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 nation looked upon him as a deserter, and he shrunk into insignificancy and an earldom.... He was fixed in the house of lords, that hospital of incurables, and his retreat to popularity was cut off; for the confidence of the public, when once great and once lost, is never to be regained.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“In my state, on the basis of the separate but equal doctrine, we have made enormous strides over the years in the education of both races. Personally, I think it would have been sounder judgment to allow that progress to continue through the process of natural evolution. However, there is no point crying about spilt milk.”
—Lyndon Baines Johnson (1908–1973)