Classical Numerical Analysis

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.

• 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.

'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