《理論數值分析（第3版）》是世界圖書出版公司2013年出版的圖書，作者是阿特肯森。
基本介紹
- 書名：理論數值分析（第3版）
- 作者：阿特肯森
- 出版社：世界圖書出版公司
- 出版時間：2013年
內容簡介,作品目錄,
內容簡介
《理論數值分析（第3版）》旨在為讀者提供一個基於泛函分析並專注於數值分析的數學框架，讓讀者更好地學習數值分析和計算數學，及早進入科研項目。本教程包括了泛函分析、逼近理論、傅立葉分析和小波等諸多基礎專題，每個專題的表述既能了解該科目，又可以達到的深度，特別專題的參考文獻都列於每章末，供讀者深入學習和研究。由於現實問題的往往是多相關的，多變數多項式在研究和套用中扮演著重要的角色，第三版中就此專題新增了一章。
作品目錄
Preface
Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact sets
Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 (V,W) as a Banach space
2.3 The geometric series theorem and its variants
2.3.1 A generalization
2.3.2 A perturbation result
2.4 Some more results on linear operators
2.4.1 An extension theorem
2.4.2 Open mapping theorem
2.4.3 Principle of uniform boundedness
2.4.4 Convergence of numerical quadratures
2.5 Linear functionals
2.5.1 An extension theorem for linear functionals
2.5.2 The Riesz representation theorem
2.6 Adjoint operators
2.7 Weak convergence and weak pactness
2.8 Compact linear operators
2.8.1 Compact integral operators on C(D)
2.8.2 Properties of pact operators
2.8.3 Integral operators on L2(a,b)
2.8.4 The Fredholm alternative theorem
2.8.5 Additional results on Fredholm integral equations
2.9 The resolvent operator
2.9.1 R(A) as a holomorphic function
Approximation Theory
3.1 Approximation of continuous functions by polynomials
3.2 Interpolation theory
3.2.1 Lagrange polynomial interpolation
3.2.2 Hermite polynomial interpolation
3.2.3 Piecewise polynomial interpolation
3.2.4 Trigonometric interpolation
3.3 Best approximation
3.3.1 Convexity,lower semicontinuity
3.3.2 Some abstract existence results
3.3.3 Existence of best approximation
3.3.4 Uniqueness of best approximation
3.4 Best approximations in inner product spaces,projection on
closed convex sets
3.5 Orthogonal polynomials
3.6 Projection operators
3.7 Uniform error bounds
3.7.1 Uniform error bounds for L2-approximations
3.7.2 L2-approximations using polynomials
3.7.3 Interpolatory projections and their convergence
Fourier Analysis and Wavelets
4.1 Fourier series
4.2 Fourier transform
4.3 Discrete Fourier transform
4.4 Haar wavelets
4.5 Multiresolution analysis
Nonlinear Equations and Their Solution by Iteration
5.1 The Banach fixed-point theorem
5.2 Applications to iterative methods
5.2.1 Nonlinear algebraic equations
5.2.2 Linear algebraic systems
5.2.3 Linear and nonlinear integral equations
5.2.4 Ordinary differential equations in Banach spaces
5.3 Differential calculus for nonlinear operators
5.3.1 Frechet and Gateaux derivatives
5.3.2 Mean value theorems
5.3.3 Partial derivatives
5.3.4 The Gateaux derivative and convex minimization
5.4 Newton's method
5.4.1 Newton's method in Banach spaces
5.4.2 Applications
5.5 Completely continuous vector fields
5.5.1 The rotation of a pletely continuous vector field
5.6 Conjugate gradient method for operator equations
Finite Difference Method
6.1 Finite difference approximations
6.2 Lax equivalence theorem
6.3 More on convergence
Sobolev Spaces
7.1 Weak derivatives
7.2 Sobolev spaces
7.2.1 Sobolev spaces of integer order
7.2.2 Sobolev spaces of real order
7.2.3 Sobolev spaces over boundaries
7.3 Properties
7.3.1 Approximation by smooth functions
7.3.2 Extensions
7.3.3 Sobolev embedding theorems
7.3.4 Traces
7.3.5 Equivalent norms
7.3.6 A Sobolev quotient space
7.4 Characterization of Sobolev spaces via the Fouriertransform
7.5 Periodic Sobolev spaces
7.5.1 The dual space
7.5.2 Embedding results
7.5.3 Approximation results
7.5.4 An illustrative example of an operator
7.5.5 Spherical polynomials and spherical harmonics
7.6 Integration by parts formulas
8 Weak Formulations of Elliptic Boundary Value Problems
8.1 A model boundary value problem
8.2 Some general results on existence and uniqueness
8.3 The Lax-Milgram Lemma
8.4 Weak formulations of linear elliptic boundary valueproblems
8.4.1 Problems with homogeneous Dirichlet boundarycon-ditions
8.4.2 Problems with non-homogeneous Dirichlet boundaryconditions
8.4.3 Problems with Neumann boundary conditions
8.4.4 Problems with mixed boundary conditions
8.4.5 A general linear second-order elliptic boundary valueproblem
8.5 A boundary value problem of linearized elasticity
8.6 Mixed and dual formulations
8.7 Generalized Lax-Milgram Lemma
8.8 A nonlinear problem
9 The Galerkin Method and Its Variants
9.1 The Galerkin method
9.2 The Petrov-Galerkin method
9.3 Generalized Galerkin method
9.4 Conjugate gradient method： variational formulation
10 Finite Element Analysis
10.1 One-dimensional examples
10.1.1 Linear elements for a second-order problem
10.1.2 High order elements and the condensation technique
10.1.3 Reference element technique
10.2 Basics of the finite element method
10.2.1 Continuous linear elements
10.2.2 Affine-equivalent finite elements
10.2.3 Finite element spaces
10.3 Error estimates of finite element interpolations
10.3.1 Local interpolations
10.3.2 Interpolation error estimates on the reference element
10.3.3 Local interpolation error estimates
10.3.4 Global interpolation error estimates
10.4 Convergence and error estimates
……
11 Elliptic Variational Inequalities and Their NumericalAp-proximations
12 Numerical Solution of Fredholm Integral Equations of the SecondKind
13 Boundary Integral Equations
14 Multivariable Polynomial Approximations
References
Index
Linear Spaces
1.1 Linear spaces
1.2 Normed spaces
1.2.1 Convergence
1.2.2 Banach spaces
1.2.3 Completion of normed spaces
1.3 Inner product spaces
1.3.1 Hilbert spaces
1.3.2 Orthogonality
1.4 Spaces of continuously differentiable functions
1.4.1 HSlder spaces
1.5 Lp spaces
1.6 Compact sets
Linear Operators on Normed Spaces
2.1 Operators
2.2 Continuous linear operators
2.2.1 (V,W) as a Banach space
2.3 The geometric series theorem and its variants
2.3.1 A generalization
2.3.2 A perturbation result
2.4 Some more results on linear operators
2.4.1 An extension theorem
2.4.2 Open mapping theorem
2.4.3 Principle of uniform boundedness
2.4.4 Convergence of numerical quadratures
2.5 Linear functionals
2.5.1 An extension theorem for linear functionals
2.5.2 The Riesz representation theorem
2.6 Adjoint operators
2.7 Weak convergence and weak pactness
2.8 Compact linear operators
2.8.1 Compact integral operators on C(D)
2.8.2 Properties of pact operators
2.8.3 Integral operators on L2(a,b)
2.8.4 The Fredholm alternative theorem
2.8.5 Additional results on Fredholm integral equations
2.9 The resolvent operator
2.9.1 R(A) as a holomorphic function
Approximation Theory
3.1 Approximation of continuous functions by polynomials
3.2 Interpolation theory
3.2.1 Lagrange polynomial interpolation
3.2.2 Hermite polynomial interpolation
3.2.3 Piecewise polynomial interpolation
3.2.4 Trigonometric interpolation
3.3 Best approximation
3.3.1 Convexity,lower semicontinuity
3.3.2 Some abstract existence results
3.3.3 Existence of best approximation
3.3.4 Uniqueness of best approximation
3.4 Best approximations in inner product spaces,projection on
closed convex sets
3.5 Orthogonal polynomials
3.6 Projection operators
3.7 Uniform error bounds
3.7.1 Uniform error bounds for L2-approximations
3.7.2 L2-approximations using polynomials
3.7.3 Interpolatory projections and their convergence
Fourier Analysis and Wavelets
4.1 Fourier series
4.2 Fourier transform
4.3 Discrete Fourier transform
4.4 Haar wavelets
4.5 Multiresolution analysis
Nonlinear Equations and Their Solution by Iteration
5.1 The Banach fixed-point theorem
5.2 Applications to iterative methods
5.2.1 Nonlinear algebraic equations
5.2.2 Linear algebraic systems
5.2.3 Linear and nonlinear integral equations
5.2.4 Ordinary differential equations in Banach spaces
5.3 Differential calculus for nonlinear operators
5.3.1 Frechet and Gateaux derivatives
5.3.2 Mean value theorems
5.3.3 Partial derivatives
5.3.4 The Gateaux derivative and convex minimization
5.4 Newton's method
5.4.1 Newton's method in Banach spaces
5.4.2 Applications
5.5 Completely continuous vector fields
5.5.1 The rotation of a pletely continuous vector field
5.6 Conjugate gradient method for operator equations
Finite Difference Method
6.1 Finite difference approximations
6.2 Lax equivalence theorem
6.3 More on convergence
Sobolev Spaces
7.1 Weak derivatives
7.2 Sobolev spaces
7.2.1 Sobolev spaces of integer order
7.2.2 Sobolev spaces of real order
7.2.3 Sobolev spaces over boundaries
7.3 Properties
7.3.1 Approximation by smooth functions
7.3.2 Extensions
7.3.3 Sobolev embedding theorems
7.3.4 Traces
7.3.5 Equivalent norms
7.3.6 A Sobolev quotient space
7.4 Characterization of Sobolev spaces via the Fouriertransform
7.5 Periodic Sobolev spaces
7.5.1 The dual space
7.5.2 Embedding results
7.5.3 Approximation results
7.5.4 An illustrative example of an operator
7.5.5 Spherical polynomials and spherical harmonics
7.6 Integration by parts formulas
8 Weak Formulations of Elliptic Boundary Value Problems
8.1 A model boundary value problem
8.2 Some general results on existence and uniqueness
8.3 The Lax-Milgram Lemma
8.4 Weak formulations of linear elliptic boundary valueproblems
8.4.1 Problems with homogeneous Dirichlet boundarycon-ditions
8.4.2 Problems with non-homogeneous Dirichlet boundaryconditions
8.4.3 Problems with Neumann boundary conditions
8.4.4 Problems with mixed boundary conditions
8.4.5 A general linear second-order elliptic boundary valueproblem
8.5 A boundary value problem of linearized elasticity
8.6 Mixed and dual formulations
8.7 Generalized Lax-Milgram Lemma
8.8 A nonlinear problem
9 The Galerkin Method and Its Variants
9.1 The Galerkin method
9.2 The Petrov-Galerkin method
9.3 Generalized Galerkin method
9.4 Conjugate gradient method： variational formulation
10 Finite Element Analysis
10.1 One-dimensional examples
10.1.1 Linear elements for a second-order problem
10.1.2 High order elements and the condensation technique
10.1.3 Reference element technique
10.2 Basics of the finite element method
10.2.1 Continuous linear elements
10.2.2 Affine-equivalent finite elements
10.2.3 Finite element spaces
10.3 Error estimates of finite element interpolations
10.3.1 Local interpolations
10.3.2 Interpolation error estimates on the reference element
10.3.3 Local interpolation error estimates
10.3.4 Global interpolation error estimates
10.4 Convergence and error estimates
……
11 Elliptic Variational Inequalities and Their NumericalAp-proximations
12 Numerical Solution of Fredholm Integral Equations of the SecondKind
13 Boundary Integral Equations
14 Multivariable Polynomial Approximations
References
Index
