# Maths - Numerical Methods

## 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?

- Slow
- 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.

What diagram is this?

A staircase diagram.

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 $x = a$?

What is the condition for $f(x)$ to have a repelling fixed point at $x = a$?

### 2022-02-10

What is the Newton-Rhapson formula for $x _ {n+1}$ in terms of $x _ n$?

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 $x = \alpha$?

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.