# Review of modern homotopy methods and practical techniques

I'm hoping someone can recommend recent literature concerning homotopy methods for solving systems of nonlinear equations. Already by the time of Layne Watson's 1986 paper there were a lot of methods, and I'm sure there have been developments since then as well as practical shifts in the field due to hardware advancements.

I'm looking to apply globally convergent methods to problems in nonlinear elasticity with ~$10^6$ degrees of freedom. The Jacobians are sparse but local methods like Newton-Raphson or quasi-Newton with line search or trust region tend to get trapped at points where the Jacobian is singular. I've had some success with a crude implementation of a homotopy method but I'm looking to get better performance and better familiarity with modern methods.

Edit: I'm mainly interested in the benefits and drawbacks of the three methods for curve tracking that Watson describes: the ODE algorithm, the normal flow algorithm, and the augmented Jacobian algorithm. Watson's papers are very nice but they are a bit outdated for assessing my problem. For example, it seems that the limited memory version of BFGS had not been developed yet, which made it impossible to fairly test a sparse version of the augmented Jacobian algorithm. For my problem, I am only interested in the solution, i.e. $x$ s.t. $F(x,\lambda)=0$ for $\lambda=1$, but I am very concerned with stability and efficiency.

The linear algebra steps seem quite straightforward, and I can test different preconditioners, etc.

• I suggest you ask Layne Watson himself. Afaik, he's still with the CS department at Virginia Tech. You can find his email address on the department's webpage. Chances are he'll get back to you with useful information. – GoHokies Jun 29 '18 at 20:00