## 1 Introduction  
There are certain continuous-time systems that are of interest to computer musicians, that contain a delay-free loop--instantaneous feedback from the output to the input. These are said to be 'non-computable loops' because in order to process the input, one needs to know the output, which hasn't been computed yet. The most straightforward way to deal with this problem is to insert a small amount of delay in the feedback loop. This was done, for example, in one of the earliest considerations of digitizing the Moog ladder filter [1]. In that case, the analog circuitry under consideration contains a delay-free path from output to input in its continuous-time block diagram. The authors add unit delay (\(z^{-1}\)) to the structure in order to make it computable (Figure 1). This has the unwanted side-effect, however, of coupling the resonance parameter of the filter to the parameter governing cutoff frequency. Figure 1: Continuous-time (**a**) and discrete time (**b**) block diagrams of the Moog ladder filter after [PERSON] and [PERSON], 1996 [1]. The circuit emulation software SPICE (Simulation Program with INtegrated Circuit Emphasis) [2] can compute systems with delay-free loops through the use of iterative solvers. SPICE is very robust and is still the gold standard against which to demonstrate accuracy in virtual analog literature. It is not, however, well suited for virtual instrument design since it is not meant to be run in real-time. It is also unsuitable for describing systems that are not represent-able as electrical circuits. A class of structures that can also emulate circuits (and other things) in real-time are wave digital filters (WDFs). WDFs were first introduced in the early 1970's [3] and have been the subject of a steady stream of study since then. More recently, there has been increased development of WDF theory in the digital audio community, particularly in circuit simulation applications [4; 5; 6], but in virtual acoustics as well [7; 8; 9]. Until very recently, however, WDF theory was not sufficient for representing systems involving delay-free loops. To this end, the K-method [10] has been suggested as a means of solving coupled, multivariate nonlinearities in WDFs [11; 12]. Interestingly, the K-method was originally proposed in [10] as a means of eliminating delay-free loops in virtual acoustic models. Nevertheless, it and its variants are frequently used in circuit simulation applications [13; 14; 15]. The K-method operates on dynamical state-space systems that are of the form:  
\[\dot{\mathbf{x}} =\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}+\mathbf{C}\mathbf{i}( \mathbf{v}) \tag{1}\] \[\mathbf{y} =\mathbf{D}\mathbf{x}+\mathbf{E}\mathbf{u}+\mathbf{Fi}(\mathbf{v})\] (2) \[\mathbf{v} =\mathbf{L}\mathbf{x}+\mathbf{M}\mathbf{u}+\mathbf{Ni}(\mathbf{v}) \tag{3}\]  
where \(\mathbf{x}\) is the'state', \(\mathbf{u}\) is an input vector, \(\mathbf{y}\) is the output and \(\mathbf{i}\) is a nonlinear vector function of \(\mathbf{v}\). After discretizing the system, the nonlinearity in Equation (3) can be solved with an iterative solver such as Newton-Raphson method. Furthermore multidimensional lookup tables can be pre-computed to avoid convergence issues, rendering the solution more suitable to real-time computation. The method also requires discretizing Equation (1) (e.g., with Backward Euler or the trapezoidal rule). A consequence of this is that if an adjustable parameter presents itself in the discretized version of matrix \(\mathbf{A}\) it may be the case that multiple matrix inversions need to be performed in order to find the correct discretized coefficients that correspond to \(\mathbf{B}\), \(\mathbf{C}\), _etc_. There are times when analysis can resolve this issue (as in [15]), but this can potentially lead to significant slowdown if controllable parameters are desired. If \(\mathbf{i}\) is linear or if it is a low order polynomial, an exact solution be found to the system and resorting to such methods may not be necessary. However, this is not always the case.