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:
“Genius detects through the fly, through the caterpillar, through the grub, through the egg, the constant individual; through countless individuals the fixed species; through many species the genus; through all genera the steadfast type; through all the kingdoms of organized life the eternal unity. Nature is a mutable cloud which is always and never the same.”
—Ralph Waldo Emerson (1803–1882)
“There never comes a point where a theory can be said to be true. The most that one can claim for any theory is that it has shared the successes of all its rivals and that it has passed at least one test which they have failed.”
—A.J. (Alfred Jules)