Homotopy Continuation in Julia

This is a project for the "Laboratorio Computazionale" exam at the University of Pisa

Implemented

• Total-degree Homotopy with "Roots of unity" start system
• Euler-Newton predictor-corrector method with adaptive step size
• Homotopy Continuation for all roots of the target system

TODO

• Parallelization
• Homogenization

Example systems

Here's some tests on 2x2 systems, with the plotted real approximate solutions

$$ \begin{align*} x^3 + 5x^2 - y - 10 &= 0 \\ 2x^2 - y - 10 &= 0 \\ \end{align*} $$

Single-threaded	Multi-threaded (nproc=6)

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$$ \begin{align*} x^2 + 2y &= 0 \\ y - 3x^3 &= 0 \\ \end{align*} $$

Single-threaded	Multi-threaded (nproc=6)

$$ \begin{align*} x^2 + y^2 - 4 &= 0 \\ xy - 1 &= 0 \\ \end{align*} $$

Single-threaded	Multi-threaded (nproc=6)

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$$ \begin{align*} x^2 + y^2 - 2 &= 0 \\ xy - 1 &= 0 \\ \end{align*} $$

Single-threaded	Multi-threaded (nproc=6)