Solving the Harmonic Balance Equations

Harmonic Balance leaves us with a problem to be solved: a nonlinear algebraic equation system with respect to the Fourier coefficients of the approximation. In this chapter, we categorize the fundamentally different strategies suited for solving such equation systems. We then focus on local numerical methods, in particular on Newton-type methods. These compute a single solution point if a good initial guess is available. A challenge is often the high computational effort, and we therefore present important means of increasing numerical efficiency. Usually, one is not only interested in just a single solution point but how the solution evolves under variation of certain parameters. We therefore address the important topic of numerical path continuation. Finally, we discuss the issues associated with finding a good initial guess, handling branching points, and uncovering isolated branches.