课程内容
- 第一讲：线性代数基础
( slides )
矩阵基础知识, 数值域, Chebyshev 多项式
参考资料：
矩阵论 (戴华, 科学出版社, 2001)
矩阵分析与应用, 第二版 (张贤达, 清华大学出版社, 2013)
Matrix Analysis, 2nd Edition (Horn & Johnson, 2013)
Topics in Matrix Analysis (Horn & Johnson, 1991)
课外阅读：
The Linear Algebra a Beginning Graduate Student Ought to Know,
3rd (Golan, 2012, 有许多历史起源介绍)
- 第二讲：非负矩阵与 M 矩阵
( slides )
非负矩阵, 不可约非负矩阵, M-矩阵和单调矩阵, 对角占优 M-矩阵
参考资料：
Nonnegative Matrices in the Mathematical Sciences (Berman & Plemmons, 1994)
Matrix Iterative Analysis, 2nd Edition (R.S. Varga, 2000)
Matrix Analysis, 2nd Edition (Horn & Johnson, 2013)
- 第三讲：定常迭代方法
( slides)
定常迭代法, 收敛性分析, 正则分裂, 交替方向迭代法, 加速技巧
参考资料：
数值线性代数 (曹志浩, 复旦大学出版社, 1996)
矩阵计算的理论与方法, (徐树方, 北京大学出版社, 1995)
线性代数方程组的迭代解法, (胡家赣, 科学出版社, 1991)
Matrix Computations, 4th Edition (Golub & van Loan, 2013)
Iterative Solution Methods, (O. Axelsson, 1994)
Matrix Iterative Analysis, 2nd Edition (R.S. Varga, 2000)
Iterative Solution of Large Linear Systems, (D.M. Young, 1971)
- 第四讲：Krylov 子空间方法
( slides)
投影法, Krylov 子空间与 Arnoldi 过程, 一般线性方程组的 Krylov 子空间方法,
对称线性方程组的 Krylov 子空间方法, 收敛性分析, 基于双正交的迭代方法,
免转置迭代方法, 正规方程迭代方法
参考资料：
Iterative Methods for Sparse Linear Systems, 2nd Edition (Y. Saad, 2003)
Iterative Krylov Methods for Large Linear Systems, (H.A. van der Vorst, 2003)
Iterative Methods for Solving Linear Systems, (A. Greenbaum, 1997)
- 第五讲：预处理方法
( slides )
参考资料：
Iterative Methods for Sparse Linear Systems, 2nd Edition (Y. Saad, 2003)
Matrix Preconditioning Techniques and Applications (K. Chen, 2005)
Preconditioning Techniques for Large Linear Systems - A Survey,
M. Benzi, J. Computational Physics, 2002.
Preconditioning, A.J. Wathen, Acta Numerica, 2015.
Preconditioners for Krylov subspace methods: An overview,
J.W. Pearson and J. Pestana, 2020.
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课外阅读：
- V. Simoncini1 and D. B. Szyld,
Recent computational developments in Krylov subspace methods for linear systems,
Numerical Linear Algebra With Applications, 14 (2007), 1--59.
- M. Benzi, G. H. Golub and J. Liesen,
Numerical solution of saddle point problems,
Acta Numerica, (2005), 1--137.
- V. Simoncini and D. B. Szyld
On the Occurrence of Superlinear Convergence of Exact and Inexact Krylov Subspace Methods,
SIAM Review, 47(2005), 247–-272.
- Jorg Liesen and Petr Tichy,
Convergence analysis of Krylov subspace methods,
GAMM-Mitteilungen, 27 (2004), 153–173.
- Yousef Saad and Henk A. van der Vorst,
Iterative solution of linear systems in the 20th century,
Journal of Computational and Applied Mathematics, 123 (2000) 1--33.
- A. Hadjidimos,
Successive overrelaxation (SOR) and related methods,
Journal of Computational and Applied Mathematics, 123 (2000) 177--199.
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