Maths - numerical methods and sequences Flashcards Preview

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Flashcards in Maths - numerical methods and sequences Deck (12)
1
Q

A

Occurs when 0 < f’(w) < 1

Where w is the root and f(x) is the convergent function

This will successfully obtain the root

2
Q

A

Occurs when f’(w) > 1

Where w is the root and f(x) is a divergent function

This won’t obtain the root

3
Q

Cobweb diagrams: Inward spiral

A

Occurs when -1 < f’(w) < 0

Where w is the root and f(x) is a convergent function

This will successfully obtain the root

4
Q

Cobweb diagrams: Outward spiral

A

Occurs when f’(w) < -1

Where w is the root and f(x) is the divergent sequence

Won’t obtain the root

5
Q

Drawing cobweb/staircase diagrams

A

Start with a point and draw a line to the curve, then across to the line then to the curve etc

6
Q

Using the derivative of the iterative function f(x) to determine if the process will converge or diverge

A

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

7
Q

Alternating sequence

Oscillating sequence

A

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

8
Q

Periodic sequence

A

A sequence that consists of a repeating pattern of numbers

9
Q

Convergent sequence

Divergent sequence

A

A sequence which approaches a definite value

A sequence which doesn’t approach a definite value

10
Q

Condition for a geometric progression to be convergent

A

the common ratio, r :

-1 < r < 1

11
Q

Exponential growth and decay

A

When the rate of growth/decay is directly proportional to the quantity present

12
Q

How can you tell which iterative sequence will converge more rapidly?

A

Magnitude of the gradient is closest to the root at the given starting point