I'm studying for a course called numerical mathematics and let me apologize in advance if some of the terminology I use is unknown to you. It's in another language and I'm having a hard time finding English equivalents to any of this (makes it hard to google as well, hence this question).
Anyway, the topic is on the errors of approximate function values (there's the first term that I may not have translated correctly).
My textbook goes like this:
Let $y$ be a function of parameters $(a_1, ..., a_n) \in G$, $y = y(a_1, ..., a_n)$ and let $\bar{y}$ be an approximate value for $y$. The absolute error of $\bar{y}$ is $$(8) \quad A(\bar{y}) = \sup_{(a_1, ..., a_n) \in G}|y(a_1, ..., a_n) - \bar{y}|$$ and the relative error is $\frac{A(\bar{y})}{|\bar{y}|}$
Okay before moving forward I already have one question here. At first this didn't make any sense but then it occurred to me that maybe this means "Hey, let's approximate a function $y$ with a constant $\bar{y}$ at every single point". Meaning whatever the parameters of the actual function $y$, I shall approximate it with one single value that I picked called $\bar{y}$. Then if I want to calculate the error of that approximation I'll just see what the differences are between the actual values (at every point) and my constant value and just take the supremum of those differences. I hope I got this part right, because after this it gets even more confusing.
If $G$ is an n-dimensional rectangle $$|a_k -\bar{a_k}| \leq \Delta(\bar{a_k}), \quad k=1, ...n$$ and $$\bar{y} = y(\bar{a_1},..., \bar{a_n})$$ and $y$ is a continuously differentiable function, then according to the Lagrange formula: $$y(a_1, ..., a_n) - \bar{y} = \sum_{k=1}^{n}\frac{dy}{da_k}\bigg(a_1 + \theta(a_1 - \bar{a_1}), ..., a_n + \theta(a_n - \bar{a_n})\bigg)(a_k - \bar{a_k}) \quad \theta \in [0, 1]$$
I'll stop here for a second as well. If I got the first part right (that the initial idea was to approximate a function by some constant) then this part should be about doing something smarter than that which is approximating a function by a function. If I got it right, the idea is to take approximate parameters and evaluate the actual function at them, hence the $\bar{y} = y(\bar{a_1},..., \bar{a_n})$ part. The question I got here is: Which Lagrange formula is this? It's just mentioned as if it's very common and I should know it, but frankly I've got no idea. I'll just take it for granted for now.
Now check out the next sentence:
Therefore, because of (8), we have $$A(\bar{y}) \leq \sum_{k=1}^n \sup_{G}\bigg|\frac{dy}{da_k}(\bar{a_1}, ... \bar{a_n})\bigg|\Delta(\bar{a_k})$$
And that's all, no further explanations. For the first two confusing parts I've, at least, had some justifications for myself. Here, on the other hand, I've got absolutely no idea what happened. So, any ideas?
Thanks.
