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Let $\hat{x}\neq 0$ be an approximation of a non null quantity $x$. Find the relation between the relative error $\epsilon = |x - \hat{x}|/|x|$ and $\tilde{E} = |x - \hat{x}|/|\hat{x}|$

I believe we need to study the conditioning number of both equations and compare them, but I am not exactly sure. Any suggestions is greatly appreciated.

Wolfy
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1 Answers1

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It holds that

$$\hat{E}= \epsilon \cdot \frac{|x|}{|\hat{x}|}$$

Do you see why?

evinda
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  • I think so, is it because in the way you set $\hat{E}$ to be and the way $\epsilon$ is defined? Not sure how to word this – Wolfy Sep 12 '15 at 22:27
  • Yes, you want to write $\hat{E}=\frac{|x-\hat{x}|}{|\hat{x}|}$ in respect to $\epsilon=\frac{|x-\hat{x}|}{|x|}$. $$$$ So you want to find a $\lambda$ such that $\frac{|x-\hat{x}|}{|\hat{x}|}= \lambda \cdot \frac{|x-\hat{x}|}{|x|} \Rightarrow \frac{1}{|\hat{x}|}= \lambda \cdot \frac{1}{|x|} \Rightarrow \lambda=\frac{|x|}{|\hat{x}|}$ – evinda Sep 12 '15 at 22:32
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    great, thanks. Did you ever study the book called Numerical Mathematics by Quarteroni, Alfio, Sacco, Riccardo, Saleri, Fausto? – Wolfy Sep 12 '15 at 22:39
  • @MorganWeiss No, I studied Numerical Analysis from an other book. – evinda Sep 12 '15 at 22:40
  • which book? I do not have access to solutions for this one which makes solving these problems difficult. – Wolfy Sep 12 '15 at 22:41
  • @MorganWeiss A greek version of this book: G. D. Akrivis, V. A. Dougalis: Introduction to Numerical Analysis. I don't know if there is a corresponding english version. – evinda Sep 12 '15 at 22:48