Solving systems of equations

The goal of solving systems of equations is to solve equations of the type \[\begin{equation} \label{eq:system_of_equations} \require{physics} \vb{f}(\vb{x}) = \vb{0} \thinspace , \end{equation}\] in which \(\vb{f}\) is a vector field as in equation \(\eqref{eq:vector_field}\). In order to solve equation \(\eqref{eq:system_of_equations}\), we use a linear approximation of \(\vb{f}\) at a specified \(\vb{x}_0\): \[\begin{equation} \label{eq:system_of_equations_jacobian} \vb{f}(\vb{x}) \approx \vb{f}(\vb{x}_0) + \vb{J}(\vb{x}_0) (\vb{x} - \vb{x}_0) \thinspace , \end{equation}\] in which \(\vb{J}(\vb{x}_0)\) is the Jacobian (cfr. equation \(\eqref{eq:jacobian}\)) of \(\vb{f}\), calculated at the point \(\vb{x}_0\), which immediately signifies the meaning of the Jacobian: it is a generalization of the concept of derivative.

There are a few ways of solving systems of equations: - Newton’s method; - Broyden’s method; - DIIS.