Extended Fixed-Point Methods for the Computation of Virtual Analog Models

A family of iterative root-finding methods for nonlinear discrete-time systems of equations is presented, with a formulation that puts it in between the Fixed-Point (FP) and Newton-Raphson (NR) methods. Applicability of this family is allowed, provided that the Jacobian matrix of the nonlinear system has a spectral radius less than one. By varying the order of a matrix geometric sum that approximates the inverse Jacobian matrix, root-finding at any iteration can be steered toward the FP or conversely toward the NR method, becoming identical to either of them if the order is equal to zero or infinitely large, respectively. Since the methods in this family do not need the solution of a linear system at each iteration as required by NR, their computational cost makes them palatable for the online digital implementation of nonlinear models. As an example of application, a Virtual Analog model of the voltage-controlled filter onboard a popular music synthesizer is tested, showing that for some orders of the aforementioned geometric sum the proposed methods perform better than FP and NR in terms of computational cost, while exhibiting the same accuracy.