Implementing an arc-length method for a robust approach in solving systems of nonlinear equations

Solving systems of nonlinear equations can be challenging and analysts are often required to provide an initial guess of the solution as a starting point for use in an iterative solver. Insight into approximate solutions leading to a good initial guess can usually be obtained if equations are representative of a physical system. However, this process may not be achievable for complex systems or when the analyst lacks familiarity or experience with the system. In this case, convergence may not be achieved if the initial guess is not close to the solution. A general nonlinear solver suite based on the arc-length method with these circumstances in mind was developed for the purpose of numerical experimentation and was found to be a useful alternative to the fsolve function inherent to the MATLAB software. Due to the additional unknown variable and supplemental constraint equation used by the arc-length method, curves representing solutions to example equation sets were found by embedding the solver in a loop. Restarts in the analysis were minimized as the arc-length method is capable of solving beyond local maxima or minima on smooth curves. Several examples are provided demonstrating the unique capabilities of arc-length solvers.