Numerical Schubert calculus by the Pieri homotopy algorithm

摘要

Based on Pieri's formula on Schubert varieties, the Pieri homotopy algorithm was first proposed by Huber, Sottile, and Sturmfels [J. Symbolic Comput., 26 (1998), pp. 767 788] for numerical Schubert calculus to enumerate all p-planes in Cm+p that meet n given planes in general position. The algorithm has been improved by Huber and Verschelde [SIAM J. Control Optim., 38 (2000), pp. 1265 1287] to be more intuitive and more suitable for computer implementations. A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm. [References: 10]