Lecture 1: Divide and Conquer

Background Material

• Recurrence Relations
• Geometric series
• Summation Notation and Manipulation
• Growth Rates and Asymptotic Analysis

Reading

• Chapters 1-6 of Cormen, Lieserson, and Rivest
• Chapter 9 of Lipson
• Cooley, J. M. and J. W. Tukey, "An algorithm for the machine calculation of complex Fourier series", Math. Comp., 19, 297-301, 1965.
• A. Karatsuba and Y. Ofman, "Multiplication of multidigit numbers on automata," Dokl. Akad. Nauk SSSR 145, 293-294, 1962.
• V. Strassen, "Gaussian elimination is not optimal", Numerische Mathematik, 13, 354-356, 1969.

Topics

• Overview of the course.
• Objectives
• Topics
• Grading policy and expectations
• Three Problems
  • Integer multiplication and Karatsuba's algorithm
  • Discrete Fourier transform (DFT) and the fast Fourier transform (FFT)
  • Matrix Multiplication and Strassen's algorithm

Lecture Slides

• Integer multiplication and Karatsuba's algorithm imult.ppt
• The FFT fft.ppt
• The 3 primes algorithm fast.ppt

Maple worksheets and programs

• karatsuba.mws - Implementation and analysis of Karatsuba's algorithm and generalizations
• fft.mws - Implementation of the FFT
• threeprimes.mws - Implementation of the 3 primes algorithm

Resources

• www.maplesoft.com (Maplesoft Web site)
• GNU Multiple Precision Arithmetic LibraryFast implementation of integer and rational arithmetic.

Assignments

• Assignment 1