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

非线性系统求解的数值方法及其在大规模集成电路设计中的应用研究

• 批准号: 11650395
• 负责人: YAMAMURA Kiyotaka
• 金额: $ 2.3万
• 依托单位: Chuo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11650395/
• 关键词: nonlinear system; numerical analysis; nonlinear circuit; LSI design; circuit simulation; polymer chemistry; linear programming; interval analysis; 多相平衡

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.

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。

项目成果

山村清隆: "線形計画法を用いた区分的線形回路の全解探索法"電子情報通信学会論文誌(A). J83-A・8. 965-975 (2000)

Kiyotaka Yamamura：“使用线性编程的分段线性电路的完整解决方案搜索方法”电子信息通信工程师学会学报（A）965-975（2000）。

Kiyotaka Yamamura: "Finding all solutions of nonlinear equations using inverses of approximate Jacobian matrices"IEICE Trans.Fundamentals. E84-A・11. 2950-2952 (2001)

Kiyotaka Yamamura：“使用近似雅可比矩阵的逆求非线性方程的所有解”IEICE Trans.Fundamentals。 2950-2952 (2001)。

Kiyotaka Yamamura and Yoshii Hata: "Finding all solutions of weakly nonlinear equations using linear programming"IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences. Vol. E83-A, no. 12. 2758-2761 (2000)

Kiyotaka Yamamura 和 Yoshii Hata：“使用线性规划查找弱非线性方程的所有解”IEICE Trans。

Kiyotaka Yamamura, Masayuki Shimada and Takuya Yuasa: "An algorithm for finding all solutions of piecewise-trapezoidal circuits described by set-valued mappings"IEICE Trans. on Fundamentals of Electronics, Communications and Computer Sciences. Vol. J84-A,

Kiyotaka Yamamura、Masayuki Shimada 和 Takuy​​a Yuasa：“一种用于查找由集值映射描述的分段梯形电路的所有解的算法”IEICE Trans。

Institute of Electrical Engineers of Japan (ed.), partly written by Kiyotaka Yamamura: "Electrical Engineering Handbook"Ohmsa, Tokyo. (2001)

日本电气工程师学会（主编），部分由 Kiyotaka Yamamura 撰写：“电气工程手册”Ohmsa，东京。