Expanding the region of convergence for second order methods to solve nonlinear equations via the generalized moment generating function approach

Iterative methods are used to approximately solve equations if an analytical treatment is not feasible. While many publications focus on high convergence order and numerical efficiency, the dependence of the convergence behavior on the initial value of such algorithms persists. To mitigate this issue, recently the generalized moment generating function algorithm was developed at the hand of Newton’s method. This approach reformulates the root-finding problem via infinitely many generalized moment generating functions and picks the representation which is approximated most effectively by a linear function. For many problems, this algorithm converges more rapidly than other methods, especially if the initial value and the sought root are far apart from each other. This publication suggests a systematic way to derive a generalized moment generating function approach for an arbitrary second order method. We exemplify the treatment for a concrete method. The convergence order is discussed and a fixed-point analysis is given. We also treat two exactly soluble problems and compare the performance of our approach with higher order methods for seven non-trivial numerical problems. Especially in the case that the initial value and the root are far apart from each other, our method offers benefits over the other algorithms in terms of stability and computation time. As such, it may serve as an enabler for highly dynamical control problems with a real-time target.