Numerical analysis in modern scientific computing现代科学计算中的数据分析
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ISBN：9780387954103
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简介
This introductory book directs the reader to a selection of useful elementary numerical algorithms on a reasonably sound theoretical basis, built up within the text. The primary aim is to develop algorithmic thinking -- emphasizing long living computational concepts over fast changing software issues. The guiding principle is to explain modern numerical analysis concepts applicable in complex scientific computing at much simpler model problems. For example, the two adaptive techniques in numerical quadrature elaborated here carry the germs for either extrapolation methods or multigrid methods in differential equations, which are not treated here. The presentation draws on geometrical intuition wherever appropriate, supported by a large number of illustrations. Numerous exercises are included for further practice and improved understanding. This text will appeal to undergraduate and graduate students as well as researchers in mathematics, computer science, science, and engineering. At the same time it is addressed to practical computational scientists who, via self-study, wish to become acquainted with modern concepts of numerical analysis and scientific computing on an elementary level. Sole prerequisite is undergraduate knowledge in Linear Algebra and Calculus.
目录
Preface
Outline
1 Linear Systems
1.1 Solution of Triangular Systems
1.2 Gaussian Elimination
1.3 Pivoting Strategies and Iterative Refinement
1.4 Cholesky Decomposition for Symmetric Positive Definite Matrices
Exercises
2 Error Analysis
2.1 Sources of Errors
2.2 Condition of Problems
2.2.1 Normwise Condition Analysis
2.2.2 Componentwise Condition Analysis
2.3 Stability of Algorithms
2.3.1 Stability Concepts
2.3.2 Forward Analysis
2.3.3 Backward Analysis
2.4 Application to Linear Systems
2.4.1 A Zoom into Solvability
2.4.2 Backward Analysis of Gaussian Elimination
2.4.3 Assessment of Approximate Solutions
Exercises
3 Linear Least-Squares Problems
3.1 Least-Squares Method of Gauss
3.1.1 Formulation of the Problem
3.1.2 Normal Equations
3.1.3 Condition
3.1.4 Solution of Normal Equations
3.2 Orthogonalization Methods
3.2.1 Givens Rotations
3.2.2 Householder Reflections
3.3 Generalized Inverses
Exercises
4 Nonlinear Systems and Least-Squares Problems
4.1 Fixed-Point Iterations
4.2 Newton Methods for Nonlinear Systems
4.3 Gauss-Newton Method for Nonlinear Least-Squares Problems
4.4 Nonlinear Systems Depending on Parameters
4.4.1 Solution Structure
4.4.2 Continuation Methods
Exercises
5 Linear Eigenvalue Problems
5.1 Condition of General Eigenvalue Problems
5.2 Power Method
5.3 QR-Algorithm for Symmetric Eigenvalue Problems
5.4 Singular Value Decomposition
5.5 Stochastic Eigenvalue Problems
Exercises
6 Three-Term Recurrence Relations
6.1 Theoretical Background
6.1.1 Orthogonality and Three-Term Recurrence Relations
6.1.2 Homogeneous and Inhomogeneous Recurrence Relations
6.2 Numerical Aspects
6.2.1 Condition Number
6.2.2 Idea of the Miller Algorithm
6.3 Adjoint Summation
……
7 Interpolation and Approximation
8 Large Symmetric Systems of Equations and Eigenvalue Problems
9 Definite Integrals
References
Software
Index
Outline
1 Linear Systems
1.1 Solution of Triangular Systems
1.2 Gaussian Elimination
1.3 Pivoting Strategies and Iterative Refinement
1.4 Cholesky Decomposition for Symmetric Positive Definite Matrices
Exercises
2 Error Analysis
2.1 Sources of Errors
2.2 Condition of Problems
2.2.1 Normwise Condition Analysis
2.2.2 Componentwise Condition Analysis
2.3 Stability of Algorithms
2.3.1 Stability Concepts
2.3.2 Forward Analysis
2.3.3 Backward Analysis
2.4 Application to Linear Systems
2.4.1 A Zoom into Solvability
2.4.2 Backward Analysis of Gaussian Elimination
2.4.3 Assessment of Approximate Solutions
Exercises
3 Linear Least-Squares Problems
3.1 Least-Squares Method of Gauss
3.1.1 Formulation of the Problem
3.1.2 Normal Equations
3.1.3 Condition
3.1.4 Solution of Normal Equations
3.2 Orthogonalization Methods
3.2.1 Givens Rotations
3.2.2 Householder Reflections
3.3 Generalized Inverses
Exercises
4 Nonlinear Systems and Least-Squares Problems
4.1 Fixed-Point Iterations
4.2 Newton Methods for Nonlinear Systems
4.3 Gauss-Newton Method for Nonlinear Least-Squares Problems
4.4 Nonlinear Systems Depending on Parameters
4.4.1 Solution Structure
4.4.2 Continuation Methods
Exercises
5 Linear Eigenvalue Problems
5.1 Condition of General Eigenvalue Problems
5.2 Power Method
5.3 QR-Algorithm for Symmetric Eigenvalue Problems
5.4 Singular Value Decomposition
5.5 Stochastic Eigenvalue Problems
Exercises
6 Three-Term Recurrence Relations
6.1 Theoretical Background
6.1.1 Orthogonality and Three-Term Recurrence Relations
6.1.2 Homogeneous and Inhomogeneous Recurrence Relations
6.2 Numerical Aspects
6.2.1 Condition Number
6.2.2 Idea of the Miller Algorithm
6.3 Adjoint Summation
……
7 Interpolation and Approximation
8 Large Symmetric Systems of Equations and Eigenvalue Problems
9 Definite Integrals
References
Software
Index
Numerical analysis in modern scientific computing现代科学计算中的数据分析
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