Classical Numerical Analysis: A Comprehensive Course

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.

Table of Contents:
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.

About the Author :
Abner J. Salgado is Professor of Mathematics at the University of Tennessee, Knoxville. He obtained his PhD in Mathematics in 2010 from Texas A&M University. His main area of research is the numerical analysis of nonlinear partial differential equations, and related questions. Steven M. Wise is Professor of Mathematics at the University of Tennessee, Knoxville. He obtained his PhD in 2003 from the University of Virginia. His main area of research interest is the numerical analysis of partial differential equations that describe physical phenomena, and the efficient solution of the ensuing nonlinear systems. He has authored over 80 publications.

Review :
'This impressive volume covers an unusually broad range of topics in the field of numerical analysis, including numerical linear algebra, polynomial and trigonometric interpolation, best approximation, numerical quadrature, the approximate solution of nonlinear equations and convex optimization, and the numerical solution of ordinary and partial differential equations by finite difference, spectral and finite element methods. A particularly appealing feature of the text is the way in which it integrates a mathematically rigorous exposition with a wealth of illustrative examples, including numerical simulations, sample codes, and exercises. I warmly recommend the book to students and lecturers as an advanced undergraduate or introductory graduate level text.' Endre Süli, University of Oxford 'This long-awaited graduate text covers every topic of a traditional, one-year Numerical Analysis course in an accessible, rigorous, and comprehensive way. It's an indispensable assistant for anyone offering the class and a precious source of knowledge for a junior researcher in the field.' Maxim Olshanskii, University of Houston 'This is the book I have been waiting for: a textbook of numerical analysis fit for the Twenty-First Century. It sketches a path from the mathematical foundations of the subject to the wide range of its modern methods and algorithms, compromising on neither rigour nor clarity.' Arieh Iserles, University of Cambridge 'The book is the only textbook I know that covers the current topics for beginning graduate students in numerical analysis. The chosen topics in the book match exactly what one wishes to cover in a two-semester course sequence in computational mathematics, as the selection of the numerical methods is in align with the modern treatment of the subjects. Many instructors in the field have struggled to find two or more textbooks for the same coverage, but you can have all of them in this book.' Xiaofan Li, Illinois Institute of Technology