# Maths - Numerical Methods

> Source: https://ollybritton.com/notes/a-level/maths/topics/numerical-methods/ · Updated: 2022-02-07 · Tags: maths, year-2, school

## Flashcards
### 2022-02-07
##### What is the first step in using the decimal search method??
Rearranging to get 0 on one side.

##### How can you find a good estimate to start with for decimal search??
Draw a sketch.

##### What should you say when an identifying a root using decimal search??
"Gee wiz! The function $f(x)$ is continuous in the interval $[a, b]$ and there is a change of sign, so a root lies near $x$".

##### How would you show that $1.398$ is the best approximation to 3 d.p. when using decimal search??
Substitute in the upper and lower bounds $1.3975$ and $1.3985$ and show there is a change of sign.

##### What's a common criticism of decimal search??
1. Slow
2. Can misclassify asymptotes as roots

### 2022-02-08
##### How can you solve $f(x) = a$ using fixed-point iteration??
Rearrange into some $x = g(x)$ and see if repeating the expression converges.

##### ![PHOTO STAIRCASE DIAGRAM](staircase-diagram.png) What diagram is this??
A staircase diagram.

##### ![PHOTO COBWEB DIAGRAM](cobweb-diagram.png) What diagram is this??
A cobweb diagram.

##### How do you draw a staircase or cobweb diagram??
Draw the rearranged function along with the line $y = x$.

##### What is the condition for $f(x)$ to have an attracting fixed point at $x = a$??
$$
|f'(a) < 1|
$$

##### What is the condition for $f(x)$ to have a repelling fixed point at $x = a$??
$$
|f'(a) > 1|
$$

### 2022-02-10
##### What is the Newton-Rhapson formula for $x_{n+1}$ in terms of $x_n$??
$$
x_{n+1} = x - \frac{f(x_n)}{f'(x_n)}
$$

##### When doesn't the Newton-Rhapson method work at $x_n$??
When $x_n$ is a stationary point.

### 2022-06-06
##### For the curve $$x = f(x)$$ what are the conditions for iteration to work at a point $x = \alpha$??
$$
-1 < f(\alpha) < 1
$$

##### How can you remember whether to go to the curve or line first in a staircase or cobweb diagram??
Curve then line, alphabetically

##### How do you begin sketching at $x_0$ a staircase or cobweb diagram??
Draw a vertical line from $x_0$ upwards to the curve, then to the line, etc.

##### When do you get a staircase diagram in terms of the derivative of the rearranged function at the point??
When it is positive.

##### When do you get a cobweb diagram in terms of the derivative of the rearranged function at the point??
When it is negative.

##### Is it possible to get a mix of a staircase and cobweb diagrams??
Yes.

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