Concrete mathematics : a foundation for computer science

428 ASYMPTOTIC3
How about the functio:n efi; where does it live in the hierarchy? We
can answer questions like this by using the rule
which follows in two steps from definition (9.3) by taking logarithms. Consequently
1 + f(n) 4 s(n) ==+ eiflnll + e191”ll .
And since 1 4 log logn 4 \/logn 4 c logn, we have logn + ee + n6.
When two functions f(n) and g(n) have the same rate of growth, we
write ‘f(n) x g(n)‘. The ofhcial definition is:
f(n) =: s(n) W f(n)1 < Clg(n)l and Is(n)1 6 Clf(nll,
for some C and for all sufficiently large n. (9.8)
This holds, for example, if f(n) is constant and g(n) = cos n + arctan n. We
will prove shortly that it h.olds whenever f(n) and g(n) are polynomials of
the same degree. There’s al.so a stronger relation, defined by the rule
In this case we say that “f(n) is asymptotic to g(n)!’
G. H. Hardy [148] introduced an interesting and important concept called
the class of logarithmico-exponential functions, defined recursively as the
smallest family C of functions satisfying the following properties:
. The constant function f(n) = 01 is in C, for all real 01.
. The identity function f(n) = n is in C.
. If f(n) and g(n) are in 2, so is f(n) - g(n).
. If f(n) is in 2, so is efcni.
. If f(n) is in C and is “eventually positive,” then lnf(n) is in C.
A function f(n) is called “eventually positive” if there is an integer no such
that f(n) > 0 whenever n 2: no.
We can use these rules to show, for example, that f(n) + g(n) is in C
whenever f(n) and g(n) are, because f(n) + g(n) = f(n) - (O-g(n)). If f(n)
and g(n) are eventually positive members of C, their product f(n) g(n) =
elnf(n)+lnsin) and quotient f(n)/g(n) = elnf(nlm lnsini are in C; so are functions
like m = eilnf(nl, etc. Hardy proved that every logarithmicoexponential
function is eventually positive, eventually negative, or identically
zero. Therefore the product and quotient of any two C-functions is in 2,
except that we cannot divide by a function that’s identically zero.