Akra-Bazzi theorem: The polynomial growth condition #
This file defines and develops an API for the polynomial growth condition that appears in the
statement of the Akra-Bazzi theorem: for the theorem to hold, the function g
must
satisfy the condition that c₁ g(n) ≤ g(u) ≤ c₂ g(n)
, for u between b*n and n for any constant
b ∈ (0,1)
.
Implementation notes #
Our definition requires that the condition must hold for any b ∈ (0,1)
. This is equivalent to
requiring it for b = 1 / 2
(or any other particular value in (0, 1)
). While this could, in
principle, make it harder to prove that a particular function grows polynomially, this issue does
not seem to arise in practice.
The growth condition that the function g
must satisfy for the Akra-Bazzi theorem to apply.
It roughly states that c₁ g(n) ≤ g(u) ≤ c₂ g(n)
, for u
between b * n
and n
, for any
constant b ∈ (0, 1)
.