Using bipartite and multidimensional matching to select the roots of a system of polynomial equations

  • Authors:

  • H. Bekker;E. P. Braad;B. Goldengorin

  • Affiliations:

  • Department of Mathematics and Computing Science, University of Groningen, Groningen, The Netherlands;Department of Mathematics and Computing Science, University of Groningen, Groningen, The Netherlands;Faculty of Economic Sciences, University of Groningen, Groningen, The Netherlands

  • Venue:
  • ICCSA'05 Proceedings of the 2005 international conference on Computational Science and Its Applications - Volume Part IV
  • Year:
  • 2005

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Abstract

Assume that the system of two polynomial equations f(x,y) = 0 and g(x,y) = 0 has a finite number of solutions. Then the solution consists of pairs of an x-value and an y-value. In some cases conventional methods to calculate these solutions give incorrect results and are complicated to implement due to possible degeneracies and multiple roots in intermediate results. We propose and test a two-step method to avoid these complications. First all x-roots and all y-roots are calculated independently. Taking the multiplicity of the roots into account, the number of x-roots equals the number of y-roots. In the second step the x-roots and y-roots are matched by constructing a weighted bipartite graph, where the x-roots and the y-roots are the nodes of the graph, and the errors are the weights. Of this graph the minimum weight perfect matching is computed. By using a multidimensional matching method this principle may be generalized to more than two equations.