34 research outputs found
Certifying reality of projections
Computational tools in numerical algebraic geometry can be used to
numerically approximate solutions to a system of polynomial equations. If the
system is well-constrained (i.e., square), Newton's method is locally
quadratically convergent near each nonsingular solution. In such cases, Smale's
alpha theory can be used to certify that a given point is in the quadratic
convergence basin of some solution. This was extended to certifiably determine
the reality of the corresponding solution when the polynomial system is real.
Using the theory of Newton-invariant sets, we certifiably decide the reality of
projections of solutions. We apply this method to certifiably count the number
of real and totally real tritangent planes for instances of curves of genus 4.Comment: 9 page
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
Computing the common zeros of two bivariate functions via Bézout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and Bézout matrices with polynomial entries. Using techniques including domain subdivision, Bézoutian regularization, and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (≥ 1000). We analyze the resultant method and its conditioning by noting that the Bézout matrices are matrix polynomials. Two implementations are available: one on the Matlab Central File Exchange and another in the roots command in Chebfun2 that is adapted to suit Chebfun’s methodology
PHClab: A MATLAB/Octave interface to PHCpack
•PHCpack [2] offers no scripting language; • Automatic input/output format conversions for systems and solutions. PHClab is a collection of m-files which callphc, the executable built with PHCpack. It applies the idea of OpenXM [1], needs only executable program
Numerical Irreducible Decomposition using PHCpack
Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components