A study on consistent preconditioners for iterative solutions of large-scale linear systems

2016 Fiscal Year Final Research Report

• Project/Area Number: 25390145
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Computational science
• Research Institution: Tokyo Denki University (2015-2016); The University of Tokyo (2013-2014)
• Principal Investigator: Itoh Shoji 東京電機大学, 理工学部, 研究員 (70333482)
• Co-Investigator(Renkei-kenkyūsha): SUGIHARA Masaaki 青山学院大学, 理工学部, 教授 (80154483)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 大規模行列計算 / クリロフ部分空間法 / 前処理系

Outline of Final Research Achievements

In this study, we analyzed preconditioned bi-Lanczos iterative algorithms, which assume the existence of a dual system. Various iterative methods are often used in conjunction with preconditioning that improve the properties of linear equations; such typical algorithms are preconditioned CGS (PCGS) or preconditioned BiCG (PBiCG). Especially, we discussed a variety of PCGS algorithms including improved PCGS, by comparing two typical PCGS, and we analyzed the structure of the solution vector for each Krylov subspace. Further, we analyzed the structures on the polynomials of the PBiCG algorithms that correspond to the above PCGS. The improved PCGS has been verified as a coordinative to the left-PCGS, and it has the advantages of both the conventional and the left-PCGS. And, we showed that the direction of the preconditioned system of the bi-Lanczos algorithms can be switched by the construction and setting of the initial residual vector of the dual system.

Free Research Field

数値解析学