Classical Numerical Analysis
A Comprehensive Course

A thorough introduction to graduate classical numerical analysis, with all important topics covered rigorously.

Abner J. Salgado (Author), Steven M. Wise (Author)

9781108837705, Cambridge University Press

Hardback, published 20 October 2022

937 pages
26 x 18.2 x 5.2 cm, 1.81 kg

'This is an impressive and comprehensive textbook that covers virtually every topic that could be found in a one-year course in numerical analysis. Despite the word 'classical' in the title, modern methods and algorithms are included. The authors, experienced teachers and researchers in the area, have organised the vast material from their lecture notes … the book is highly recommendable, primarily to students of mathematics looking for rigorous founding of numerical analysis, but also to students of applied sciences in need of methods of scientific computing.' Martin Lukarevski, The Mathematical Gazette

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.

Part I. Numerical Linear Algebra: 1. Linear operators and matrices
2. The singular value decomposition
3. Systems of linear equations
4. Norms and matrix conditioning
5. Linear least squares problem
6. Linear iterative methods
7. Variational and Krylov subspace methods
8. Eigenvalue problems
Part II. Constructive Approximation Theory: 9. Polynomial interpolation
10. Minimax polynomial approximation
11. Polynomial least squares approximation
12. Fourier series
13. Trigonometric interpolation and the Fast Fourier Transform
14. Numerical quadrature
Part III. Nonlinear Equations and Optimization: 15. Solution of nonlinear equations
16. Convex optimization
Part IV. Initial Value Problems for Ordinary Di fferential Equations: 17. Initial value problems for ordinary diff erential equations
18. Single-step methods
19. Runge–Kutta methods
20. Linear multi-step methods
21. Sti ff systems of ordinary diff erential equations and linear stability
22. Galerkin methods for initial value problems
Part V. Boundary and Initial Boundary Value Problems: 23. Boundary and initial boundary value problems for partial di fferential equations
24. Finite diff erence methods for elliptic problems
25. Finite element methods for elliptic problems
26. Spectral and pseudo-spectral methods for periodic elliptic equations
27. Collocation methods for elliptic equations
28. Finite di fference methods for parabolic problems
29. Finite diff erence 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.

Subject Areas: Numerical analysis [PBKS]