Introduction to Algorithms (Square Roots, Newton's Method)

Error Analysis of Newton's Method

Suppose 

Then, 

Therefore, 

Multiplication Algorithms

1. Naive Divide & Conquer method: Θ(d^2) time
2. Karatsuba: Θ(d^1.584...)
3. Toom-Cook generalizes Karatsuba (break into k ≥ 2 parts), Θ(d^1.465...)
4. Schönhage-Strassen - almost linear! Θ(d lgd lglgd) using FFT.

High-Precision Division

We want high precision rep of a/b

• Compute high-precision rep of 1/b first
• High-precision rep of 1/b means,  where R is large value s.t. it is easy to divide by R

Division

Newton's Method for computing R/b

 

Quadratic convergence; # digits doubles at each step