ONotation

1
O-Notation

• Definition Let f and g be real-valued functions
  defined on the same set of real numbers.
• Then f is of order g, written f(x) is O(g(x)),
• iff ? M?R and x0?R such that
• f(x) ? Mg(x) , whenever xgtx0 .
• f(x) is O(g(x)) is also read
• g is a big-oh approximation for f .
• Example 2x3 ? 5x for xgt1.
• Thus, 2x3 is O(x) .

2
Orders of Power Functions

• If xgt1
• then x lt x2 , x2 lt x3, , xk lt xk1
• for any k1 .
• Thus, by transitivity of lt ,
• if xgt1 and rlts,
• then xr lt xs .
• Proposition For any rational numbers r and s,
• if rlts,
• then xr is O(xs) .
• Examples x7 is O(x11) x4/3 is O(x3/2).

3
Orders of General Polynomial Functions

• Theorem (On Polynomial Orders)
• If a0, a1, a2, , an are real numbers and an?0,
• then anxn an-1xn-1a1xa0 is O(xm)
• for all m n.
• Example 4x7 - 15x5 6x2 - 3x 5 is O(x7) .

4
Orders for Functions of Integer Variables

• Example Show that
• 135(2n1) is O(n2) .
• Solution 135(2n1)
• (01)(21)(41)(2n1)
• (0242n) (n1)
• 2(12n) (n1)
• 2n(n1)/2 (n1)
• (n2n) (n1)
• n22n1
• which is O(n2) by the theorem on
• polynomial orders

5
Best Big-Oh Approximations

• Definition
• Suppose f is a function defined on R
• S is a set of functions defined on R.
• Then function g?S is
• a best big-oh approximation for f in S iff
• 1. f(x) is O(g(x))
• 2. for any h?S, if f(x) is O(h(x))
• then g(x) is also O(h(x)) .
• In other words,
• g is the smallest function in S that is an
  order of f.
• Example Let f(x) 3x52x34
• S be the set of all integral power
  functions xk, k?Z .
• Then g(x)x5 is a best big-oh
  approximation for f in S

6
Big-Oh Approximations on Rational Power
Functions

• Proposition If a0, a1, , an are real numbers
  and an?0,
• and r0, r1, , rn are rational numbers with r0
  lt r1 lt lt rn ,
• then
• Example Find a best big-oh approximation from
  the set of all rational power functions for
• Solution
• Then, by the proposition,
• g(x)x3/2 is a best approximation for f(x).