Math 300 – Numerical Analysis, Fall 2008.

MW 4-5:50, Randell 238B.

Instructor: Anatolii Grinshpan
Office hours: Tue, Thu 4-5 (Korman 247) or by appointment.

Course information. Math resource center. Academic Calendar.

Elementary Numerical Analysis (K. Atkinson and W. Han).

Numerical Computing with MATLAB (C. Moler).

Outline

Sept 22. Introduction to the course. Taylor polynomials.

Sept 24. Taylor polynomials (continued). Examples. Polynomial evaluation.

Operations count. A&H notes: 1.1, 1.2, 1.3. Homework 1, due October 1. Week 1 exercises.

Sept 29. Nested multiplication and long division. Approximation of irrational numbers.

Oct 1. Quiz 1 (Taylor polynomials). Decimal and binary arithmetic. Maya numerals.

          Week 1 answers.  Homework 2, due October 8. Week 2 exercises.

Oct 6. Babylonian numerals. Octal and hexadecimal systems. Error, absolute and relative error. Significant digits.

           A&H notes: 2.2

Oct 8. Quiz 2 (binary/denary arithmetic, polynomial evaluation). Error classification.

Modes of rounding. Loss of significance error. Propagation of error in function evaluation.
           A&H notes: 2.3. Week 2 answers. Homework 3, due October 15. Week 3 exercises.

Oct 13. No class: Columbus day.

Oct 15. Propagation of error in arithmetic operations. Noise in evaluation. Underflow/overflow.

Fixed-point and floating-point representations of numbers.

A&H notes: 2.1. Week 3 answers. Homework 4, due October 22. Week 4 exercises.

Oct 20. Subnormal numbers and biased exponents. Single (32-bit), double (64-bit), and extended (80-bit) precision models.

“1-byte” floating-point model. Rounding estimates. A&H notes: 2.4 (summation example).

Example: summing in 1-byte format. Additional material: Goldberg’s paper on IEEE.

Oct 22. Quiz 3 (error arithmetic, floating-point format). Discussion. Practice test. Week 4 answers. Homework 5: none.

Oct 27. Midterm 1 (lectures of Sept 22 – Oct 22, homework, exercises, quizzes). Answers. Extra credit: Kahan’s algorithm.

Oct 29. Iterations and rootfinding. Rate and order of convergence. Bisection and Newton’s method.

A&H notes: 3.1, 3.2. Homework 6, due November 5. Week 6 exercises.

Additional reading: introduction to MATLAB (1.1, 1.2), zeros and roots.

Nov 3. Midterm 1 discussion. Examples on Newton’s method. Error analysis for Newton’s method.

Nov 5. Division via Newton’s iterations. An example on Newton’s method. The secant method, error analysis.

Handout: a chaotic search for i (hardcopy). A&H notes: 3.3. Week 6 answers.

Nov 10. Quiz 4 (rootfinding). Fixed point iterations.

A&H notes: 3.4. Homework 7, due November 17 .

Nov 12. Behavior of fixed point iterates. Types of fixed points. Contractive mapping theorem: proof.

Examples. Week 7 exercises.

Nov 17. Quiz 5 (fixed point iterations). Aitken’s extrapolation formula. Logistic map. An example.

Week 7 answers. Week 8 exercises. Homework 8: none.

Nov 19. Logistic map (continued). Stability of roots. Wilkinson’s example.

Something to play with. A&H notes: 3.5 . Practice test. Answers.

Nov 24. Midterm 2 (lectures of Oct 29 – Nov 19, homework, exercises, quizzes). Answers. Problem 3B. Problem 4B.

Dec 1. Lagrange and Newton interpolation.

A&H notes: 4.1, example.

Dec 3. The minimax problem. Chebyshev polynomials.

Animations for Chebyshev polynomials.

A&H notes: 4.2, 4.4, 4.5. Exercises on interpolation.

Dec 5. Office hours 2-3 P.M., Korman 255.

Dec 9. Office hours: 12-2 P.M., Korman 247. Practice exam. Answers.

Dec 10. Office hours: 1:30-2:15 P.M. Korman 255. Answers to exercises. Graphs.

Dec 11. Office hours: 1-2 P.M., Korman 255.

Dec 12. Final exam (all lectures, homework, quizzes, exercises): 8-10 A.M., Randell 231.

Answers. Problem 6.