### 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 Norms and condition numbers. 11. 2.2 More on norms and condition numbers. 12. 2.3 Rounding error analysis for Gaussian elimination. 13. 2.3 Systems of interval equations. 14. 2.4 Sparse systems of equations. 15. 3.1 Polynomial interpolation. 16. 3.1 More on polynomial interpolation. 17. 3.2 Numerical differentiation. 18. 3.3 Cubic splines. 19. 3.5 Rational interpolation. 20. 3.6 Pade' approximations. 21. 3.7 Acceleration of convergence. 22. 3.8 Shape preserving rational interpolation. 23. 4.1 Midpoint, bisection, and secant method. 24. 4.2 Newton's method; convergence order. 25. 4.3 Error analysis for Newton's method. 26. 4.5 Complex zeros: rigorous error bounds. 27. 4.6 Minimization of univariate functions (line searches). 28. 5.1 Systems of nonlinear equations: preliminaries. 29. 5.2 Theory of Newton's method. 30. 5.3 Affine invariant Newton's method. 31. 5.4 Error analysis for the multivariate Newton's method. 32. 5.4 Interval Newton method; supplement with section 1.5 of Rigorous Global Search: Continuous Problems. 33. 5.5 Multivariate optimization. 34. 5.5 Global optimization; possibly supplement with Chapter 5 of Rigorous Global Search: Continuous Problems. 35. 5.6 Overdetermined nonlinear systems. 36. 5.7 The conjugate gradient method. 37. 5.7 More on the conjugate gradient method. 38. Ch. 6 Time permitting, topics from data analysis will be covered. Otherwise, the remainder of the text will be covered in the second semester. (There are 45 meeting periods in the fall semester.)
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