Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose that $p_{1},p_{2}>0$ and that $F_{i}(h)=L_{i}+O(h^{p_{i}})$ as $h \to {0^ + }$ for $i=1,2$. What are the rates of convergence of $F_{1}(h)F_{2}(h)$ for various values of $L_{1},L_{2}$?

share|improve this question

1 Answer 1

up vote 0 down vote accepted

Just plugging in blindly, if $p = \min(p_1, p_2)$,

$\begin{align} F_1(h) F_2(h) &= (L_1 + O(h^{p_1})(L_2 + O(h^{p_2})) \\ &= L_1 L_2 + L_1 O(h^{p_2}) + L_2 O(h^{p_1}) + O(h^{p_1+p_2}) \\ &= L_1 L_2 + O(h^{p_2}) + O(h^{p_1}) + O(h^{p_1+p_2}) \\ &= L_1 L_2 + O(h^{p}) \\ \end{align} $

This depends on knowing that $O(h^a) + O(h^b) = O(h^a)$ as $h \to 0$ if $0 < a < b$. For example, $O(h)+O(h^2) = O(h)$ since $h^2$ is smaller than $h$ as $h \to 0$.

This, of course, could be considered somewhat abusive, but I enjoy it.

share|improve this answer
    
Ah, thanks loads for that! –  drawar Feb 11 '13 at 11:11

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.