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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?

  1. Do I have the right idea about Homotopy or am I talking in gibberish?

  2. Does Homotopy give all the roots?

  3. 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?

  4. What is a good rule of thumb for choosing $G(x)$?

  5. Is there other iterative methods out there that is better or more efficient than N-R or Homotopy?

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    Just as an aside, there's a whole theory of convex optimization to consider. You can use interior point methods, semi-Newton methods, gradient methods, or even basic line search methods to find roots. I'm not suggesting that your interest in homotopy methods is bad, but you'll probably be much better served by considering the half-century old ideas on root-finding with linear algebra before going off to homotopy.2012-04-03
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    Thanks EMS, I will look them up. I don't know much about solving nonlinear system other than using N-R, gauss-seidel and fixed-point iterative method. And as far as I know, N-R is the "best".2012-04-03

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