Preconditioning polynomial systems using Macaulay dual spaces
| dc.contributor.author | Ihde, Steven L., author | |
| dc.contributor.author | Bates, Daniel J., advisor | |
| dc.contributor.author | Peterson, Chris, committee member | |
| dc.contributor.author | Hulpke, Alexander, committee member | |
| dc.contributor.author | Young, Peter, committee member | |
| dc.date.accessioned | 2015-08-28T14:35:26Z | |
| dc.date.available | 2015-08-28T14:35:26Z | |
| dc.date.issued | 2015 | |
| dc.description.abstract | Polynomial systems arise in many applications across a diverse landscape of subjects. Solving these systems has been an area of intense research for many years. Methods for solving these systems numerically fit into the field of numerical algebraic geometry. Many of these methods rely on an idea called homotopy continuation. This method is very effective for solving systems of polynomials in many variables. However, in the case of zero-dimensional systems, we may end up tracking many more solutions than actually exist, leading to excess computation. This project preconditions these systems in order to reduce computation. We present the background on homotopy continuation and numerical algebraic geometry as well as the theory of Macaulay dual spaces. We show how to turn an algebraic geometric preconditioning problem into one of numerical linear algebra. Algorithms for computing an H-basis and thereby preconditioning the original system to remove extraneous calculation are presented. The concept of the Closedness Subspace is introduced and used to replace a bottleneck computation. A novel algorithm employing this method is introduced and discussed. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Ihde_colostate_0053A_13158.pdf | |
| dc.identifier.uri | http://hdl.handle.net/10217/167183 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.subject | dual space | |
| dc.subject | homotopy | |
| dc.subject | algebraic geometry | |
| dc.subject | Macaulay | |
| dc.subject | H-basis | |
| dc.title | Preconditioning polynomial systems using Macaulay dual spaces | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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