Systems with parameters

How to track parametrized systems of equations

Parameter Homotopies

Consider the situation in which one has to solve a specific instance of a parametrized family of polynomial systems

$$ P = \{F(x,p) = (f_1(x,p), \ldots, f_n(x,p)) \mid p \in \mathbb{C}^m\}. $$

To not destroy the solution structure it is desirable to not leave $P$ during the homotopy. This can be accomplished by using the homotopy $$H(x,t) := F(x, (1-t)p + tq)$$ where $p$ and $q$ are parameters in $\mathbb{C}^m$. Note that you have to provide the start solutions for this kind of homotopy.

The syntax in HomotopyContinuation.jl to construct such a homotopy is as follows.

solve(F, startsolutions; parameters=params, start_parameters=q, target_parameters=p)

where p and q are vectors of parameter values for F. params is a vector of variables that specify the parameters of F. Necessarily, length(params), length(p)and length(q) must all be equal.

A simple example

$$F(x,y,a,b) = \begin{bmatrix} x^2-a \\ xy-a+b \end{bmatrix}.$$

For tracking the solution $(x,y) = (1,1)$ from $(a,b) = (1,0)$ to $(a,b) = (2,5)$ we do the following.

julia> @polyvar x y a b
julia> F = [x^2 - a, x * y - a + b]
julia> startsolution = [[1, 1]]
julia> solve(F, startsolution; parameters=[a, b], start_parameters=[1, 0], target_parameters=[2, 5])
Result with 1 solutions
• 1 non-singular solution (1 real)
• 0 singular solutions (0 real)
• 1 paths tracked
• random seed: 772337