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A group of parallel algorithms, and their implementation is discussed, for solving a special class of large sparse nonlinear equations. The type of sparsity occurring in these problems, which arise in VLSI design, structural engineering and many other areas, is called a block bordered structure. An explicit method and several implicit methods are described for solving block bordered nonlinear problems, and a mathematical analysis and computational comparisons are made for the two types of methods. Several variations and globally convergent modifications of the implicit method are also presented. Parallel algorithms for solving block bordered nonlinear equations are described and experimental results presented on the Intel hypercube that show the effectiveness of the parallel implicit algorithms. These experiments include a fairly large circuit simulation that leads to a multi-level block bordered system of nonlinear equations. Keywords: Nonlinear equations; Block bordered; Parallel; Implicit methods; Circuit simulation.

This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F(x) by a quadratic model where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two-dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations.