Project/Area Number  13650444 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
System engineering

Research Institution  Tokyo Institute of Technology 
Principal Investigator 
KOJIMA Masakazu Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Professor, 大学院・情報理工学研究科, 教授 (90092551)

CoInvestigator(Kenkyūbuntansha) 
FUJISAWA Katsuki Tokyo Denki University, Faculty of Science and Engineering, Associate Professor, 理工学部, 助教授 (40303854)
MATSUOKA Satoshi Tokyo Institute of Technology, Global Scientific Information and Computing Center, Professor, 学術国際情報センター, 教授 (20221583)

Project Period (FY) 
2001 – 2002

Project Status 
Completed(Fiscal Year 2002)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 2002 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 2001 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  System of Polynomial Equations / Polyhedral Homotopy Method / PredictorCorrector Method / Parallel Computation / Ninf 
Research Abstract 
The purpose of this research project is to develop practical numerical methods for all real and complex solutions of large scale systems of polynomial equations. The polyhedral homotopy continuation method used in this research consists of the following three phases : Phase 1 : Construction of polyhedral homotopy systems. Phase 2 : Numerical tracing of homotopy paths by the predictorcorrector method. Phase 3 : Verification of solutions. In 2001, we designed and developed basic algorithms for each phase above. In 2002, we studied the following issues. 1. Improvement of computational efficiency in each phase. In phase 1, we proposed an efficient construction of homotopy systems arising from symmetric systems of polynomial equations. We incorporated a linear algebra library LAPACK into phase 2, and developed a new method for verifying and classifying solutions of the cyclic polynomial. 2. Improvement of numerical stability in each phase. Linear systems to be solved in phase 2 become often so illconditioned that computation of their accurate solutions are difficult. We devised new dynamic scaling techniques to resolve this difficulty. We confirmed through numerical experiments that the use of these scaling techniques together with the singular value decomposition worked very effectively to improve the numerical stability of phase 2. 3. We combined the three phases into a software package PHoM, and released it through Internet. This software solved some large scale systems of polynomial equations that had not been solved before. In conclusion, this research project has accomplished its purpose mentioned above. 4. We have started a parallel implementation of PHoM, which will continue in the next year.
