Numerical-Algebra

Codes for Numerical Algebra course (Programming Language: C++)

Textbook: 北京大学数学教学系列丛书 数值线性代数 第2版 (徐树方，高立，张平文编著)

Chapter 1 Direct Solutions to Systems of Linear Equations

1.1 Triangular equations and trigonometric decomposition

1.2 Elective triangular decomposition of the principal element

1.3 Square Root Method

1.4 Chunked triangulation

Chapter 2 Sensitivity Analysis of Systems of Linear Equations and Rounding Error Analysis of Elimination Methods

2.1 Vector norms and matrix norms

2.2 Sensitivity analysis of systems of linear equations

2.3 Rounding error analysis for basic operations

2.4 Round-off error analysis of column principal Gauss elimination method

2.5 Accuracy estimation and iterative improvement of computational solutions

Chapter 3 Solving the Least Squares Problem

3.1 Least Squares problems

3.2 Elementary orthogonal transformations

3.2.1 Householder transform

3.2.2 Givens Transform

3.3 Orthogonal transformation method

Chapter 4 Classical Iterative Solutions to Systems of Linear Equations

4.1 Single-step linear constant iterative method

4.1.1 Jacobi Iterative Method

4.1.2 Gauss-Seidel Iterative Method

4.1.3 Single-step linear constant iterative method

4.2 Convergence theory

4.2.1 Sufficient necessary conditions for convergence

4.2.2 Sufficient conditions for convergence and error estimation

4.2.3 Convergence of Jacobi Iterative Method and G-S Iterative Method

4.3 Convergence speed

4.3.1 Average and asymptotic convergence speeds

4.3.2 Model issues

4.3.3 Asymptotic convergence speed of Jacobi and G-S iteration

4.4 Super-relaxation iterative method

Chapter 5 Conjugate Gradient Method 138

5.1 Fastest descent method

5.2 The conjugate gradient method and its basic properties

5.2.1 Conjugate gradient method

5.2.2 Basic properties

5.3 Practical conjugate gradient method and its convergence

5.4 Preoptimal conjugate gradient method

5.5 Krylov Subspace Method

Chapter 6 Methods for Calculating Asymmetric Eigenvalue Problems

6.1 Basic Concepts and Properties

6.2 Power method

6.3 Inverse power

6.4 QR Methods

Chapter 7 Methods for Calculating Symmetric Eigenvalue Problems

7.1 Basic properties

7.2 Symmetric QR method

7.3 Jacobi Method

7.3.1 Classical Jacobi Methods

7.3.2 Cyclic Jacobi methods and their variants

7.3.3 Parallel scheme of Jacobi methods

7.4 Dichotomy

7.5 The Divide and Conquer Law

7.6 Calculation of singular value decomposition

7.6.1 Diagonalization

7.6.2 SVD iteration

7.6.3 SVD algorithm