Partial Inverse Heuristic for the Approximate Solution of Non-linear Equations

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. This methods are only suitable for computer algebra systems, where the equations to be solved can be manipulated in symbolic form. 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, ex and In 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.