1. Asymptotics & Stirling’s Approximation
Mathematics for Computer Science MIT 6.042J/18.062J
Asymptotics & Stirling’s Approximation
Copyright © Albert Meyer, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. Factorial defines a product:
Stirling’s Formula
Factorial defines a product:
Turn product into a sum taking logs:
ln(n!) = ln(1 · 2 · 3 · · · (n – 1) · n)
= ln 1 + · · · + ln(n – 1) + ln(n) =

3. … bound by integral method Stirling’s Formula ln (n) … ln (n+1) ln2 3 1 4 5
n–2
n–1 n … …
ln 2
ln 3
ln 4
ln 5 ln
n-1
ln n

4. Integral method to bound ln(n!)
Stirling’s Formula
Integral method to bound ln(n!)

5. Stirling’s Formula
“~” means:

6. Asymptotic Notation
f(x) = O(g(x)) +c c
g(x)
blue stays below red
f(x)

7. Definition 3.1: Asymptotic Equivalence
f(n) is asymptotically equivalent to g(n) when:
Denoted by: f(n) ~ g(n)

8. Definition 3.1: Asymptotic Equivalence
f(n) is asymptotically equivalent to g(n) when:
Denoted by: f(n) ~ g(n)

9. Denoted by: f(n) = o(g(n))
Definition 7.1: Little Oh
f(n) is asymptotically smaller to g(n) when:
Denoted by: f(n) = o(g(n))

10. Denoted by: f(n) = O(g(n)) Note: f~g or f = o(g)  f=O(g)
Definition 7.2: Big Oh
f(n) is asymptotically bounded by g(n) when:
Denoted by: f(n) = O(g(n))
Note: f~g or f = o(g)  f=O(g)

11. f(n) = O(g(n)) iff c > 0, n0 n  n0 |f(n)|  c·g(n) Equivalently,
Definition 7.3: Big Oh
Equivalently,
f(n) = O(g(n))
iff
c > 0, n0 n  n0
|f(n)|  c·g(n)

12. f(n) = (g(n)) iff f(n) = O(g(n)) and g(n) = O(f(n))
Definition 7.7: Theta
f(n) = (g(n))
iff
f(n) = O(g(n))
and
g(n) = O(f(n))

13. Lemma: ln x = o(x)
Proof:
variable substitution: