Research on Numerical Methods for Solving Nonlinear Systems and Their Applications

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In this project, we first proposed an efficient homotopy method for solving nonlinear circuits, and prove its global convergence property for modified nodal equations that describe nonlinear circuits. By this method, bipolar analog integrated circuits with more than 10000 elements were solved efficiently with the theoretical guarantee of global convergence. We further proposed an improved version of the homotopy method using a new homotopy function and an effective initial solution algorithm, which is much more efficient than the above method.We next proposed an efficient algorithm for finding all solutions of piecewise-linear resistive circuits using the dual simplex method. Using this algorithm, all solutions of large scale problems, including those where the number of variables is 2000 and the number of linear regions is 1000^{2000}, could be found in practical computation time, which is a remarkable development in this field.We also proposed SPICE-oriented numerical methods for solving nonlinear problems using path following circuits. The path following circuits (PFC's) are circuits for solving nonlinear problems on the circuit simulator SPICE. In the method of PFC's, formulas of numerical methods are described by circuits, which are solved by SPICE. Using PFC's, numerical analysis without programming is possible, and various techniques implemented in SPICE will make the numerical analysis very efficient. In this project, we applied the PFC's of the homotopy method to various nonlinear problems where the homotopy method is proven to be globally convergent. This approach makes SPICE applicable to a broader class of scientific problems.