Maths - numerical methods and sequences Flashcards
Staircase diagrams: Inward ladder
Occurs when 0 < f’(w) < 1
Where w is the root and f(x) is the convergent function
This will successfully obtain the root
Staircase diagrams: Outward ladder
Occurs when f’(w) > 1
Where w is the root and f(x) is a divergent function
This won’t obtain the root
Cobweb diagrams: Inward spiral
Occurs when -1 < f’(w) < 0
Where w is the root and f(x) is a convergent function
This will successfully obtain the root
Cobweb diagrams: Outward spiral
Occurs when f’(w) < -1
Where w is the root and f(x) is the divergent sequence
Won’t obtain the root
Drawing cobweb/staircase diagrams
Start with a point and draw a line to the curve, then across to the line then to the curve etc
Always start with the curve!
Using the derivative of the iterative function f(x) to determine if the process will converge or diverge
In order for the sequence to converge then:
-1 < f’(w) < 1
Where f’(w) is the derivative of the iterative function at w
Where w is the root
Alternating sequence
Oscillating sequence
An oscillating sequence which oscillates around 0 - so the terms are alternately positive and negative
A sequence that is alternately greater then small than a given value
Periodic sequence
A sequence that consists of a repeating pattern of numbers
Convergent sequence
Divergent sequence
A sequence which approaches a definite value
A sequence which doesn’t approach a definite value
Condition for a geometric progression to be convergent
the common ratio, r :
-1 < r < 1
Exponential growth and decay
When the rate of growth/decay is directly proportional to the quantity present
How can you tell which iterative sequence will converge more rapidly?
Magnitude of the gradient is closest to the root at the given starting point
