approximate solution of non-linear equations

**Gaston H. Gonnet and Allan Bonadio
Informatik E.T.H. Zurich, Switzerland
and
Waterloo Maple, San Francisco
**

We show how to generate many fix-point iterators
of the form
*x*_{i+1} = *F*(*x*_{i})which could solve a given non-linear equation.
In particular, these iterators tend to have good global
convergence, and we show examples whereby obscure solutions
can be discovered.
Also, a systematic method for finding most or all solutions to
nonlinear equations that have multiple solutions is described.
The most successful iterators are constructed to have a small number of
occurrences of *x*_{i} in *F*.
We use grouping of polynomial terms and expressions in *x*, *e*^{x}and
using known inverse relations to obtain better iterators.
Each iterator is tried in a limited way, in the expectation that
at least one of them will succeed.
This heuristic shows a very good behaviour in most cases,
in particular when the answer involves extreme ranges.

- Introduction
- Algorithm of partial inversion
- Convergence of partial inverses
- Building additional iterators
- Inverting functions
- Conclusions
- Bibliography
- About this document ...