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?
What diagram is this?A staircase diagram.
What diagram is this?
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.