数值分析

1 Error Analysis

1.1 Introduction

1.2 Sources of Erro

1.3 Erro and Significant Digits

1.4 Error Propagation

1.5 Qualitative Anal

1 Error Analysis

1.1 Introduction

1.2 Sources of Erro

1.3 Erro and Significant Digits

1.4 Error Propagation

1.5 Qualitative Analysis and Control of Erro

1.5.1 Ill-condition Problem and Condition Number

1.5.2 The Stability of Algorithm

1.5.3 The Control of Erro

1.6 Computer Experiments

1.6.1 Functio Needed in The Experiments by Mathematica

1.6.2 Experiments by Mathematica

1.6.3 Functio Needed in The Experiments by Matlab

1.6.4 Experiments by Matlab

2 Interpolating

2.1 Introduction

2.2 Basic Concepts

2.3 Lagrange Interpolation

2.3.1 Linear and Parabolic Interpolation

2.3.2 Lagrange Interpolation Polynomial

2.3.3 Interpolation Remainder and Error Estimate

2.4 Divided-differences and Newton Interpolation

2.5 Differences and Newton Difference Formulae

2.5.1 Differences

2.5.2 Newton Difference Formulae

2.6 Hermite Interpolation

2.7 Piecewise Low Degree Interpolation

2.7.1 Ill-posed Properties of High Degree Interpolation

2.7.2 Piecewise Linear Interpolation

2.7.3 Piecewise Interpolation of Hermite with Degree Three

2.8 Cubic Spline Interpolation

2.8.1 Definition of Cubic Spline

2.8.2 The Cotruction of Cubic Spline

2.9 Computer Experiments

2.9.1 Functio Needed in The Experiments by Mathematica

2.9.2 Experiments by Mathematica

3 Best Approximation

3.1 Introduction

3.2 Norms

3.2.1 Vector Norms

3.2.2 Matrix Norms

3.3 Spectral Radius

3.4 Best Linear Approximation

3.4.1 Basic Concepts and Theories

3.4.2 Best Linear Approximation

3.5 Discrete Least Squares Approximation

3.6 Least Squares Approximation and Orthogonal Polynomials

3.7 Computer Experiments

3.7.1 Functio Needed in The Experiments by Mathematica

3.7.2 Experiments by Mathematica

3.7.3 Functio Needed in The Experiments by Matlab

3.7.4 Experiments by Matlab

4 Numerical Integration and Differentiation

4.1 Introduction

4.2 Interpolatory Quadratures

4.2.1 Interpolatory Quadratures

4.2.2 Degree of Accuracy

4.3 Newton-Cotes Quadrature Formula

4.4 Composite Quadrature Formula

4.4.1 Composite Trapezoidal Rule

4.4.2 Composite Simpson's Rule

4.5 Romberg Integration

4.5.1 Recuive Trapezoidal Rule

4.5.2 Romberg Algorithm

4.5.3 Richardson's Extrapolation

4.6 Gaussian Quadrature Formula

4.7 Numerical Differentiation

4.7.1 Numerical Differentiation

4.7.2 Differentiation Polynomial Interpolation

4.7.3 Richardson's Extrapolation

4.8 Computer Experiments

4.8.1 Functio Needed in The Experiments by Mathematica

4.8.2 Experiments by Mathematica

4.8.3 Experiments by Matlab

5 Solution of Nonlinear Equatio

5.1 Introduction

5.2 Basic Theories

5.3 Bisection Method

5.4 Iterative Method and Its Convergence

5.4.1 Fixed Point and Iteration

5.4.2 Global Convergence

5.4.3 Local Convergence

5.4.4 Order of Convergence

5.5 Accelerating Convergence

5.6 Newton's Method

5.6.1 Newton's Method and Its Convergence

5.6.2 Reduced Newton Method and Newton's Descent Method

5.6.3 The Case of Multiple Roots

5.7 Secant Method and Muller Method

5.7.1 Secant Method

5.7.2 Muller Method

5.8 Systems of Nonlinear Equatio

5.9 Computer Experiments

5.9.1 Functio Needed in The Experiments by Mathematica

5.9.2 Experiments by Mathematica

5.9.3 Experiments by Matlab

6 Direct Methods for Solving Linear Systems

6.1 Introduction

6.2 Gaussian Elimination

6.2.1 Basic Gaussian Elimination

6.2.2 Triangular Decomposition

6.3 Gaussian Elimination with Column Pivoting

6.4 Methods of The Triangular Decomposition

6.4.1 The Direct Methods of The Triangular Decomposition

6.4.2 The Square Root Method

6.4.3 The Speedup Method

6.5 Analysis of Round-off Erro

6.5.1 Condition Number

6.5.2 Iterative Refinement

6.6 Computer Experiments

6.6.1 Functio Needed in The Experiments by Mathematica

6.6.2 Experiments by Mathematica

6.6.3 Functio Needed in The Experiments by Matlab

6.6.4 Experiments by Matlab

7 Iterative Techniques for Solving Linear Systems

7.1 Introduction

7.2 Basic Iterative Methods

7.2.1 Jacobi Method

7.2.2 Gauss-Seidel Method

7.2.3 SOR Method .

7.3 Iterative Method Convergence

7.3.1 Basic Theorems

7.3.2 Some Special Systems of Equatio

7.4 Computer Experiments

7.4.1 Functio Needed in The Experiments by Mathematica

7.4.2 Experiments by Mathematica

8 Numerical Solution of Ordinary Differential Equatio

8.1 Introduction

8.2 The Existence and Uniqueness of Solutio

8.3 Taylor-Series Method

8.4 Euler's Method

8.5 Single-step Methods

8.5.1 Single-step Methods

8.5.2 Local Truncation Error

8.6 Runge-Kutta Methods

8.6.1 Second-Order Runge-Kutta Method

8.6.2 Fourth-Order Runge-Kutta Method

8.7 Multistep Methods

8.7.1 General Formulas of Multistep Methods

8.7.2 Adams Explicit and Implicit Formulas

8.8 Systems and Higher-Order Differential Equatio

8.8.1 Vector Notation

8.8.2 Taylor-Series Method for Systems

8.8.3 Fourth-Order Runge-Kutta Formula for Systems

8.9 Computer Experiments

8.9.1 Functio Needed in The Experiments by Mathematica

8.9.2 Experiments by Mathematica

附录

附录A Mathematica基本操作

附录B Matlab基本操作

附录C Awe to Selected Problems

参考文献

《数值分析》内容包括：数值分析的基本概念、非线性方程的数值解法、线性方程组的直接解法、线性方程组的迭代解法、数值积分与数值微分、常微分方程的数值解法及相应的上机实验内容。

本书是在作者多年来开设“数值分析”双语教学讲义的基础上，吸收国内外同类教材的精华，采用中英文双语编写而成。主要介绍了计算机中常用的、有效的各类数值问题的计算方法及相关数学理论。全书共分八章，包括误差分析、插值、函数逼近、数值积分和数值微分、非线性方程的数值解法、线性方程组的直接解法、线性方程组的迭代解法、常微分方程的数值解法等内容。每章由开篇、理论、应用和习题四个部分构成。