Continued Fractions, Euclidean Algorithm and Lehmer’s Algorithm Applied Symbolic Computation CS 567 Jeremy Johnson TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A

Outline Fast Fibonacci Continued Fractions Lehmer’s Algorithm Analysis of the Euclidean Algorithm (bit complexity) Assignment 1

Euclidean Algorithm g = gcd(a,b) a1 = a; a2 = b; while (a2  0) a3 = a1 mod a2; a1 = a2; a2 = a3; } return a1;

Remainder Sequence a 1 = a, a 2 = b a 1 = q 3  a 2 + a 3, 0  a 3 < a 2    a i = q i  a i+1 + a i+2, 0  a i+2 < a i+1    a l = q l  a l +1 gcd(a,b) = a l +1

Extended Euclidean Algorithm g = gcd(a,b,*x,*y) a1 = a; a2 = b; x1 = 1; x2 = 0; y1 = 0; y2 = 1; while (a2  0) a3 = a1 mod a2; q = floor(a1/a2); x3 = x1 – q*x2; y3 = y1 – q*y2; a1 = a2; a2 = a3; x1 = x2; x2 = x3; y1 = y2; y2 = y3; } return a1;

Lehmer’s Algorithm u = , v = u’ v’ q’ u’’ v’’ q’’ u’/v’ < u/v < u’’/v’’

Maximum Number of Divisions Theorem. Let a  b  0 and n = number of divisions required by the Euclidean algorithm to compute gcd(a,b). Then n < 2lg(a).

Maximum Number of Divisions Theorem. The smallest pair of integers that require n divisions to compute their gcd is F n+2 and F n+1. Theorem. Let a  b  0 and n = number of divisions required by the Euclidean algorithm to compute gcd(a,b). Then n < 1.44lg(a).

Average Number of Divisions Theorem. Let a  b  0 and n = average number of divisions required by the Euclidean algorithm to compute gcd(a,b). Then n  12ln(2) 2 /  2 lg(a)  lg(a). Theorem [Dixon] D(a,b)  ½ ln(a) for almost all pairs u  a  b  1 as u  

Dominance and Codominance Definition. Let f, g be real valued functions on a common set S. [Dominance] [Codominance] [Strict Dominance]

Basic Properties Theorem. Let f, f 1, f 2, g, g 1, and g 2 be nonnegative real-valued functions on S and c>0.

Integer Length Definition. A =  i=0..m a i  i, L  (A) = m

Basic Arithmetic Computing Times Theorem. Let A, M, D be the classical algorithms for addition, multiplication and division.

Maximum Computing Time Theorem.

Average Computing Time Theorem.

Probability of Relative Primality  p/d 2 = 1  p = 1/  (2)  (z) =  1/n z  (2) =  2 /6

Formal Proof Let q n be the number of 1 a,b  n such that gcd(a,b) = 1. Then lim n  q n /n 2 = 6/  2

Mobius Function  (1) = 1  (p 1    p t ) = -1 t  (n) = 0 if p 2 |n  (ab) =  (a)  (b) if gcd(a,b) = 1.

Mobius Inversion  d|n  (d) = 0 (  n  (n)n s )  (  n 1/n s ) = 1

Formal Proof q n =  n  (k)  n/k  2 lim n  q n /n 2 =  n  (n)/n 2 = 1/(  n 1/n 2 ) = 6/  2

Assignment 1 Empirically investigate distribution of quotients in Euclidean algorithm Implement and analyze classical division algorithm Implement and analyze Lehmer’s algorithm Study and summarize gcd algorithms in GMP