# Further Maths - Numerical Methods

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/numerical-methods/ · Updated: 2021-12-08 · Tags: school, further-maths, year-2, further-pure-1

## See Also
- [Maths - Numerical Methods](https://ollybritton.com/notes/a-level/maths/topics/numerical-methods/)

Try out an interactive visualisation of Euler's method here: [Euler's method](https://projects.ollybritton.com/eulers).

## Flashcards
### 2021-12-08
##### How could you summarise Euler's method for solving first-order differential equations??
Start with some point on the curve and then follow the direction of the curve.

##### If a gradient is given by $\frac{\text{d}y}{\text{d}x}$, how much would you increase the $y$-coordinate for a step size of $h$??
$$
y_1 = y_0 + \frac{\text{d}y}{\text{d}x} h
$$

##### What does using the assumption $$\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y_0}{h}$$ mean you should do in a question??
Use Euler's method.

##### What assumption are you making for Euler's method??
$$
\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y_0}{h}
$$

##### What is the formula for $y_{r + 1}$ given $\frac{\text{d}y}{\text{d}x}$ and $y_r$ using Euler's method??
$$
y_{r + 1} = y_r + \left( \frac{\text{d}y}{\text{d}x} \right)h
$$

##### If you were asked to use $5$ iterations to approximate a solution at $x = 2$ given the point $(1, 2)$ using Euler's method, what would your step size be??
$$
h = 0.2
$$

##### What does using the assumption $$\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y{-1}}{2h}$$ mean you should do in a question??
Use the midpoint method.

##### What is the improvement on Euler's method called??
The midpoint method.

##### What assumption are you making for the midpoint method??
$$
\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y{-1}}{2h}
$$

##### What is the formula for $y_{r+1}$ in terms of $y_r$ and $y_{r-1}$ using the midpoint method??
$$
y_{r+1} \approx y_{r-1} + 2h\left(\frac{\text{d}y}{\text{d}x}\right)_{y_r}
$$

##### What do you often need to do in order to use the midpoint method and given one point??
Use Euler's method to find the next point.

##### What's a nice way of answering numerical method questions??
Using a table.

### 2021-12-10
##### What assumption are you making for approximating solutions to second order differential equations??
$$
(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right)_0 - \left(\frac{\text{d}x}{\text{d}y}\right)_{-1}}{h}
$$

##### $$(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right)_0 - \left(\frac{\text{d}x}{\text{d}y}\right)_{-1}}{h}$$ What is this assumption in terms of $y_1$, $y_0$ and $y_{-1}$??
$$
(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{y_1 - 2y_0 + y_{-1}}{h^2}
$$

##### What's the formula for $y_{r+1}$ in terms of $y_r$ and $y_{r-1}$ using the approximation $$(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{y_1 - 2y_0 + y_{-1}}{h^2}??
$$
y_{r + 1} \approx 2y_4 - y_{r-1} + h^2(\frac{\text{d}^2y}{\text{d}x^2})_{r}
$$

##### When do you have to use simultaneous equations when approximating a second-order differential equation??
When they ask you to use the midpoint method.

##### What is Simpson's rule used for??
Approximating integrals.

##### How does Simpson's rule work??
Splitting curves up into quadratics.

##### What is Simpson's rule for $$\int^b_a f(x) dx$$ in terms of even values, odd values and endpoints??
$$
\int^b_a f(x) dx \approx \frac{1}{3}h((\text{endpoints}) + 4(\text{odd values}) + 2(\text{even values}))
$$

##### Why can't you use Simpson's rule when splitting up a curve into 7 strips/intervals??
Simpson's rule only works with an even number of strips/intervals.

### 2022-04-24
##### What's a stupid mnemonic for remembering Simpson's rule??
On the third of Haugust, it's the end 4 Mr Odd. He was 2 even.

##### "On the third of Haugust, it's the end 4 Mr Odd. He was 2 even." Can you turn this into Simpson's rule??
$$
\frac{1}{3} h (\text{end} + 4\text{odd} + 2\text{even})
$$

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