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数值分析

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通过学习该课程，学生将了解各种数值方法的构建过程，掌握其设计原理、能够编程实现并判断其效率和准确性，能够应用于个人研究领域中的相关科学与工程计算问题。

Syllabus

Course Introduction

0-1 Introductory example
0-2 Round-off Errors and Computer Arithmetics
0-3 Applications and Course Organization

Chapter 01 Approximation of Data

1-1 Introduction
1-2 Discrete Least Squares Approximation
1-3 Model Selection and Linearization

Chapter 02 Interpolation of Data

2-1 Introduction
2-2 Newton’s Divided Differences
2-3 Lagrange Polynomials
2-4 Piecewise Polynomials and Cubic Spline Interpolation
2-5 Osculating Polynomials and Hermite Interpolations

Chapter 03 Numerical Solution of Nonlinear Equations

3-1 Introduction and Bisection Method
3-2 Newton Raphson Method
3-3 Fixed Point Iteration
3-4 Muller’s Method

Chapter 04 Numerical Solution of Nonlinear Equation Systems

4-1 Introduction and Newton’s Method
4-2 Quasi Newton’s Method
4-3 Steepest Descent Technique
4-4 A Comprehensive Project

Chapter 05 Numerical Integration

5-1 Introduction
5-2 Newton-Cotes Formulas
5-3 Composite Quadratures
5-4 Gaussian Quadratures
5-5 Improper Integrals

Chapter 06 Numerical Methods for Initial Value Problems

6-1 Introduction
6-2 Euler’s Method
6-3 Taylor’s Method of Higher Order
6-4 Runge-Kutta Methods
6-5 Multistep Methods
6-6 Stiff Equations

Chapter 07 Numerical Differentiation

7-1 Numerical Differentiation Formulas
7-2 Application to Boundary Value Problems
7-3 Stability and Optimal Step Sizes
7-4 Application to Partial Differential Equations
7-5 Comments on Non-Uniform Meshes

Chapter 08 Direct Methods for Linear Equation Systems

8-1 Gaussian Elimination
8-2 Operations Count
8-3 LU Factorization
8-4 PLU Factorization
8-5 Factorization of Special Matrices

Chapter 09 Iterative Methods for Linear Equation Systems

9-1 Iterative Techniques
9-2 Convergence and SOR Methods
9-3 Error Bound and Iterative Refinement

Chapter 10 Approximating Eigenvalues

10-1 Introduction
10-2 Power Method
10-3 Inverse Power Method

Final Exam

Taught by

China University of Petroleum,Beijing

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