Classical Numerical Analysis

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.

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• Análise numérica