Estimating unique solutions of DC transistor circuits

For each natural n let Fⁿ denote the collection of mappings of Rⁿ onto itself defined by: F ∈Fⁿ if and only if there exist n strictly monotone increasing functions f_k mapping R onto itself such that for each x =[x₁, …, x_n]^T ∈ Rⁿ, F(x) = [f₁(x₁), …, f_n(x_n)]^T. The following new property of the class P₀ of matices is proved: a real n × n matrix A belongs to P₀ if and only if for every G, H ∈ Fⁿ the set S₀ = { x ∈ Rⁿ : − G(x) ≤Ax ≤ − H(x) } is bounded. As an illustration of this property a method of estimating the unique solution of the nonlinear equation F(x) + A(x) =b describing the large class of DC transistor circuits is developed. This can improve the efficiency of known computation algorithms. Numerical examples of transistor circuits illustrate in detail how the method works in practice.