polynomials with normally distributed coefficients, each
polynomial having total degree less or equal to a fixed maximum degree.
The single-threaded machine and multi-threaded tests (which used the \texttt{@threads}
macro from the \textit{Threads.jl} package on the root tracking \texttt{for} loop in the file \hyperref[sec:listing]{solve.jl}) were run in order to visualize the real solutions of
small (2x2) systems: here, multi-threaded runs didn't improve the
performance on these smaller systems, as the overhead of multi-threading was too big compared to the actual computation time.
However, when testing a parallel implementation on larger randomly generated systems we observed an improvement in execution times on larger systems compared to the single-node
runs, as we show in the \hyperref[sec:parallel]{Results} section.
The Julia implementation for the tests described above can be found in Appendix \hyperref[sec:listing]{B}, while
the hardware specifications are listed in Appendix \hyperref[sec:hw]{A}.
Since our start systems have the maximum number of solutions for its degree, some of them might converge to a point at infinity of our original system. In our current
implementation, we waste time by tracking them until reaching the maximum number of iterations.
To better treat such cases, we could view the system inside an affine patch of the projective plane, and using homogenized coordinates detect when a solution is going to infinity. This would involve homogenizing both systems and modifying the path-tracking algorithm for the detection of a point going to infinity.
Our (un)specific choice of predictor could be unsuitable for badly-conditioned systems; other software implementations of the homotopy continuation method use more accurate and numerically stable predictors, such as Runge-Kutta methods
\caption{Performance comparison of parallel path tracking on a cluster.}
\end{figure}
As we can see from the plot, the parallel implementation appears to scale well with the number of tracked roots, and is faster than the single-node implementation for larger
For the single-threaded runs, the code was executed on a laptop with an Intel Core i7-3520M CPU @ 3.60GHz and 6 GB of RAM.
The multithreaded runs were tested on a desktop with an AMD FX-8350 CPU @ 4.00GHz with 4 cores and 8 threads, and 12 GB of RAM.
Finally, the parallel computations were run on a cluster with 20 nodes, each having a CPU @ 1.008GHz with 4 Performance cores, 2 efficiency cores and 4 GB of RAM.
\bibitem{BertiniBook} Bates, Daniel J. \textit{Numerically solving polynomial systems with Bertini}. SIAM, Society for Industrial Applied Mathematics, 2013.
