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Math Help - Orders of convergence

  1. #1
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    May 2008
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    Orders of convergence

    Hi Guys,

    I have a small question on orders of convergence.
    I'm happy with the idea that for $N\in\mathbb{N}$ if,
    $\text{error} \le C_1N^{-\alpha}+C_2N^{-\beta}$, where $\alpha<\beta$, $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$, $C_1, C_2 \in\mathbb{R}$ are constants
    $\Rightarrow \text{error} \le (C_1+C_2)N^{-\alpha}$

    or

    $\text{error} \le C_1N^{\alpha}+C_2N^{\beta}$, where $\alpha<\beta$, $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$,  $C_1, C_2 \in\mathbb{R}$ are constants
    $\Rightarrow \text{error} \le (C_1+C_2)N^{\beta}$

    but for
    $\text{error} \le C_1N^{-\alpha}+C_2N^{\beta}$, where $\alpha, \beta \in\mathbb{R}$, $\alpha,\beta>0$,  $C_1, C_2 \in\mathbb{R}$ are constants.
    is this the right outcome
    $\text{error} \le (C_1+C_2)N^{\beta}$ ?
    Other than when 0 < N < 1 is there a time when $N^{-\alpha}$ would win?
    Thanks :-)
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  2. #2
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    Joined
    May 2008
    Posts
    9

    Re: Orders of convergence

    I think I might have just figured out what I needed to know. I think the size of the coefficients $C_1$ and $C_2$ play an important role here as to which rate dominates the other. Please let me know if you think I'm wrong or I should be considering other factors as well.

    Thanks.
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