Constructing Dixon Matrix for Sparse Polynomial Equations Based on Hybrid and Heuristics Scheme

Solving polynomial equations inevitably faces many severe challenges, such as easily occupying storage space and demanding prohibitively expensive computation resources. There has been considerable interest in exploiting the sparsity to improve computation efficiency, since asymmetry phenomena are prevalent in scientific and engineering fields, especially as most of the systems in real applications have sparse representations. In this paper, we propose an efficient parallel hybrid algorithm for constructing a Dixon matrix. This approach takes advantage of the asymmetry (i.e., sparsity) in variables of the system and introduces a heuristics strategy. Our method supports parallel computation and has been implemented on a multi-core system. Through time-complexity analysis and extensive benchmarks, we show our new algorithm has significantly reduced computation and memory overhead. In addition, performance evaluation via the Fermat-Torricelli point problem demonstrates its effectiveness in combinatorial geometry optimizations.