## 1 Introduction
Although it is known that some circuits with positive feedback can maintain multiple operating points, it is not possible to reliably predict multiple DC solutions just by inspecting the circuit topology and component values. Since the standard DC analysis of a circuit in the SPICE software does not automatically detect the possibility of multiple operating points, it is essential to look for them systematically. As shown in the paper [7], the number of operating points in a simulation depends on how the circuit is modeled and may require detailed modeling in regions of operation that are often not accurately modeled. The presence of positive feedback with a loop gain greater than unity is a necessary (though not sufficient) condition for equilibrium point instability [7]. The Newton method and its variants are widely used to calculate solutions to nonlinear equations, for example, in circuit and power grid simulators. These methods can achieve quadratic or super-linear convergence, provided that an initial guess close enough to the solution is specified. However, the methods have some fundamental disadvantages. First, convergence is only guaranteed if the user can provide a sufficiently accurate initial guess. In addition, the lack of convergence of the method does not imply the lack of circuit solutions. Finally, once a solution is found, the methods, due to the specific nature of the numerical algorithm used, do not provide information on the existence of other solutions. Thus, the methods are unsuitable for computing multiple solutions [11]. Another problem is that we have to recalculate the Jacobian matrix at the beginning of each iteration. This matrix contains all the partial derivatives of the nonlinear equations with respect to the circuit variables, which is computationally expensive [3]. Algorithms based on the Newton method, with the default settings of the simulator, often find a solution corresponding to the unstable equilibrium point, e.g., for the circuit shown in Figure 1 using IsSpec4 [12; 13], such a solution was determined (the nodal voltages are shown in Figure 1). The SPICE circuit simulator has become an industry standard, and many SPICE-like tools are now in use. Various efficient techniques such as sparse matrix techniques, LU decomposition, modified Newton's method, and stable numerical integration methods are implemented in SPICE. In addition, SPICE contains various know-how accumulated over 50 years. Therefore, in the field of circuit simulation, from a practical point of view, it seems a good practice to develop a new method using SPICE as the core [14]. Therefore, this paper also decided to follow this approach. Typically, SPICE-simulator-based programs offer sets of different methods rather than a single method for DC analysis, allowing one to choose a new procedure when a tried algorithm fails. For example, MicroCap 10 offers five basic methods: standard Newton method, source stepping, diagonal Gmin stepping, junction Gmin stepping, and pseudo transient. The user controls the methods and the order in which they are tried. IsSpice4 offers three methods: standard Newton, source stepping, and Gmin stepping [15]. Since traditional methods of solving nonlinear equations describing transistor circuits have often shown convergence difficulties, more sophisticated mathematical techniques and tools, such as parameter embedding, continuation, and homotopy methods, have been implemented. Some of these techniques are discussed later. These methods make it possible to analyze circuits that could not be simulated using conventional techniques and can also be used to find multiple DC operating points [5]. Finding all operating points (DC solutions) of a nonlinear circuit is a fundamental problem in the computer-aided design of analog circuits [4; 16; 17; 18; 19; 20; 21]. Many approaches have been proposed to solve the problem. The most-commonly used methods to find all DC solutions are those based on piecewise linear (PWL) approximation and computational techniques [4; 14; 17; 18; 22].