Limit (mathematics) - Convergence and Fixed Point

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

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