@Article{Hao:2012:AAS,
  author =       "Wenrui Hao and Andrew J. Sommese and Zhonggang Zeng",
  title =        "Algorithm 931: An algorithm and software for computing
                  multiplicity structures at zeros of nonlinear systems",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       "40",
  number =       "1",
  year =         "2013",
  month =        oct,
  pages =        "5:1--5:16",
  url =          "http://doi.acm.org/10.1145/2513109.2513114",
  accepted =     "5 September 2012",
  abstract =     "
                  A Matlab implementation, MULTIPLICITY, of a
                  numerical algorithm for computing the multiplicity
                  structure of a nonlinear system at an isolated zero is
                  presented. The software incorporates a newly developed
                  equation-by-equation strategy that significantly improves
                  the efficiency of the closedness subspace algorithm
                  and substantially reduces the storage requirement. The
                  equation-by-equation strategy is actually based on a
                  variable-by-variable closedness subspace approach. As a
                  result, the algorithm and software can handle much larger
                  nonlinear systems and higher multiplicities than their
                  predecessors, as shown in computational experiments on
                  the included test suite of benchmark problems.",
}