In today's music industry and among musicians, instead of using analog hardware effects to alter sound, digital counterparts are increasingly being used, often in the form of software plugins. The circuits of musical devices often contain nonlinear components (diodes, vacuum tubes, etc.), which complicates their digital modeling. One of the approaches to address this is the use of state-space methods, such as the Euler or Runge-Kutta methods. To guarantee stability, implicit state-space methods should be used; however, they require the numerical solution of an implicit equation, leading to large computational complexity. Alternatively, the K-method can be used that avoids the need of numerical methods if the system meets certain conditions, thus significantly decreasing the computational complexity. Although the K-method has been invented almost three decades ago, the authors are not aware of a thorough computational complexity analysis of the method in comparison to the more common implicit state-space approaches, such as the backward Euler method. This paper introduces these two methods, explores their advantages, and compares their computational load as a function of model size by using a scalable circuit example.
