How to track the paths as t varies from 1 to 0, using total-degree homotopy?

[IJ05]

I am very new to Julia and its data structures. I want to track the path evolution, for the following Straightline homotropy
$H(x,t) = tG(x)+(1-t)F(x)$.
The minimum working example, from the total degree homotopy page is:
julia> using HomotopyContinuation
julia> @var x y;
julia> f = System([x^2 + 2y, y^2 - 2])
julia>result= solve(f; start_system = :total_degree)
There would be 4 paths for the above system, and my requirement is to track all the paths as t varies from 1 to 0. I would be deeply grateful if I could get some pointers regarding this.

[PBrdng]

Hi,

the code you copied into your post should do exactly what you want: execute total degree homotopy on x^2 + 2y = y^2 - 2 = 0. Or do you want to access the 4 paths directly?

[IJ05]

Hello,
I want to access the 4 paths directly, so that I can see (or plot, or export) the evolution of all the four paths from a starting solution (when t=1) to the end solution (when t=0). Thank you.

[PBrdng]

Hi,

there is no function which returns all intermediate steps, because they are discarded by default. You can, however, set up a tracker and track the solutions from 1 to t for any t. See here for a small example.

The solver_startsolutions functions can set up a tracker for you.

You should also check out the path_info function.

[IJ05]

Hello,
Apologies for the late response. Thank you very much for your insights. I have managed to obtain the intermediate iterations using tracker. Below is my implementation, although I am not quite sure whether this is the most efficient way to do so.
using HomotopyContinuation
@var x y;
f = System([x^2 + 2y, y^2 - 2])
G = System((0.2 + 0.4im) .* [x^2 - 1, y^2 - 1])
H = StraightLineHomotopy(G, f);
S = [[1,1],[1,-1],[-1,1],[-1,-1]];
tracker = Tracker(H, options = (automatic_differentiation = 3,))

Now, I run the evolution of variables x and y for the first path (starting point: [1,1]), using a for loop
timevar = LinRange(1,0,21);
newvar=Array{ComplexF64}(undef,length(timevar),length(timevar))
for i=1:length(timevar)
sol = track(tracker, S[1], 1, timevar[i]);
newvar[i,1]=sol.solution[1];
newvar[i,2]=sol.solution[2];
end

[PBrdng]

Hi, one could possibly use track! instead of track to save some memory. But all in all this is similar to what I would do. Hope it helps!

[IJ05]

Thank you very much for your insights. I deeply appreciate it.