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Next: Introduction

Partial inverse heuristic for the
approximate solution of non-linear equations

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


We show how to generate many fix-point iterators of the form xi+1 = F(xi)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 xi in F. We use grouping of polynomial terms and expressions in x, exand $\ln x$ 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.


Gaston Gonnet