Convergence rate vs convergence order

Question

I'm a bit confused about the concepts of convergence rate and convergence order. Let me first give you the definitions we use: [sorry for the English, it's all self translated]

Let $x^{*}$ be our solution.

Definition 1: The sequence $x^{(k)}$ is called linearly converging towards $x^{*}$, if $$\exists L<1 \text{ so that } \|x^{(k+1)}-x^{*}\|\leq L\| x^{(k)}-x^{*}\|,\quad \forall k\geq k_0.$$

We call the constant $L$ rate of convergence.

Definition 2: The order of convergence of a numerical method is $p$, if: $$\exists C > 0 \text{ so that }\|x^{(k+1)}-x^{*}\|\leq C\| x^{(k)}-x^{*}\|^p,\quad \forall k\in \mathbb N \quad\text{with $C<1$ for $p=1$}.$$

Note: We assumed, that chose the starting value so that we get an converging sequence.

Questions:

Question 1: Can someone explain me the difference between $C$ and $L$ here?

Question 2: Can someone explain me the concept/idea behind the rate/order of convergence? [Just so I heard it from another perspective]

Question 3: Also, I often see that we use linglog and loglog plots, but I don't really get why we do that. E.g., if we have linear convergence, we can see a linear function if we lin-log plot our errors. (Why do we need the lin-log plot here)

Answer 1

$\|x^{(k)}-x^*\|$ is the error in the $k$th term, call it $E_k$. For a "good" numerical method, we want the approximation to get closer and closer to the desired result so $E_k$ has to decrease to zero. If the error is guaranteed to reduce to at least a certain fraction $L$ of the previous step, you have linear convergence: $$E_{k+1} \le L E_k.$$ This essentially guarantees that the error drops at least as fast as a geometric series. An example would be the sequence $0.9, 0.99, 0.999 \ldots$ which converges to 1. The error at the $k$th step is $10^{-k}$ so that $L=0.1$.

But the error could fall faster than linear. Consider the sequence $$0.9, 0.99, 0.9999, 0.99999999\ldots$$ This converges to 1 too, but at each step the error reduces as the square of the previous step so that $p=2$. When you have a faster than linear convergence order ($p>1$), you are not really worried about $C$: it could even be greater than 1. But when $p=1$, $C$ becomes the same as $L$ and you want it to be less than 1.

Now why the log-linear plot? Assuming linear convergence with geometrically reducing error, we would have $$E_k = L^{-k} E_0$$ replacing $\le$ with equality. Taking the log of this, $$\log(E_k) = \log(E_0) - k \log(L)$$ which is a linear curve.