A-Iterations

Question 1

If a function is not continuous, what does that mean about it in regards to differentiability?

Answer 1

If it is not continuous, it is not differentiable. However, there are continuous functions which also aren’t differentiable.

Question 2

How might you find the limit X of a sequence?

Answer 2

X is defined by

f(X)=X

Solve for this.

Question 3

What does is mean for a function to be monotonic?

Answer 3

Question 4

What does ⊂ mean?

Answer 4

If A and B are sets and every element of A is also an element of B, then:

A is a subset of B, denoted by A⊂B

Question 5

How do you find the order of convergence?

Answer 5

Order of convergence=p

Question 6

How do you find the asymtopic convergence factor?

Answer 6

Question 7

What does it mean for an interval to be bracketing for X

Answer 7

The end points of the interval will had opposite signs.

Question 8

What does it mean for a function to be strictly decreasing over I?

Answer 8

Its derivative must be below 0 over the internal I

Question 9

What does it mean for a function to be strictly monotonic?

Answer 9

Question 10

How do you use the fixed point interation method?

Question 11

How do you find the rate of convergence?

Answer 11

rate of convergence= |g’(X)|

Question 12

What is the equation for Newton’s method?

Answer 12

g(x_n)= x_n-1-f(x_n-1)/f’(x_n-1)

Question 13

What does it mean for a function g to be a contraction over an internval I?

Answer 13

|g’(x)|<1 over the entire interval.

Question 14

What is a fixed point of g(x)?

Answer 14

Denoted by X

x=g(x)

Question 15

what does f ε C^k(I) mean ?

Answer 15

Question 16

What condition must a fixed point satisfy so as to be stable?

Answer 16

|g’(X)|<1 to be stable

If |g’(X)|>1 then is is unstable

Question 17

If a sequence converges lineraly to X, what is ρ?

Answer 17

ρ<1

Question 18

If a sequence is to converge super linearly to X, what must ρ be?

Answer 18

ρ=0

Question 19

If a sequence converges sub-linealy to X, what is ρ ?

Answer 19

ρ=1

Question 20

What is the difference between the interval (0,1] and [0,1] ?

Answer 20

(0,1] goes from just above 0 to 1.

0

[0,1] goes from 0 to 1.

0≤x≤1

Question 21

What is the CCT and what does it assume?

Answer 21

Contractive convergence theorem

Question 22

If the CCT is met, what does that mean for convergence?

Answer 22

It will converge.

However, it can converge if the CCT is not met.

Question 23

What is a useful equation to find ρ?

Answer 23

Question 24

How do you determine if a root is simple?

Answer 24

X is a simple root of f if:

f(X)=0

f’(X)≠0