Continued Fractions, Euclidean Algorithm and Lehmer’s Algorithm Applied Symbolic Computation CS 567 Jeremy Johnson TexPoint fonts used in EMF. Read the.

Continued Fractions, Euclidean Algorithm and Lehmer’s Algorithm

Applied Symbolic ComputationCS 567

Jeremy Johnson

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

a1 = a, a2 = b

a1 = q3 a2 + a3, 0 a3 < a2

ai = qi ai+1 + ai+2, 0 ai+2 < ai+1

al= ql al+1

gcd(a,b) = al+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 = 27182818, v = 10000000

u’ v’ q’ u’’ v’’ q’’2718 1001 2 2719 1000 21001 716 1 1000 719 1 716 285 2 719 281 2 285 146 1 281 157 1 146 139 1 157 124 1 139 7 19 124 33 3 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 Fn+2 and Fn+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) 0.584 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]

f ¹ g , 9c > 0;f (x) · cg(x);8x 2 S

f » g , f ¹ g and g ¹ f

f Á g , f ¹ g and g 6¹ f

Basic Properties

Theorem. Let f, f1, f2, g, g1, and g2 be nonnegative real-valued functions on S and c>0.

f » cf

f 1 ¹ g1 and f 2 ¹ g2 ) f 1 + f 2 ¹ g1 + g2 and f 1f 2 ¹ g1g2

f 1 ¹ g and f 2 ¹ g ) f 1 + f 2 ¹ g

max(f ;g) » f + g

1 ¹ f and 1 ¹ g ) f + g ¹ f g

1 ¹ f ) f » f + c

Integer Length

Definition. A = i=0..m aii, L(A) = m

• • •

•

L ¯ (A) = dlog (jAj + 1)e= blog (jAj)c+ 1

L ¯ » L °

L(a§ b) ¹ L(a) + L(b)

L(ab) » L(a) + L(b); a;b2 I ¡ 0

L([a=b]) » L(a) ¡ L(b) + 1; jaj ¸ jbj > 0

L(Q n

i=1 ai ) »P n

i=1 L(ai ); ai 2 I ¡ ¡ 1;0;1

Basic Arithmetic Computing Times

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

tA (a;b) » L(a) + L(b)

tM (a;b) » L(a)L(b)

tD (a;b) » L(b)(L[a=b]) » L(b)(L(a) ¡ L(b) + 1)

Maximum Computing Time

Theorem. t+E (m;n;k) ¹ n(m¡ k + 1)

tE (a;b) »P `

i=1 L(qi )L(ai+1) ¹ L(b)P `

i=0 L(qi )

Average Computing Time

Theorem. t¤E (m;n;k) » n(m¡ k + 1)

Probability of Relative Primality

p/d2 = 1 p = 1/(2)

(z) = 1/nz

(2) = 2/6

Formal Proof

Let qn be the number of 1 a,b n such that gcd(a,b) = 1. Then limn qn/n2 = 6/2

Mobius Function

• (1) = 1• (p1 pt) = -1t

• (n) = 0 if p2|n

(ab) = (a)(b) if gcd(a,b) = 1.

Mobius Inversion

d|n (d) = 0

(n (n)ns)(n 1/ns) = 1

Formal Proof

qn = n (k)n/k2

limn qn/n2 = n (n)/n2 = 1/(n 1/n2) = 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