From what I remember:
Originally Posted by Tallon
For fixed point interation, say I have an equation f(x) = 0. I understand I rearrange that to make x =g(x). The root of f(x) = 0 is the same as the root where x = g(x) right?
The roots of x=g(x) will all be roots of f(x)=0; but x=g(x) won't give all of the roots.
Consider
. Obviously the roots are
; but if you rearrange f(x)=0 as:
,
; this form only gives one of the roots of f(x)=0
So I'd represent the root graphically as either f(x) crossing the x axis OR where g(x) crosses the line y = x, right?
Both should be ok; but I'd do the f(x) crossing the x axis form. Most exam questions I saw at A-level had the f(x) being easier to sketch than g(x).
Are there any others ways to show there's a root in the interval, whatever?
Decimal search.
e.g. if you had
and wanted to solve f(x)=0; you can try x=1.4, which gives f(x)<0; and x=1.5, which gives f(x)>0 - so you know that there's got to be a root in between 1.4 and 1.5
Another bit i'm fuzzy with is, how does the gradient of g(x) matter?
If the gradient of g(x) is too large where the root is, then x=g(x) won't converge to that root. I forget what interval the gradient has to be, I've got a hunch that any gradient between -1 and 1 will converge; and others don't.
whats the significance of it being posative or negative?
Positive and Negative gradients at the root converge in different ways. If you draw the sketch of x=g(x) for a g(x) with a positive gradient and draw on how x=g(x) converges, you'll get a staircase pattern. If you draw a g(x) with negative gradient then you'll get a cobweb-type pattern (alternates above and below the root whilst getting closer to it)
I think Autograph can do this for you if you have it.
does the gradient only matter for fixed point iteration?
As I remember, the gradient affects how fast Newton-Raphson iterations converge; but not whether or not it will converge.
What does the magnitude actually mean?
Similar to modulus, i.e. the size of something.
The magnitude of 3 is 3; the magnitude of -3 is also 3; the magnitude of the vector (3,4) is 5.
What is oscelate?
If you look at the
again; solving f(x)=0 with
, the subsequent values of
you get change between 1 and 2, but never get to the solution. That's an example of oscillation in this instance.
if it diverges it can if it goes posative to negative, can it oscelate and converge?
Not sure I understand the first part of your question; but it cannot oscillate and converge.
does the gradient have something to do with this?
Not sure. We spent very little time looking at gradient in the x=g(x) iteration at school.
Maths revision, coursework or discussion you will find help in here.
Useful Resources:
Numerical Methods. Fixed point iteration.
