No applied mathematician can be properly trained without some basic understanding of numerical methods, i.e., numerical analysis. And no scientist and engineer should be using a package program for numerical computations without understanding the program's purpose and its limitations. This book is an attempt to provide some of the required knowledge and understanding. It is written in a spirit that considers numerical analysis not merely as a tool fer solving applied problems but also as a challenging and rewarding part of mathematics. The main goal is to provide insight into numerical analysis rather than merely to provide numerical recipes.

此书为英文版。

1 Introduction

2 Linear Systems

2.1 Examples for Systems of Equations

2.2 Gaussian Elimination

2.3 LR Decomposition

2.4 QR Decomposition

Problems

3 Basic Functional Analysis

3.1 Normed Spaces

3.2 Scalar Products

3.3 Bounded Linear Operators

3.4 Matrix Norms

3.5 Completeness

3.6 The Banach Fixed Point Theorem

3.7 Best Approximation

Problems

4 Iterative Methods for Linear Systems

4.1 Jacobi and Gauss-Seidel Iterations

4.2 Relaxation Methods

4.3 Two-Grid Methods

Problems

5 Ill-Conditioned Linear Systems

5.1 Condition Number

5.2 Singular Value Decomposition

5.3 Tikhonov Regularization

Problems

6 Iterative Methods for Nonlinear Systems

6.1 Successive Approximations

6.2 Newton's Method

6.3 Zeros of Polynomials

6.4 Least Squares Problems

Problems

7 Matrix Eigenvahm Problems

7.1 Examples

7.2 Estimates for the Eigenvalues

7.3 The Jacobi Method

7.4 The QR Algorithm

7.5 Hessenberg Matrices

Problems

8 Interpolation

8.1 P(flynomial Interpolation

8.2 Trigonometric Interpolation

8.3 Spline Interpolation

8.4 Bdzier Polynomials

Problems

9 Numerical Integration

9.1 Interpolatory Quadratures

9.2 Convergence of Quadrature Formulae

9.3 Gaussian Quadrature Formulae

9.4 Quadrature of Periodic Functions

9.5 Romberg Integration

9.6 Improper Integrals

Problems

10 Initial Value Problems

10.1 The Picard LindelSfTheorem

10.2 Euler's Method . .

10.3 Single-Step Methods

10.4 Multistep Methods

Problems

11 Boundary Value Problems

11.1 Shooting Methods

11.2 Finite Difference Methods

11.3 The Riesz and Lax-Milgram Theorems

11.4 Weak Solutions

11.5 The Finite Element Method

Problems

12 Integral Equations

12.1 The Riesz Theory

12.2 Operator Approximations

12.3 Nystrom's Method

12.4 The Collocation Method

12.5 Stability

Problems

References

Index