Study on Numerical Methods for Solving Nonlinear Systems and Their Application to LSI Design

在这个项目中，我们首先提出了一种新的算法，该算法利用修正的节点方程的不动点同伦是全局收敛的。利用该算法，可以有效地求解15,000多个单元的双极型模拟集成电路。在较弱的条件下，我们证明了一个保证算法全局收敛的定理。在此基础上，提出了一种基于线性规划的求解分段线性电阻电路所有解的有效算法。利用该算法可以在实际计算时间内求解大规模问题的所有解，包括变量个数为300、线性区域个数为10^&lt；300的问题。我们发现了一种有趣的现象，称为聚合物溶液的可重入三相平衡，并利用这一现象，首次计算了聚合物溶液的四相平衡。

In this project, we first propose a new algorithm using the fixed-point homotopy that is globally convergent for modified nodal equations. Using this algorithm, bipolar analog integrated circuits with more than 15,000 elements are solved efficiently. We prove a theorem that guarantees the global convergence of the proposed algorithm under mild conditions. We also show that the proposed algorithm converges to a stable operating point with high possibility.We next proposed an efficient algorithm for finding all solutions of piecewise-linear resistive circuits using linear programming. Using this algorithm, all solutions of large scale problems, including those where the number of variables is 300 and the number of linear regions is 10^<300>, could be solved in practical computation time.We also studied the problem of calculating multiphase equilibria of polymer solutions, which is a very difficult problem in polymer chemistry. We discovered an interesting phenomena termed the reentrant three-phase equilibria of polymer solutions, and exploiting this phenomena, we calculated four-phase equilibria of polymer solutions for the first time.