I want to prove that any polynomial of degree $k$ is in $\Theta(n^k)$. The coefficient of $n^k$, $a_{k}$, is positive.
I know I need $0 \leq c_{1}n^k \leq a_{k}n^k + ... + a_{0} \leq c_{2}n^k$ for all $n \geq n_0$.
The upper limit is easy to prove by taking $c_{2} = \sum\limits_{i=0}^k |a_i|$
I don't know how to prove the lower limit. Any hints?
Asked By : ask
Answered By : ask
I think I found a good way to prove $p(n) = \Omega(n^k)$:
We want to show that $0 \leq cn^k \leq p(n)\ \forall{n \geq n_{0}}$
We know $\lim_{n\to\infty} p(n)/n^k = a_{k}$
This gives us some intuition to choose $c \leq a_{k}$.
Let $c = a_{k}/2$
Now choose $n_{0}$ such that $cn^k = (a_{k}/2)n^k \leq p(n)\ \forall{n \geq n_{0}}$.
or rearranging, $(a_{k}/2)n^k \geq -\sum\limits_{i=0}^{k-1} a_{i}n^i\ \forall{n \geq n_{0}}$
or we can relax the inequality and pick $n_{0}$ such that $(a_{k}/2)n^k \geq \sum\limits_{i=0}^{k-1} |a_{i}|n^i\ \forall{n \geq n_{0}}$
or $(a_{k}/2)n^k \geq n^{k-1}\sum\limits_{i=0}^{k-1} |a_{i}|n^{i-(k-1)}\ \forall{n \geq n_{0}}$
or we can relax the inequality and pick $n_{0}$ such that $(a_{k}/2)n \geq \sum\limits_{i=0}^{k-1} |a_{i}|\ \forall{n \geq n_{0}}$ since $n^{i-(k-1)} \leq 1$
Hence pick $n_{0} = 2/a_{k}\sum\limits_{i=0}^{k-1} |a_{i}|$
We now have a $c$ and $n_{0}$ such that $0 \leq cn^k \leq p(n)\ \forall{n \geq n_{0}}$
Hence proved.
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Question Source : http://cs.stackexchange.com/questions/21298
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