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修正节点方程的全局收敛的不动点同伦算法。该算法可有效求解15000元以上双极模拟集成电路。证明了算法在温和条件下的全局收敛性。我们还证明了该算法收敛到一个稳定的工作点的可能性很高。接下来，我们提出了一种利用线性规划求分段线性电阻电路所有解的有效算法。使用该算法，在实际计算时间内可以求解所有大规模问题的解，包括变量数为300和线性区域数为10^<300>的问题。我们还研究了聚合物溶液多相平衡的计算问题，这是聚合物化学中的一个难题。我们发现了一个有趣的现象，称为聚合物溶液的可重入三相平衡，并利用这一现象，我们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.