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:
“The foot of the heavenly ladder, which we have got to mount in order to reach the higher regions, has to be fixed firmly in every-day life, so that everybody may be able to climb up it along with us. When people then find that they have got climbed up higher and higher into a marvelous, magical world, they will feel that that realm, too, belongs to their ordinary, every-day life, and is, merely, the wonderful and most glorious part thereof.”
—E.T.A.W. (Ernst Theodor Amadeus Wilhelm)
“The best joke-tellers are those who have the patience to wait for conversation to come around to the point where the jokes in their repertoire have application.”
—Joseph Epstein (b. 1937)