Math. 555-01, Fall, 1997 Assignments

1. Read Section 1.1.
2. Do problems 1.2 and 1.3 on page 53 of the text.

1. Read and review all of Chapter 1 through Section 1.5
2. Do problems 1.5 and 1.8 on pp. 54-55 of the text.

1. Read through Section 2.1 of the text.
2. Read the handout, and read 2.2.1, pp. 78-82 of Rigorous Global Search: Continuous Problems (available on reserve in Dupre Library or on the Ultra system at /home/rbk5287/reports/opt-book/book.dvi or /home/rbk5287/reports/opt-book/book.ps. (Please only reproduce the relevant sections, to respect Kluwer's copyright.)

1. Do problems 1.14, 1.16, and 1.17 on pp. 57-58 of the text.
2. Study Section 2.1, and, in particular, the material on pp. 80-88 dealing with pivoting, factorizations of symmetric matrices, and iterative refinement.

1. Hand in problems 2.1 and 2.2 on p. 121 of the text.
2. Study section 2.2 (norms and condition numbers)

1. Hand in problem 2.3, p. 121 and 2.9, p. 122 of the text.

1. Study sections 2.2, 2.3 and 2.4 of the text.
2. Hand in exercises 2.17

1. Study all of Section 3.1 carefully.

1. Hand in problems 2.22 and 2.23 from page 126.

1. Study Sections 3.2, 3.3, and 3.4 carefully.
2. Hand in problems 3.1, 3.3, and 3.5 (use a divided difference table as indicated in class) from pp. 197--198
3. Do problem 3.11 (you may use Netscape instead of FTP; start at http://www.netlib.org/)

1. Study Sections 4.1, 4.2, and 4.3 of the text.
2. Hand in problems 4.1, 4.5, 4.7, and 4.12 on pp. 262-265.

1. Problem 2.21, p. 125 of text.

1. Hand in problem 4.15, p. 265 of text.
2. Study sections 5.1 and 5.2 of the text.
3. Study as much of section 5.3 as possible.

1. Make sure sections 5.1, 5.2, quasi-Newton methods and continuation methods from 5.3, 5.4, and 5.5 are thoroughly studied.
2. Hand in problems 5.1 (you may use either MATLAB or Fortran), 5.2, 5.4.

1. Hand in 5.10.

1. The forward and backward modes of automatic differentiation, for both one and more than one variable.
2. Upward rounding, downward rounding, round-to-nearest, and (for interval arithmetic) outward rounding.
3. Numerical stability of expressions (e.g. avoiding subtraction of almost equal numbers).
4. The condition number of a computation.
5. Elementary interval arithmetic.
6. The mean value form for an interval enclosure for a function.
7. Why partial pivoting is required in Gaussian elimination.
8. Definition and elementary properties of vector norms.
9. Definition and elementary properties of derived matrix norms.
10. Equivalence of norms in n-space.
11. The condition number of a matrix.
12. The form of the Lagrange interpolating polynomial.
13. The form of the Newton interpolating polynomial.
14. The error formula for an interpolating polynomial, both as a divided difference and in terms of a derivative.
15. Hermite interpolation and more general interpolation as limiting cases of Newton interpolation.
16. Computing Hermite interpolants with a divided difference table.
17. Failure of interpolation with equi-spaced points for Runge's function.
18. The midpoint bisection algorithm and its convergence.
19. Line searches. (Be prepared to do a simple one-dimensional line search as described on pp. 257-259. You may want to look up golden section search in other texts, too.)
20. The multivariate Newton method and its convergence (or lack thereof).
21. The Krawczyk operator for nonlinear systems of equations.
22. Descent methods with line searches:
• Steepest descent
• Descent using the Newton direction as search (pp. 304-306).
• The conjugate gradient method.