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Using the local geometrical properties of a given zero-dimensional squaremultivariate nonlinear system inside a box, we provide a simple but effectiveand new criterion for the uniqueness and the existence of a real simple zero ofthe system inside the box. Based on the result, we design an algorithm based onsubdivision and interval arithmetics to isolate all the real zeros of a generalreal nonlinear system inside a given box. Our method is complete for systemswith only finite isolated simple real zeros inside a box. A terminationprecision is given for general zero-dimensional systems. Multiple zeros of thesystem are output in bounded boxes. A variety of benchmarks show theeffectivity and efficiency of our implementation (in C++). It works forpolynomial systems with Bezout bound more than 100 million. It also works fornon-polynomial nonlinear systems. We also discuss the limitations of ourmethod.

Certified Numerical Real Root Isolation of Zero-dimensional Multivariate Real Nonlinear Systems