Determine the normwise condition of the operation $f_1(x,y)=\frac{x}{y}$
I unfortunately have no idea how a task like that can be solved.
There doesn't seem to be much source on the internet about it. But I found this in a book:
Let $X=\mathbb{R}^{n}, Y=\mathbb{R}^m \text{with fixed norms, } x \in X, f: X \rightarrow Y, x \neq 0, f(x) \neq 0$
$c_{abs}$ stands for absolute condition and $c_{rel}$ for relative condition.
$$c_{abs}=c_{abs}(x) = \text{lim sup}_{\tilde{x} \rightarrow x}\frac{\left \| f(\tilde{x})-f(x) \right \|_Y}{\left \| \tilde{x}-x \right \|_X}= \text{lim sup}_{\tilde{x} \rightarrow x} \frac{\text{absolute error of }f(\tilde{x})}{\text{absolute error of } \tilde{x}}$$
$$c_{rel} = c_{rel}(x) = \text{lim sup}_{\tilde{x} \rightarrow x}\frac{\frac{\left \| f(\tilde{x})-f(x) \right \|_Y}{\left \| f(x) \right \|_Y}}{\frac{\left \| \tilde{x}-x \right \|_X}{\left \| x \right \|_X}}= \text{lim sup}_{\tilde{x} \rightarrow x}\frac{\text{relative error of }f(\tilde{x})}{\text{relative error of }\tilde{x}}$$
These conditions are also called normwise conditions.
I'm not even sure if this is the correct thing that is needed to solve this task. But I hope you can show me at this example how a task like that can be solved?
I wouldn't know what to do in an exam else, because there is nothing so far I can read about it :s
By your definition we have
$$c_{abs}(x,y)= \limsup_{(\overline{x},\overline{y})\to(x,y)} \frac{\left|\frac{\overline{x}}{\overline{y}}-\frac{x}y\right|} {\sqrt{{\left(\overline{x}-x\right)}^2+{\left(\overline{y}-y\right)}^2}}$$
Another way to write this is as follows:
$$c_{abs}(x,y)= \limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$$
Does this expression look more familiar to you? Do you think you can take it from here?