Quantized state systems method

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The quantized state systems (QSS) methods are a family of numerical integration solvers based on the idea of state quantization, dual to the traditional idea of time discretization. Unlike traditional numerical solution methods, which approach the problem by discretizing time and solving for the next (real-valued) state at each successive time step, QSS methods keep time as a continuous entity and instead quantize the system's state, instead solving for the time at which the state deviates from its quantized value by a quantum. They can also have many advantages compared to classical algorithms. They inherently allow for modeling discontinuities in the system due to their discrete-event nature and asynchronous nature. They also allow for explicit root-finding and detection of zero-crossing using explicit algorithms, avoiding the need for iteration---a fact which is especially important in the case of stiff systems, where traditional time-stepping methods require a heavy computational penalty due to the requirement to implicitly solve for the next system state. Finally, QSS methods satisfy remarkable global stability and error bounds, described below, which are not satisfied by classical solution techniques.