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I'm teaching myself how to find the relative condition numbers and I am struggling with connecting it to something basic like scalar multiplication. For example, the first problem of my text book seems simple enough, $f(x_1,x_2) = X_1 * X_2$ with $i=1,2$

How would I get started with finding the relative condition number?

What does relative mean here?

I assume that I would start with an identity matrix of one column and two rows $(i=1,2)$ but I don't know what to do next. I can see how the growth of this function will be linear and thus well conditioned, correct?

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Suppose we have a function $f(x)$. Let $\delta x$ be a small perturbation of $x$ then we write.

$$ \delta f = f(x +\delta x) - f(x) \tag{1} $$

now the absolute condition number $\hat{\kappa} = \hat{\kappa}(x)$ is defined as

$$ \hat{\kappa}(x) =\lim_{\delta \to 0} \sup_{\| \delta x\| \leq \delta } \frac{\| \delta f\|}{\| \delta x\|} \tag{2}$$

we can simply write this as $$ \hat{\kappa}(x) = \sup_{ \delta x} \frac{\| \delta f\|}{\| \delta x\|} \tag{3}$$

in comparison the relative condition number $\kappa = \kappa(x)$ is defined as

$$ \kappa(x) =\lim_{\delta \to 0} \sup_{\| \delta x\| \leq \delta } \bigg(\frac{\| \delta f\|}{\| f(x)\|} \bigg/ \frac{\| \delta x\|}{\| x\|} \bigg) \tag{4}$$

in the same way as above

$$ \kappa(x) = \sup_{ \delta x } \bigg(\frac{\| \delta f\|}{\| f(x)\|} \bigg/ \frac{\| \delta x\|}{\| x\|} \bigg) \tag{5}$$

With regards to the formula. If you consider multiplying two numbers as matrix times a vector

$$ \kappa(A) = \|A\| \frac{\| x\|}{\|Ax\|} \tag{6}$$

note for numbers this just becomes the absolute value

$$ \kappa(f(x_{1},x_{2})) = |x_{1}| \frac{|x_{2}|}{|x_{1}x_{2}|}\tag{7}$$

I believe

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  • $\begingroup$ If the function is continuously differentiable, this is equal to $$\kappa_{\text{rel}} := \frac{\| f'(x) \| \| x \|}{\| f(x) \|}.$$ $\endgroup$ – Viktor Glombik Mar 3 at 18:03

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