Lecture 4: A Fast Algorithm to Compute Fibonacci Numbers

Background Material

• Fibonacci Numbers
• Asymptotic Computing Time Analysis
• Matrix notation, eigenvalues, and eigenvectors
• Limits

Reading

• Lecture 3 notes on Fibonacci numbers.
• Maple Help - LinearAlgebra package

Topics

• Matrix interpretation of the Fibonacci recurrence.
• Alternative proof that the ratio of Fib_n/Fib_{n-1} approaches a limit.
• Binary powering algorithm (see power.mws)
• A fast algorithm to compute Fibonacci numbers
  Use binary powering to compute F^n, where F = Matrix([[1,1],[0,1]]).

Maple worksheets and documentation

• fib.mw - Maple worksheet investigating the Fibonacci numbers. Also contains a recursive function to generate all configurations of dominoes that fill a 2 X n rectangle.
• power.mw - Maple worksheet to compute powers. Includes binary powering algorithm.

Resources

• www.maplesoft.com (Maplesoft Web site)
• www.mapleapps.com (Maple Application Center Web site)
• www.maple4students.com (Maple Student Center Web site)