High-Performance Numeric Solution of Polynomial Systems 

Mathematica 10 includes a new homotopy-based numerical polynomial solver. This method is automatically selected when appropriate. The following charts compare the timing of this new algorithm with Mathematica 9's Gröbner-basis method and the faster of Maple 18's solve or Homotopy commands. All tests were performed on a 16-core, 2.40 GHz 64-bit Linux system with Hyper-Threading enabled and a time limit of 12 hours.

Comparison for a standard economics system in variables, total degree , and distinct solutions, given by the following formula.

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For example, for the system takes the following form.

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Comparison for the standard Noonburg neural-network system, given by the following formula. For variables, this system has total degree and different solutions.

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For example, in five variables the system takes the following form.

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Comparison for the standard Katsura- system of ferromagnetic lattice probabilities, which in variables has total degree and different solutions. The order- system takes the following form.

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For example, for the system has six equations in six unknowns.

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