http://interval.louisiana.edu/courses/555/555-fall-2000-outline.html

### Math. 555 Course Outline

This outline is a tentative guide. Exercises will be assigned as the topics are covered. Section numbers are from the text. This outline is subject to change.

 Day Section Description 1. 1.1 Numbers, evaluation of expressions, and automatic differentiation. 2. 1.1 Explanation of software for automatic differentiation (excerpts from Rigorous Global Search: Continuous Problems, Chapter 2.) 3. 1.2 Floating point numbers, roundoff error. 4. 1.3 numerical stability. 5. 1.4 Error propagation and condition numbers. 6. 1.5 Interval arithmetic. 7. 1.5 Interval evaluation of expressions; software tools (excerpts from Rigorous Global Search: Continuous Problems, Chapter 2.) 8. 2.1 Gaussian elimination. 9. 2.1 More on Gaussian elimination. 10. 2.2 Structured systems of equations 11. 2.3 Rounding error analysis for Gaussian elimination; pivoting 12. 2.4 Vector and matrix norms 13. 2.4 More on vector and matrix norms 14. 2.5 Condition numbers 15. 2.7 Error bounds for linear systems 16. 3.1 Polynomial interpolation 17. 3.1 More on polynomial interpolation. 18. 3.2 Numerical differentiation. 19 3.3 Cubic splines. 20. 3.4 Approximation by splines 21. 3.5 Radial basis functions 22. 5.1 The secant method; linear, quadratic, and superlinear convergence 23. 5.2-5.3 Bisection methods and bisection methods for eigenvalue problems 24. 5.4 Convergence order 25. 5.5 Error analysis; interval Newton method 26. 5.7 Newton's method 27. Line searches:  supplement 28. 6.1 Systems of nonlinear equations: preliminaries. 29. 6.2 Theory of Newton's method. 30. 6.3 Error analysis for the multivariate Newton's method; interval Newton method; supplement with section 1.5 of Rigorous Global Search: Continuous Problems. 31. 6.4 Other methods 32. 4.1 Quadrature formula theory 33. 4.2 Gaussian quadrature 35. 4.3 The trapezoidal rule and extrapolation.  Supplement: The trapezoidal rule for periodic functions. 36. 4.4 Adaptive integration. 37. Ch. 6 Time permitting, additional topics will be covered.  The plan for next semester is to cover optimization, ordinary and partial differential equations.  (There are 41 meeting periods in the fall semester.)
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