Greatest common divisors & Fibonacci Numbers--Algorithms

Greatest Common Divisors:

Naive Algorithm:  

  

Function NaiveGCD(a,b)

best ←0
for dfrom1toa+b:

  if d|aandd|b: 

    best←d

return best

Runtime approximatelya+b.

Very slow for 20 digit numbers. 

Euclidean Algorithm

Function EuclidGCD(a,b)

if b=0:

  return a

a′←the remainder whenais divided byb

return EuclidGCD(b,a′)

Each step reduces the size of numbersby about a factor of 2.

Takes about log(ab)steps.
GCDs of 100 digit numbers takes about 600 steps.

Each step a single division. 

Fibonacci Number:

FibRecurs(n)

if n≤1:

  return n

else:
  return FibRecurs(n−1)+FibRecurs(n−2)

Too slow! 

Better:

FibList(n)

create an array F[0...n]

F[0]←0
F[1]←1
for ifrom2ton:

    F [i]←F[i−1] +F[i−2]

return F[n]

T(n)=2n+2. SoT(100)=202.

Easy to compute.