# Notes - Continuous Mathematics HT23, Convergence

> Source: https://ollybritton.com/notes/uni/prelims/ht23/continuous-mathematics/notes/convergence/ · Updated: 2023-02-02 · Tags: uni, notes

- [Course - Continuous Mathematics HT23](https://ollybritton.com/notes/uni/prelims/ht23/continuous-mathematics/)

### Flashcards
Given an iterative numerical algorithm that attempts to approximate a true value $x^\star$, where $x_n = f(x_{n-1})$, what is the expression for the error $\epsilon$ at step $n$?::
$$
\epsilon_n = x_n - x^\star
$$

In an iterative numerical algorithm, the error at step $n$ is given by
$$
\epsilon_n = x_n - x^\star
$$
What is the goal of any iterative algorithm?::
$$
\lim_{n \to \infty} \epsilon_n \to 0
$$

Can you compare what it means for an iterative numerical algorithm to converge ($\epsilon_n \to 0$) linearly or converge sublinearly?::
Linearly:
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+1}|}{|\epsilon_n|} = a
$$
where $0 < a < 1$. Sublinearly:
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+1}|}{|\epsilon_n|} = 1
$$

What special case of an iterative numerical algorithm converging sublinearly, i.e.
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+1}|}{|\epsilon_n|} = 1
$$
exists that means it is very slow?::
Logarithmic convergence,
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+2} - \epsilon_{n+1}|}{|\epsilon_{n+1} - \epsilon_n|} = 1
$$

@Define what it means for an iterative numerical algorithm to converge ($\epsilon_n \to 0$) superlinearly.::
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+1}|}{|\epsilon_n|} = 0
$$

What does it mean for an iterative numerical algorithm to converge ($\epsilon_n \to 0$) with order $q$ convergence (careful; this definition is slightly different)?::
$$
\lim_{n \to \infty} \frac{|\epsilon_{n+1}|}{|\epsilon_n|^q} > a
$$
where $a > 0$ and $q > 1$.

@Define what it means for an iterative numerical algorithm to have quadratic convergence.::
It has order-2 convergence.

If $x_n$ converges with order-$q$ convergence, how fast does $x_{2n}$ converge?::
Order-$q^2$.

What is the region around which an iterative algorithm tends towards a solution called?::
The “basin of convergence”.

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