You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
67 lines
1.6 KiB
Markdown
67 lines
1.6 KiB
Markdown
# 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) |
|
|
|-------------------|---------------------------------|
|
|
|  |  |
|
|
|
|
---
|
|
|
|
$$
|
|
\begin{align*}
|
|
x^2 + 2y &= 0 \\
|
|
y - 3x^3 &= 0 \\
|
|
\end{align*}
|
|
$$
|
|
|
|
| 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) |
|
|
|-------------------|---------------------------------|
|
|
|  |  |
|
|
|
|
---
|
|
|
|
$$
|
|
\begin{align*}
|
|
x^2 + y^2 - 2 &= 0 \\
|
|
xy - 1 &= 0 \\
|
|
\end{align*}
|
|
$$
|
|
|
|
| Single-threaded | Multi-threaded (nproc=6) |
|
|
|-------------------|---------------------------------|
|
|
|  |  |
|