251 research outputs found
Modernizing PHCpack through phcpy
PHCpack is a large software package for solving systems of polynomial
equations. The executable phc is menu driven and file oriented. This paper
describes the development of phcpy, a Python interface to PHCpack. Instead of
navigating through menus, users of phcpy solve systems in the Python shell or
via scripts. Persistent objects replace intermediate files.Comment: Part of the Proceedings of the 6th European Conference on Python in
Science (EuroSciPy 2013), Pierre de Buyl and Nelle Varoquaux editors, (2014
Galois groups of Schubert problems via homotopy computation
Numerical homotopy continuation of solutions to polynomial equations is the
foundation for numerical algebraic geometry, whose development has been driven
by applications of mathematics. We use numerical homotopy continuation to
investigate the problem in pure mathematics of determining Galois groups in the
Schubert calculus. For example, we show by direct computation that the Galois
group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes
non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde
Sampling algebraic sets in local intrinsic coordinates
Numerical data structures for positive dimensional solution sets of
polynomial systems are sets of generic points cut out by random planes of
complimentary dimension. We may represent the linear spaces defined by those
planes either by explicit linear equations or in parametric form. These
descriptions are respectively called extrinsic and intrinsic representations.
While intrinsic representations lower the cost of the linear algebra
operations, we observe worse condition numbers. In this paper we describe the
local adaptation of intrinsic coordinates to improve the numerical conditioning
of sampling algebraic sets. Local intrinsic coordinates also lead to a better
stepsize control. We illustrate our results with Maple experiments and
computations with PHCpack on some benchmark polynomial systems.Comment: 13 pages, 2 figures, 2 algorithms, 2 table
- …