Towards a Lanczos’ \(\tau \) -Method Toolkit for Differential Problems
详细信息 查看全文
- 作者：M. Trindade ; J. Matos ; P. B. Vasconcelos
- 关键词：Spectral methods ; Preconditioners ; Boundary value problems ; Software
- 刊名：Mathematics in Computer Science
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：10
- 期：3
- 页码：313-329
- 全文大小：1,403 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics
Computer Science, general - 出版者：Springer Basel
- ISSN：1661-8289
- 卷排序：10
文摘
The aim of this work is to build a numerical software library based on the \(\tau \)-method to solve differential problems using MATLAB. The \(\tau \)-method can be very effective in the solution of certain type of these problems, and therefore, the existence of a numerical library for its dissemination is of major importance. Furthermore, the method has been used for the solution of particular problems but has not yet been explored as a general technique. Focus will be on stability issues, namely those issued from the solution of algebraic linear systems required for the process. Additionally, preconditioners for the solution with the \(\tau \)-method will be tackled, with emphasizes on incomplete LU factorizations and (block) Jacobi preconditioners. We also propose an iterative approach, build upon an LU factorization over a moderate initial size, generating better approximations and providing a priori error estimate at each iteration. Numerical results enlightening the efficiency of the proposed methods will be presented.