Convergence and Fixed Point
A formal definition of convergence can be stated as follows. Suppose as goes from to is a sequence that converges to a fixed point, with for all . If positive constants and exist with
then as goes from to converges to of order, with asymptotic error constant
Given a function with a fixed point, there is a nice checklist for checking the convergence of p.
- 1) First check that p is indeed a fixed point:
- 2) Check for linear convergence. Start by finding . If....
then there is linear convergence | |
series diverges | |
then there is at least linear convergence and maybe something better, the expression should be checked for quadratic convergence |
- 3) If it is found that there is something better than linear the expression should be checked for quadratic convergence. Start by finding If....
then there is quadratic convergence provided that is continuous | |
then there is something even better than quadratic convergence | |
does not exist | then there is convergence that is better than linear but still not quadratic |
Read more about this topic: Limit (mathematics)
Famous quotes containing the words fixed and/or point:
“These earthly godfathers of Heavens lights,
That give a name to every fixed star,
Have no more profit of their shining nights
Than those that walk and wot not what they are.”
—William Shakespeare (15641616)
“People talk about the conscience, but it seems to me one must just bring it up to a certain point and leave it there. You can let your conscience alone if youre nice to the second housemaid.”
—Henry James (18431916)