HOM4PS-2.0 is a software package in FORTRAN 90 which implements the polyhedral homotopy continuation method for solving polynomial systems. It updates its original version HOM4PS in three key aspects: (1) New method for finding mixed cells, (2) Combining the polyhedral and linear homotopies in one step, (3) New way of dealing with the curve jumping. Numerical results show that this revision leads to a spectacular speed-up, ranging up to the 50's, over its original version on all bench mark systems, especially for large ones. It surpasses established packages in this category, such as PHCpack [25] and PHoM [8], in speed by huge margins.

Abstract. The root count developed by Bernshtein, Kushnirenko and Khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus (C * ) n . In this paper, we modify this bound slightly so that it counts the number of isolated zeros in C n . Our bound is, apparently, significantly sharper than the recent root counts found by Rojas and in many cases easier to compute. As a consequence of our result, the Huber-Sturmfels homotopy for finding all the isolated zeros of a polynomial system in (C * ) n can be slightly modified to obtain all the isolated zeros in C n .

In practice, finding mixed cells in certain polyhedral subdivisions plays a dominating role when a polyhedral homotopy is employed to approximate all isolated zeros of polynomial systems. This paper gives a new algorithm for the mixed cell computation via a new formulation of the underlying linear programming problems. Numerical results show that the algorithm provides a major advance in the speed of computation with much less memory requirements.

Summary.The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of new theoretical results. In practice, it is rarely of any interest to compute all the eigenvalues. The homotopy method, having the order preserving property, can provide any specific eigenvalue without calculating any other eigenvatues. Besides this advantage, we note that the homotopy algorithm is to a large degree a parallel algorithm. Numerical experimentation shows that the homotopy method can be very efficient especially for graded matrices.

Numerical experiments are also given to support the theoretical analysis.