A-Iterations Flashcards
If a function is not continuous, what does that mean about it in regards to differentiability?
If it is not continuous, it is not differentiable. However, there are continuous functions which also aren’t differentiable.
How might you find the limit X of a sequence?
X is defined by
f(X)=X
Solve for this.
What does is mean for a function to be monotonic?
What does ⊂ mean?
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
How do you find the order of convergence?
Order of convergence=p
How do you find the asymtopic convergence factor?
What does it mean for an interval to be bracketing for X
The end points of the interval will had opposite signs.
What does it mean for a function to be strictly decreasing over I?
Its derivative must be below 0 over the internal I
What does it mean for a function to be strictly monotonic?
How do you use the fixed point interation method?
How do you find the rate of convergence?
rate of convergence= |g’(X)|
What is the equation for Newton’s method?
g(xn)= xn-1-f(xn-1)/f’(xn-1)
What does it mean for a function g to be a contraction over an internval I?
|g’(x)|<1 over the entire interval.
What is a fixed point of g(x)?
Denoted by X
x=g(x)
what does f ε Ck(I) mean ?
What condition must a fixed point satisfy so as to be stable?
|g’(X)|<1 to be stable
If |g’(X)|>1 then is is unstable
If a sequence converges lineraly to X, what is ρ?
ρ<1
If a sequence is to converge super linearly to X, what must ρ be?
ρ=0
If a sequence converges sub-linealy to X, what is ρ ?
ρ=1
What is the difference between the interval (0,1] and [0,1] ?
(0,1] goes from just above 0 to 1.
0
[0,1] goes from 0 to 1.
0≤x≤1
What is the CCT and what does it assume?
Contractive convergence theorem
If the CCT is met, what does that mean for convergence?
It will converge.
However, it can converge if the CCT is not met.
What is a useful equation to find ρ?
How do you determine if a root is simple?
X is a simple root of f if:
f(X)=0
f’(X)≠0
