课程内容
- 第零讲：课程介绍
（slides）
参考资料：
IEEE 浮点运算标准，
数值计算中的误差
课外阅读：
Numerical Analysis (Trefethen, 2008)
The Best of the 20th Century: Editors Name Top 10 Algorithms (SIAM News, 2000)
Is Numerical Analysis Boring (Sullivan, 2006)
科学计算：科技创新的第三种方法 (陈志明, 2012)
- 第一讲：线性代数基础
（slides）
参考资料：
矩阵论 (戴华, 科学出版社, 2001)
Matrix Analysis, 2nd Edition (Horn & Johnson, 2013)
Topics in Matrix Analysis (Horn & Johnson, 1991)
Linear Algebra and Its Applications, 5th Edition (Lay et al., 2016)
Introduction to Linear Algebra, 5th Edition (Strang, 2016)
The decompositional approach to matrix computation (Stewart, 2000)
Map of Matrix Decomposition I (Lu, 2022)
Map of Matrix Decomposition II (Lu, 2022)
- 第二讲：线性方程组的直接解法
参考资料：
Matrix factorizations and direct solution of linear systems (Beattie, Handbook of LA, 2014)
Gaussian Elimination (Higham, 2011)
课外阅读：
Mathematicians of Gaussian Elimination (Grcar, Notices of the AMS, 2011)
A survey of direct methods for sparse linear systems (Davis et al., 2016)
Direct Methods for Sparse Matrices, 2nd Edition (Duff et al., 2017)
Direct Methods for Sparse Linear Systems (Davis, 2006)
Numerical algorithms for high-performance computational science (Dongarra, Grigori & Higham, 2020)
Symmetric Indefinite Matrices: Linear System Solvers and Modified Inertia Problems (Cheng, 1998)
测试矩阵：
The Suitesparse Matrix Collection (Formerly The University Of Florida Sparse Matrix Collection)
- 第三讲：线性最小二乘问题
参考资料：
Numerical Methods for Least Squares Problems (Bjorck, 1996)
Singular Values and Singular Value Inequalities (Mathias, Handbook of LA, 2014)
课外阅读：
We Recommend a Singular Value Decomposition (AMS Feature Column, 2009)
Professor SVD (Moler, 2006)
视频：The Singular Value Decomposition Saves the Universe (by Moler, 2016)
应用举例：
信号去噪 LS_Denoise.zip,
图像压缩 SVD_Image.zip
- 第四讲：非对称矩阵的特征值问题
课外阅读：
Francis's Algorithm (Watkins, American Mathematical Monthly, 2011)
The QR Algorithm Revisited (Watkins, SIAM Review, 2008)，
Unsymmetric Matrix Eigenvalue Techniques (Watkins, Handbook of LA, 2014)，
Eigenvalue computation in the 20th century (Golub& van der Vorst, JCAM, 2000)
应用举例：
The $25,000,000,000 Eigenvector: The Linear Algebra behind Google (SIAM Review, 2006)
- 第五讲：对称矩阵的特征值问题
参考资料：
箭型 DC 算法
The Symmetric Eigenvalue Problem, Classic version (Parlett, SIAM, 1998)
Symmetric Matrix Eigenvalue Techniques (Slapnicar, Handbook of LA, 2014)
Computation of the SVD (Cline & Dhillon, Handbook of LA,2014)
课外阅读：
The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale
(SIAM Rev. 2018)
New Fast and Accurate Jacobi SVD Algorithm I (SIMAX, 2008)
New Fast and Accurate Jacobi SVD Algorithm II (SIMAX, 2008)
Faster SVD via Polar Decomposition (Higham, 2022)
- 第六讲：线性方程组定常迭代方法
参考资料：
Matrix Iterative Analysis (R.S. Varga, Springer, 1962 & 2000)
Iterative Solution of Large Linear Systems (D.M. Young, Academic Press, 1971)
课外阅读：
Iterative solution of linear systems in the 20th century (Saad & van der Vorst, JCAM, 2000)
- 第七讲：Krylov 子空间迭代算方法
参考资料：
Iterative Methods for Sparse Linear Systems (2nd, Y. Saad, SIAM, 2003)
Iterative Methods for Solving Linear Systems (A. Greenbaum, SIAM, 1997)
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (SIAM, 1994)
课外阅读：
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain, Shewchuk
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- 参考课程：
Matrix Computations/Numerical Linear Algebra, by James Demmel, U.C. Berkeley, 2020
- 参考课程：
Numerical Methods for Solving Large Scale Eigenvalue Problems, by P. Arbenz, ETH, 2018.
(Lecture Notes)
- 参考课程：
Ten Great Algorithms of Scientific Computing, by Golub & Diaconis, Stanford, 2001.
- 参考课程：
Computational Aspects of Matrix Theory, by Y. Saad, 2022.
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参考资料
- 曹志浩,
数值线性代数, 复旦大学出版社, 1996.
- 徐树方,
矩阵计算的理论与方法, 北京大学出版社, 1995.
- 徐树方,
矩阵计算六讲, 高等教育出版社, 2011.
- Golub and van Loan,
Matrix Computations, 4th, Johns Hopkins University Press, 2013.
- Demmel,
Applied Numerical Linear Algebra, SIAM, 1997.
- Trefethen and Bau, III,
Numerical Linear Algebra, SIAM, 1997.
- Bjorck,
Numerical Methods in Matrix Computations, Springer, 2015.
- David S. Watkins,
The Matrix Eigenvalue Problem_GR and Krylov subspace methods, SIAM, 2007.
- Corless and Fillion,
A Graduate Introduction to Numerical Methods
-- From the Viewpoint of Backward Error Analysis,
Springer, 2014.
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课外读物
- The History of Numerical Analysis and Scientific Computing, SIAM
- G. H. Golub,
History of Numerical Linear Algebra, a Personal View, 2007
- Lloyd N. Trefethen,
The Definition of Numerical Analysis, SIAM News, Nov 1992
- Lloyd N. Trefethen,
Predictions for scientific computing 50 years from now, Mathematics Today, 2000
- Lloyd N. Trefethen,
Who invented the great numerical algorithms?, 2005
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Last modified: February 2021
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