Instructor: Anatolii Grinshpan

Office hours: TWR 4-6, Korman 247, or
by appointment.

Course
information Elementary Numerical
Analysis Academic
Calendar Math resource
center

Apr 4. Introduction to the course. Number systems:
decimal and binary arithmetic.

A & H: Overview, binary numbers.

Apr 6. Octal and hexadecimal systems. Babylonian and Mayan numerals.

Approximation of irrational numbers. Error, absolute and relative error. Significant digits.

Apr 11. Propagation of error in arithmetic. Examples. Error in function evaluation.

A & H: Propagation of error.

Apr 13. Quiz 1 (number systems and conversion). Mean-value theorem.

A & H: Polynomial evaluation. Nested multiplication and operation count.

Apr 18. Nested multiplication (continued). Modes of rounding. Example on rounding. Example on summation.

Fixed-point and floating-point representations.

Apr 20. Quiz 2 (significant digits, error arithmetic). IEEE single-precision format: subnormals and exponent bias. 1-byte floating-point model.

A & H: Error accumulation in sums. Notes on IEEE: Overton, Goldberg. Arithmetic simulator: addition/subtraction, multiplication/division.

Apr 25. Computer arithmetic. Examples. The relative error of floating-point representation.

Apr 27. Midterm I (lectures of April 4 – April 25).

May 2. Taylor polynomials. Lagrange remainder formula.

May 4. Taylor polynomials (continued). Irrationality of e.

A & H: Calculation of functions.

May 9. Computing derivatives and integrals via Taylor polynomials.

Error sequences: the order and rate of convergence. Linear convergence. Bisection.

A & H: Rootfinding.

May 11. Quiz 3 (Taylor approximation). Newton’s method. Examples and error analysis.

Division via Newton’s iteration. A & H: Rootfinding (continued).

May 16. Newton’s method (continued): error analysis, cycles, basins of attraction. Example from class.

Secant method: error analysis. When to stop an iteration.

A & H: The secant method. A quartic with a thousand roots.

May 18. Quiz 4 (Bisection and Newton’s method). Multiple root. Stability of roots.

Fixed-point iteration. A & H: Multiple roots. Perturbation analysis.

May 23. Contractive mapping. Examples. Classification of fixed points. Logistic map. An applet.

A & H: Fixed-point iteration. Chaotic search for i.

May 25. Midterm II (lectures of May 2 – May 23).

May 30. Bit shifting and the logistic map. Lagrange interpolation formula.

A & H: Interpolation.

Jun 1. Divided differences and polynomial interpolation. Interpolation error. The minimax problem.

A & H: An error formula. Homework 7, due June 8.

June 6. Quiz 5 (Lagrange interpolation, divided differences). Newton interpolation formula. Chebyshev’s theorem.

A & H: Best approximation, Chebyshev polynomials.

June
8. Applications to numerical
differentiation and integration.

Jun
13. Final Exam (all lectures): 10:30-12:30 Drexel Plaza GL 20.