So far I have been using Newton-Raphson (N-R) to solve nonlinear systems. However N-R might run into the problem of singularity depending on the initial guess.
I found an alternate approach which is Homotopy. From what I have found on the internet, $H(x,t) = tF(x) + (1-t)G(x),$ where $F(x)$ is the original system and $G(x)$ is the auxiliary function.
First, starts off by solving $G(x),$ something that is either known or easy to solve. Then vary the variable $t$ from $0$ to $1,$ and then eventually it will take you to $F(x).$ But there are so many types of Homotopy, i.e., Homotopy Continuation, Newton Homotopy, Fixed-Point Homotopy etc. Frankly, I'm confused and I'm not a mathematician, and I tried to understand what they are by reading math papers, they gave me headaches. Does anyone know enough about Homotopy that can answer a couple of quesitons I have?
Do I have the right idea about Homotopy or am I talking in gibberish?
Does Homotopy give all the roots?
If so, when I plug in some value of $x,$ does $H(x,t)$ give the closest root or some random roots like N-R?
What is a good rule of thumb for choosing $G(x)$?
Is there other iterative methods out there that is better or more efficient than N-R or Homotopy?