New families of algorithms for solving nonlinear circuit equations

In this paper we describe a method of generating families of iterative algorithms which are suitable for solving nonlinear systems of equations. These families of algorithms, one of which includes the Newton-Raphson algorithm as a special case, are novel in that they could use the type and behavior of each of the individual equations to advantage; in effect, the algorithms are able to tailor themselves to the behavior of each function. In addition, by suitably choosing from among the members of one of these families of iterative schemes, a variable-order algorithm emerges. For one such family, this variable-order algorithm is equivalent to the heuristic modifications of the Newton-Raphson algorithm that have been proposed which do not update the Jacobian at every iteration. The question of how often the Jacobian should be updated can thus be discussed from a theoretical as well as an experimental viewpoint. Preliminary results indicate that the variable-order algorithms can provide significant computational savings in a transient simulation when compared with the conventional Newton-Raphson algorithm.