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The question of convergence of the different iterators can be
partially answered.
For functions with two occurrences of the unknown, ,
in general either one or the other derived iterator converges.
This is proven as follows.
Let
*f*(*x*_{1},*x*_{2})=0 be the equation to be solved, identifying
each occurrence of *x* by *x*_{1} and *x*_{2}.
Let *F*_{i}(*x*) be the iterator derived from inverting *f* on *x*_{i}.
Formally this means that

*f*(*F*_{1}(*x*),*x*) = 0

and

*f*( *x*, *F*_{2}(*x*) ) = 0

Let
be a root of *f*,
.
The convergence of the iterators depends on
.
By computing derivatives with respect to *x* of the above
identities, we find

For this system of equations to have a non-null solution in
,
its determinant must be zero, or

and from this

So unless both absolute values are equal to one, then one
iterator converges while the other diverges.
This is a very encouraging result, it guarantees that one of the
iterators will succeed for *each* of the roots.
For
the results are much weaker.
For general *x*, the condition becomes

Then, if *k*-1 of the values
,
the
remaining one must be between 0 and 1 and hence the corresponding
iterator converges.
This does not guarantee convergence for every case, as all
,
i.e.
is also a possible solution and in this case all iterators
will diverge.
The above theorem for two occurrences, strongly
suggests that grouping variables (if possible) while
building the iterators, is a good strategy.

** Next:** Building additional iterators
** Up:** Partial inverse heuristic for
** Previous:** Evaluation by simulation.
*Gaston Gonnet*

*1998-07-08*