  Numerical algebraic geometry provides a number of efficient tools for
approximating the solutions of polynomial systems. One such tool is the
parameter homotopy, which can be an extremely efficient method to solve
numerous polynomial systems that differ only in coefficients, not monomials.
This technique is frequently used for solving a parameterized family of
polynomial systems at multiple parameter values. Parameter homotopies have
recently been useful in several areas of application and have been implemented
in at least two software packages. This article describes Paramotopy, a new,
parallel, optimized implementation of this technique, making use of the Bertini
software package. The novel features of this implementation, not available
elsewhere, include allowing for the simultaneous solutions of arbitrary
polynomial systems in a parameterized family on an automatically generated (or
manually provided) mesh in the parameter space of coefficients, front ends and
back ends that are easily specialized to particular classes of problems, and
adaptive techniques for solving polynomial systems near singular points in the
parameter space. This last feature automates and simplifies a task that is
important but often misunderstood by non-experts.
