Re: Convergence Tests - Numerical Algorithms

[Francesco Potortì]

>I have been looking at convergence tests, for example, in computing the root
>or the fixed point of a function. 

You have already been correctly answered.  This is to try a more verbose
answer.

>BETTER: 
>IF  absolute_value(  x_k  -  x_k-1  )  <= 
>4.0*machine_epsilon*absolute_value(x_k)

This works if we have an efficient iterative procedure, which manages to
converge to a value that is very near the smallest representable
difference.  In this case "very near" is assumed to be four times.

>BEST:
>IF  absolute_value(  x_k  -  x_k-1  )  <= E_tol 
>4.0*machine_epsilon*absolute_value(x_k)

Here we do not make such an assumption and rely to a further multiplier,
which we will set, depending on the iterative algortihm, to a value that
may be 1 (as in the above case) to values much bigger than 1.  For
example, if we have an algorithm which converges badly we may want
E_tol=1e4, or if we have an algorithm that converges slowly and we don't
want to wait and we don't need so much precision, we may want
E_tol=sqrt(eps).

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