Using FindRoot on a system of equations with a singular Jacobian

Question

I have a system of $n$ ODEs $\frac{dx_i}{dt} = f_i(x_1.\ldots,x_n)$, which I would like to solve at steady state.

However, $\sum_{i} x_i = 1$, and so it's actually only an $n-1$-dimensional system. As a result, the jacobian for my system is singular everywhere (which FindRoot tells me when I try to use it). Is there anyway I can get Find Root to work with this, without rewriting the equations in terms of the $n-1$ variables?

For example, what I would like to do is the following:

eqn1 = y-2x*y;
eqn2 = 2x*y-y;
FindRoot[{eqn1==0,eqn2==0,x+y==1},{{x,0.2},{y,0.2}}].

I know I can use NSolve instead in this scenario, but for the system I'm actually working with, NSolve doesn't compute it in a reasonable time.

This system doesn't actually throw the Jacobian error, but it is a similar system, so I imagine if I can do it for this system, my system of interest will work with the same method.

Answer 1

Introduce an additional variable (here a) that you use nowhere in the equations. (It's just to provide three variables for three equations.)

eqn1 = y - 2 x*y;
eqn2 = 2 x*y - y;
FindRoot[{eqn1 == 0, eqn2 == 0, (x + y - 1) == 0}, 
    {{x, 0.2}, {y, 2}, {a, 2}}]

(*   {x -> 0.5, y -> 0.5, a -> 2.}   *)

eqn1 = y - 2 x*y;
eqn2 = 2 x*y - y;
FindRoot[{eqn1 == 0, eqn2 == 0, (x + y - 1) == 0}, 
   {{x, 1.2}, {y, 2}, {a, 1}}]

(*   {x -> 1., y -> -7.38374*10^-28, a -> 1.}   *)