Detecting independent parts in non-linear system of equations

Question

When solving systems of non-linear equations using Newton's method, it is often observed that the system has an independent sub-system, e.g. :

$$ f(x,y)=0 $$ $$ g(x,y)=0 $$ $$ h(x,y,z)=0 $$

If I am not mistaken, you can (and should) solve first for x and y and then subsequently solve for z.

Now when the dimension of the system increases, it is not so easy to spot such independent sub-systems. Are there algorithms adapted to this issue ?

In my specific case, the Jacobian of the system has a fixed structure so even a slow algorithm for detection would be largely beneficial when solving the non-linear system using Newton's method. I feel that this is a common topic in solving systems of equations, so I am open to be redirected to additional ressources that are relevant.

Answer 1

A non-optimum, but generally time saving algorithm:

1. Rank the variables from most common to least common
2. Rank the equations from most variables to least variables
3. Start with the most common (or one of the most if equal) variable not previously addressed
4. Start with the equation with the current variable that has the the least variables
5. Use the result on all equations with that variable. thus eliminating it.
6. Go to step 3 using the next most common variable