Maths - Numerical Methods
Pearson Edexcel Mathematics 2022
Flashcards
Decimal search
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
Fixed-point iteration and diagrams
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?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 $x = a$?
Newton-Raphson method
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.
Convergence conditions and diagram shape
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.