http://interval.louisiana.edu/courses/556/556-spring-2001-outline.html

### Math. 556 Course Outline

This outline is a tentative guide. Exercises will be assigned as the topics are covered. Section numbers are from the texts, as indicated in the outline. This outline is subject to change.
 Day Section Description 1. 5.1-5.2 and 5.7 of Neumaier A review of univariate methods for nonlinear equations, including the secant method, bisection method, and Newton's method 2. 5.4 of Neumaier Continuation of the univariate review and interpretation of convergence order 3. 5.5 of Neumaier Error analysis and interval Newton methods 4. 6.1 of Neumaier Multivariate derivatives, Jacobi matrices, automatic differentiation and reverse automatic differentiation; contraction mappings and the Banach fixed point theorem. 5. Continuation of Multivariate Newton methods 6. Continuation of Multivariate Newton methods 7. Multivariate Newton methods, interval Newton methods 8. 9. 10. 11. 2.1 and 2.2 of the CSRP online text for optimization Basic structure of local methods methods for unconstrained optimization.  Note: we won't cover details in class, but I will point out main ideas and reasons for the developments. 12. 2.3 and 2.4 of the CSRP online text Nonderivative methodsand gradient methods 13. 2.4.3 to 2.4.5 of the CSRP online text Also, see G. H. Golub and van C. F. van Loan, Matrix Computations for a good explanation of the conjugate gradient method. Also see this web material on the Gram-Schmidt process. Follow this link for a more complete and advanced explanation. The conjugate gradient method. 14. 2.5 of the CSRP online text Various flavors of the newton method 15. 3 of the CSRP text and a supplement on Lagrange multipliers (from Gill /Murray/Wright, pp. 68-70 and pp. 77--81; and and §5.2.5, pp. 195-198 of Rigorous Global Search: Continuous Problems) Methods for continuous constrained problems 16. 17. Chapter 5 of the CSRP online text, as well as the talk An Overview of the GlobSol Package. Global Search Methods and Interval Newton methods 18. 19 20. 21. 22. The ODE book from the CSRP online text. Methods for ordinary differential equations. 23. 24. 25. Stiff problems, §2.3.2 of the online text 26. 27. 28. 29. 30. 31. We will be using primarily my own lectures for PDE. It will be computationally oriented, since other courses in the department deal with theory. Online references include the PDE books from the CSRP online text, both part 1 and part 2, as well as a series of lectures on Computational Physics by Angus MacKinnon.  Also see W. F. Ames, Numerical Methods for Partial Differential Equations, third edition, Academic Press, Boston, 1992, QA374.A46 1992 (in Dupré Library). Also see E. Issacson and H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966 (for finite difference methods). PDE: Elliptic 32. In-class notes.  For books on the subject, see Books on the Finite Element Method Finite element method 33. 35. W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, Boston, 1992 The method of characteristics for hyperbolic equations 36. 37. 38. D. Kincaid and E. W. Cheney, Numerical Analysis, Brooks / Cole, Pacific Grove, California, 1992, pp. 622--631 Multigrid methods 39. 40. D. Kincaid and E. W. Cheney, Numerical Analysis, Brooks / Cole, Pacific Grove, California, 1992, pp. 108--115 Homotopy and continuation methods 41. 42. 43. Class notes Review for the comprehensive
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