# Lecture: Confinued Fractions and the Euclidean Algorithm

This lecture reviews continued fractions and relates them to the Euclidean algorithm. This material provides the background to understanding the paper by Wang et al. on the reconstruction of rational numbers from their modular image, Lehmer's improvement to the Euclidean algorithm for multiprecision inputs, and provides a fast algorithm for computing Fibonacci numbers.
arithmetic.
### Background

- Lecture on the Euclidean algorithm.

### Reading

- Paul S. Wang, M. J. T. Guy, and J. H. Davenport,
P-adic Reconstruction of Rational Numbers, ACM SIGSAM Bulletin, Volume 16 , Issue 2, 1982, pp. 2-3.
- George E. Collins, The Computing Time of the Euclidean Algorithm, SIAM Journal of Computing, Vol. 3, No. 1, 1974, pp. 1-10.
- Handout (Integer Division and Lehmer's algorithm) from Donald E. Knuth, The Art of Computer Programming, Vol. 2, Seminumerical Algorithms.

### Topics

- Introduction to continued fractions
- Convergents and the extended Euclidean algorithm
- Fast algorithm to compute Fibonacci numbers
- Bit complexity
- Dominance and co-dominance and their basic properties
- Properties of length function
- Classical integer arithmetic
- Analysis of modular algorithm for integer multiplication
- Maximum computing time of the Euclidean algorithm

- The probability that two random integers are relatively prime.
- Average computing time of the Euclidean algorithm

### Maple worksheets and other resources

- euclid.mw - Maple worksheet on the Euclidean algorithm.
- cf.pdf - Lecture notes on continued fractions.
- fastfib.pdf - Lecture notes on fast algorithm for computing Fibonacci numbers.
- (cfeuc.pptx,cfeuc.pdf) - Lecture slides on continued fractions and the Euclidean algorithm.

### Assignments

Created: Oct. 13, 2010 by jjohnson AT cs DOT drexel DOT edu