**Course No.****：**S070102ZJ004

**Course Category****：**Professional Basic Course

**Period/Credits****：**40/2

**Prerequisites****：**mathematical analysis or advanced mathematics, linear algebra, elementary functional analysis.

**Aims & Requirements****：**

This course is a speciality foundation course for Master degree candidates majoring in computational mathematics and applied mathematics. It can also be taken as an elective course for Master degree candidates majoring in Physics, Mechanics, Chemistry, and Engineering Sciences. This course mainly includes: 1. direct and iterative solutions for linear algebra equation systems; 2. numerical methods for least square problems；3. numerical methods for eigenvalue problems. Participants of the course are expected to have a command of the basic theories and methods of numerical linear algebra and have a fundamental understanding of matrix calculation. This course will enable its participants to employ the methods they learn to conduct practical scientific computing and lay a solid foundation for future research.

**Primary Coverage****：**

Chapter 1 Foundations of matrix algebra

Vector norms and matrix norms; Schur decomposition and singular value decomposition; subspace distance; Perron-Frobenius theorem; Birkhoff theorem; Bauer-Fike theorem；Hoffman-Wielandt theorem; max-min theorem for Hermitian matrix; ill-conditioned problems and numerical stability of algorithms; Householder transformation; Givens transformation; Gauss transformation.

Chapter 2 Direct solution of linear equation systems

Gauss elimination method; Cholesky decomposition; LDLT decomposition; Solution of special matrices.

Chapter 3 Iterative methods for linear equation systems

Basic iterative methods and their convergence; H matrix and convergence of iteration; Chebyshev semi-acceleration method; conjugate gradient method; incomplete Cholesky factorization; introduction to multi-mesh methods and domain decomposition methods.

Chapter 4 Numerical solution of least square problem

Normalization and orthogonalization methods; QR decomposition with column pivoting.

Chapter 5 QR algorithms for solving eigenvalue problems

Double-shift QR algorithm; QZ methods for generalized eigenvalues; iterative methods of powers; symmetric QR algorithms; Jacobi methods; tridiagonal methods; computation of SVD.

Chapter 6 Lanczos methods

Lanczos iteration and its basic properties; K-P-S theory; Lanczos algorithms; Arnoldi methods; GMRES methods.

**Textbook****：**

1. XU Shufang, Theories and Methods of Matrix Calculations, Peking University Press, Beijing, 1995.

2. Gene H. Golub, Charles F. Van Loan (translated by YUAN Yaxiang et al.), Matrix Computations, Science Press, Beijing, 2001.

**References****：**

[1] R.A. Horn and C.R. Johnson, Matrix Analysis, Vol.1-2, Posts and Telecom Press, 2005.

[2] Y. Saad, Iterative Methods for Sparse Linear Systems, Second Edition, Science Press, 2009.

**Author****：**Liqun Cao (Academy of Mathematics and Systems Science)

**Date****：**June, 2009