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:
“Nothing stands out so conspicuously, or remains so firmly fixed in the memory, as something which you have blundered.”
—Marcus Tullius Cicero (106–43 B.C.)
“And William had dudgeon for the sightless beadle
Who worshipped a God like a grandmother on ice-skates,
For William saw two angels on the point of a needle
As nobody since except W. B. Yeats.”
—Allen Tate (1899–1979)