简介
This volume is intended as an introduction into numerical analysis for students in mathematics, physics, and engineering. Instead of attempting to exhaustively cover all parts of numerical analysis, the goal is to guide the reader towards the basic ideas and general principles by way of considering main and important numerical methods. Given the rapid development of numerical algorithms, a reasonable introduction to numerical analysis has to confine itself to presenting a solid foundation by restricting the presentation to the basic principles and procedures. The book includes the necessary basic functional analytic tools for the solid mathematical foundation of numerical analysis. These are indispensable for any deeper study and understanding of numerical methods, in particular, for differential equations and integral equations. Particular emphasis will be given to the question of stability--especially to well-posedness and ill-posedness. The text is presented in a concise and easily understandable fashion and can be successfully mastered in a one-year course.
目录
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
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 Polynonfials
6.4 Least Squares Problems
Problems
7 Matrix Eigenvalue 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 Polynomial 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-LindelSf Theorem
10.2 Euler's Method
10.3 Single-Step Methods
10.4 Multistep Methods
Problems
……
11 Boundary Value Problems
12 Integral Equations
References
Index
2 Linear Systems
2.1 Examples for Systems of Equations
2.2 Gaussian Elimination
2.3 LR Decomposition
2.4 QR Decomposition
Problems
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 Polynonfials
6.4 Least Squares Problems
Problems
7 Matrix Eigenvalue 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 Polynomial 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-LindelSf Theorem
10.2 Euler's Method
10.3 Single-Step Methods
10.4 Multistep Methods
Problems
……
11 Boundary Value Problems
12 Integral Equations
References
Index
Numerical analysis数据分析
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