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Numerical Analysis is a broad field, and coming to grips with all of it may seem like a daunting task. This text provides a thorough and comprehensive exposition of all the topics contained in a classical graduate sequence in numerical analysis. With an emphasis on theory and connections with linear algebra and analysis, the book shows all the rigor of numerical analysis. Its high level and exhaustive coverage will prepare students for research in the field and become a valuable reference as they continue their career. Students will appreciate the simple notation, clear assumptions and arguments, as well as the many examples and classroom-tested exercises ranging from simple verification to qualifying exam-level problems. In addition to the many examples with hand calculations, readers will also be able to translate theory into practical computational codes by running sample MATLAB codes as they try out new concepts.
Preface
Acknowledgments
Symbols
Numerical Linear Algebra
Linear Operators and Matrices
The Singular Value Decomposition
Systems of Linear Equations
Norms and Matrix Conditioning
Linear Least Squares Problem
Linear Iterative Methods
Variational and Krylov Subspace Methods
Eigenvalue Problems
Constructive Approximation Theory
Polynomial Interpolation
Minimax Polynomial Approximation
Polynomial Least Squares Approximation
Fourier Series
Trigonometric Interpolation and the Fast Fourier Transform
Numerical Quadrature
Nonlinear Equations and Optimization
Solution of Nonlinear Equations
Convex Optimization
Initial Value Problems for Ordinary Differential Equations
Initial Value Problems for Ordinary Di erential Equations
Single-Step Methods
Runge-Kutta Methods
Linear Multi-step Methods
Stiff Systems of Ordinary Differential Equations and Linear Stability
Galerkin Methods for Initial Value Problems
Boundary and Initial Boundary Value Problems
Boundary and Initial Boundary Value Problems for Partial Di erential Equations
Finite Difference Methods for Elliptic Problems
Finite Element Methods for Elliptic Problems
Spectral and Pseudo-Spectral Methods for Periodic Elliptic Equations
Collocation Methods for Elliptic Equations
Finite Difference Methods for Parabolic Problems
Finite Difference Methods for Hyperbolic Problems
Appendix A Linear Algebra Review
Appendix B Basic Analysis Review
Appendix C Banach Fixed Point Theorem
Appendix D A (Petting) Zoo of Function Spaces
References
Index