This outline is a tentative guide. Exercises will be assigned as the topics are covered. Section numbers through the study of ordinary differential equations are from the text . Section numbers for the study of boundary value problems and of partial differential equations are from D. Kincaid and W. Cheney, Numerical Analysis, second edition, Brooks/Cole, 1996.
| Day | Section | Description |
| 1. | --- | Hand back and discuss final exam; discuss problem 5.10. |
| 2. | 6.6, 6.7 | (A review of regularization with the singular value decomposition,
done outside of class.)
Fourier series (An overview and proof of Theorem 6.37 in class; equation numbers will be filled in) |
| 3. | 6.8 | Trigonometric approximation and discrete Fourier series |
| 4. | 6.9 | The Fast Fourier Transform (FFT) |
| 5. | 3.4 ,
supplement |
Wavelets |
| 6. | 7.1 | Underlying theory of eigenvalues (Concentrate on the standard eigenvalue problem, the Schur normal form, and definite eigenvalue problems.) |
| 7. | 7.2 | The method of bisection for determining eigenvalues |
| 8. | 7.2 | Residue iteration and inverse iteration |
| 9. | 7.3 | The QR algorithm |
| 10. | 7.5 | Error analysis for eigenvalue computations. |
| 11. | 9.1 | Quadrature rules |
| 12. | 9.2, 9.3 | Trapezoidal rule; Simpson's rule; Romberg integration |
| 13. | 9.4 | Gaussian quadrature |
| 14. | 9.5 | Singular integrals; oscillating integrals |
| 15. | 10.0 | Introduction to ordinary differential equations ODE |
| 16. | 10.1 | Convergence concepts of methods for ODE initial value problems. |
| 17. | 10.2 | Convergence analysis of one-step methods. |
| 18. | 10.3 | Runge Kutta methods and other higher-order methods |
| 19. | 10.4 | Stiff differential equations |
| 20. | 10.5 | Differential-algebraic equations |
| 21. | 10.5 | More on differential-algebraic equations |
| 22. | 10.6 | Multistep methods |
| 23. | 10.7 | The Nordsieck Form |
| 24. | 10.8,10.9 | Convergence analysis (just examine the statements of the main theorems) for ODE methods, practical considerations |
| 25. | 10.9 | More on practical considerations for ODE methods |
| 26. | Kincaid /
Cheney: 9.3 |
Finite difference methods for elliptic partial differential equations (PDE) |
| 27. | Kincaid /
Cheney: 9.4 |
Galerkin and Ritz methods for elliptic PDE |
| 28. | Kincaid /
Cheney: 9.1 |
Explicit methods for parabolic PDE |
| 29. | Kincaid /
Cheney: 9.2 |
Implicit methods for parabolic PDE |
| 30. | Kincaid /
Cheney: 9.5 |
Characteristic curves |
| 31. | Kincaid /
Cheney: 9.6 |
Method of characteristics for hyperbolic PDE |
| 32. | Kincaid /
Cheney: 9.7 |
Miscellaneous methods for hyperbolic problems |
| 33. | Kincaid /
Cheney: 9.8 |
The Multigrid method |
| 34. | Kincaid /
Cheney: 9.8 |
More on the multigrid method |
| 35. | Kincaid /
Cheney: 9.9 |
Use of FFT for solving Poisson's equation |
| 36. | -- | -- |
| 37. | -- | -- |
| 38. | -- | -- |