Applied and Computational Mathematics and Statistics
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 Creator(s):
 Jonathan Hauenstein
 Description:
StewartGough 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 StewartGough platform is rigid with 40 assembly configurations (over the complex numbers) while exceptional StewartGough platforms have infinitely many assembly configurations and thus have selfmotion. We define a family of exceptional StewartGough platforms called Segredependent StewartGough platforms which aris…
 Date Created:
 20170117
 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:
 20170107
 Record Visibility:
 Public

15
Article
 Author(s):
 Daniel Bates, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Numerical algebraic geometry is the area devoted to the solution and manipulation of polynomial systems by numerical methods, which are mainly based on continuation. Due to the extreme intrinsic parallelism of continuation, polynomial systems may be successfully dealt with that are much larger than is possible with other methods. Singular solutions require special numerical methods called endgames, and the endgames currently used do not take advantage of parallelism. This article gives an ove…
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16
Article
 Author(s):
 Charles Wampler II, Daniel Bates, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Dedicated to our collaborator, mentor, and friend, Andrew Sommese, by Bates, Hauenstein, and Wampler on the occasion of his sixtieth birthday.
When numerically tracking implicitlydefined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy fo…
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17
Article
 Author(s):
 Charles Wampler, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operati…
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18
Article
 Author(s):
 Daniel Bates, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictorcorrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for…
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 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:
 20160831
 Record Visibility:
 Public