To calculate BigO and constant for the following function : I have don't know how to simplify it further. Any advise ? Thx
T(n) = (n!n+n^3)(n^2+7logn)
<= (n!n+n^3)(n^2 +7n) (since n>= log n)
<= (n!n+n^3)(n^2 +7n) (n^3 >= 7n if n > 3)
<= (n!n+n^3)(n^2 + n^3)
<= (n!n+n^3)(n!n + n^3) (n!n >= n^2)
:
The highest-order term can be found by expanding:
T(n) = (n!n+n^3)(n^2+7logn)
= n!n^3 + 6n!nlogn + n^5 + 7n^3logn
At this point we can simply compare terms and see which one is biggest:
n!n^3 vs 6n!nlogn
since n^2 > 7logn for n >= 4, n!n^3 > 7n!nlogn for n >= 1
n!n^3 vs n^5
since n! > n^2 for n >= 4, n!n^3 > n^5 for n >= 4
n!n^3 vs 7n^3logn
since n! > 7logn for n >= 4, n!n^3 > 7n^3logn for n >= 4
Based on all this, for n >= 4,
T(n) = (n!n+n^3)(n^2+7logn)
= n!n^3 + 6n!nlogn + n^5 + 7n^3logn
<= n!n^3 + n!n^3 + n!n^3 + n!n^3
= 4n!n^3
= O(n!n^3)
If you want to find another expression that bounds n!n^3 that's well and good, but I'd recommend another question for that.