The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotop...

CMPSm is a MATLAB program using a predictor-corrector path-following method for tracing homotopy curves to approximate all isolated solutions of a system of polynomial equations. The program was originally developed for cheater’s homotopy, but it can be also used for linear and polyhedral homotopies. The latest version CMPSm2 of this software, this manual and some numerical examples are availab...

In this paper, the four-precision-point geometric design problem of serial-link robot manipulators with three revolute joints is solved using a polynomial continuation method. At each precision point, the end-effector spatial locations are defined. The dimensions of the geometric parameters of the 3R manipulator are computed so that the manipulator’s end-effector will be able to reach these fou...

Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is a technique alternative to the classical symbolic approaches of computational algebraic geometry. We present a package , whose primary purpose is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approxim...

Motivated by Wilmshurst’s conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy continuation to compute a numerical approximation of each zero and Smale’s alpha-theory to certify the results. Using this approach, we provide new examples of harmo...