### 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. -- -- 37. -- -- 38. -- --
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