A parametric method for computing the steady-state characteristics of electric power-supply systems (Q1842460)

For a parameter-dependent problem \(\Phi (x,b) = 0\), defined by the smooth nonlinear mapping \(\Phi : \mathbb{R}^ n \times \mathbb{R}^ m \to \mathbb{R}^ n\), the authors consider approximations of solutions \((x_ 1, b_ 1)\) for any given \(b_ 1\) near the parameter value \(b_ 0\) of a known solution \((u_ 0, b_ 0)\). This is done on a local parametrization of solution curves of an augmented system \(\widetilde {\Phi} (x,b) := (\Phi (x,b)\), \(P^ T(b - b_ 0)) = 0\) in \(\mathbb{R}^{n + m - 1}\), where \(P \in \mathbb{R}^{(m-1) \times m}\) is an orthogonal matrix and \(P^ T (b_ 1 - b_ 0) = 0\). The augmented system depends on a single parameter and its solution curve is approximated via calculations of the Taylor expansion of its local parametrization, leading to an approximation of the solution of the original problem.