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  • Creator(s):
    Jonathan Hauenstein, Margaret Regan
    Description:

    Polynomials which arise via elimination can be difficult to compute explicitly. By using a pseudo-witness set, we develop an algorithm to explicitly compute the restriction of a polynomial to a given line. The resulting polynomial can then be used to evaluate the original polynomial and directional derivatives along the line at any point on the given line. Several examples are used to demonstrate this new algorithm including examples of computing the critical points of the discriminant locu…

    Date Created:
    2020-03-27
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein, Martin Helmer
    Description:

    Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of sol…

    Date Created:
    2020-03-03
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive …

    Date Created:
    2020-01-18
    Record Visibility:
    Public
  • Creator(s):
    Dan Bates, David Eklund, Jonathan Hauenstein, Chris Peterson
    Description:

    A fundamental problem in algebraic geometry is to decompose the solution set of a given polynomial system. A numerical description of this solution set is called a numerical irreducible decomposition and currently all standard algorithms use a sequence of homotopies forming a dimension-by-dimension approach. In this article, we pair a classical result to compute a smooth point on every irreducible component in every dimension using a single homotopy together with the theory of isosingular s…

    Date Created:
    2019-04-26
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein, Jose Rodriguez
    Description:

    In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalize the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.

    Date Created:
    2018-12-29
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein, Alan Liddell, Sanesha McPherson, Yi Zhang
    Description:

    Standard interior point methods in semidefinite programming can be viewed as tracking a solution path for a homotopy defined by a system of bilinear equations. By considering this in the context of numerical algebraic geometry, we employ numerical algebraic geometric techniques such as adaptive precision path tracking, endgames, and projective space to accurately solve semidefinite programs. We develop feasibility tests for both primal and dual problems which can distinguish between the fou…

    Date Created:
    2018-04-10
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    A common problem when analyzing models, such as a mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that th…

    Date Created:
    2018-03-09
    Record Visibility:
    Public
  • Creator(s):
    Samantha Sherman, Jonathan Hauenstein
    Description:

    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 sol…

    Date Created:
    2018-03-08
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For…

    Date Created:
    2017-04-07
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    Stewart-Gough platforms are mechanisms which consist of two rigid objects, a base and a platform, connected by six legs via spherical joints. For fixed leg lengths, a generic Stewart-Gough platform is rigid with 40 assembly configurations (over the complex numbers) while exceptional Stewart-Gough platforms have infinitely many assembly configurations and thus have self-motion. We define a family of exceptional Stewart-Gough platforms called Segre-dependent Stewart-Gough platforms which aris…

    Date Created:
    2017-01-17
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    We define tensors, most of which correspond with cubic polynomials, which have the same exponent w as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor defined on an nxn matrix A by trace(A^3). The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent w.

    Date Created:
    2017-01-07
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projecti…

    Date Created:
    2016-08-31
    Record Visibility:
    Public