Math. 556 Course Outline

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. -- --
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