Convergence Tests - Numerical Algorithms

[Joza]

This doesn't apply strictly to Octave, but I think it is relevant. 

I have been looking at convergence tests, for example, in computing the root
or the fixed point of a function. Often, a rather crude test is used, such
as 

IF  absolute_value(  x_k  -  x_k-1  )  <=  10^-6  STOP

where x_k is the kth term in the series. But this is dangerously naive, for
if say x_k = 10^12, then the next number around x_k is
machine_epsilon*absolute_value( x_k ) = 10^-4, so the test can never be
true.

I understand this quite well. Yet I've come two tests which are used as the
best for general numerical algorithms, and I cannot understand them:

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

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

where E_tol is a tolerance value. Why the factor of 4? Why the E_tol, and
what is it? And why is the last one the best?

I hope someone can explain this, and these tests mystify me!

Thanks,
Joza



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