113 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
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
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
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