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?
Best Answer
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