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I'm struggling to determine the order of error when considering the error value denoted by $\text{err}$ in relation to the variable $h$. Specifically, I aim to ascertain the value of $x$ in the expression $\text{err}=O(h^x)$. Assuming $\text{err}=ch^x$, I attempted to use the logarithm with base $h$ as follows:

$$\log_h{\text{err}}=x+\log_h{c}$$

However, I'm uncertain about the constraints that should be imposed on the constant $c$. For instance, should it be that $c<h$ or $c<1$ or $c<1$ or what? I'm seeking clarification on this matter.

For a practical example, let's consider $\text{err}=4.380691283796996\times10^{-9}$ and $h=0.0625$. What would be the correct order of error? Here are some possibilities:

  • If we assume $\text{err}=O(h^5)$, then we get $\text{err}\approx 0.0045\times h^5$.
  • If we assume $\text{err}=O(h^6)$, then we get $\text{err}\approx 0.0735\times h^6$.
  • If we assume $\text{err}=O(h^7)$, then we get $\text{err}\approx 1.1759\times h^7$.

Could you help me determine which assumption is correct?

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    $\begingroup$ Please avoid double and cross posts. The error order is a number that describes the shape of a function, the error as function of $h\approx 0$. You need more than one point to get useful estimates. $\endgroup$ Commented Jan 8 at 6:23
  • $\begingroup$ Taking the log of $ch^x$ does not result in $x + \log_h c$... $\endgroup$ Commented Jan 8 at 23:49
  • $\begingroup$ @WolfgangBangerth : The formula is correct, as $\log_hh=1$. But it is not optimal for starting a regression analysis. $\endgroup$ Commented Jan 10 at 10:39

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