u-generation: solving systems of polynomials equation-by-equation

We develop a new method that improves the efficiency of equation-by-equation homotopy continuation methods for solving polynomial systems. Our method is based on a novel geometric construction and reduces the total number of homotopy paths that must be numerically continued. These improvements may b...

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Bibliographic Details
Published inNumerical algorithms Vol. 95; no. 2; pp. 813 - 838
Main Authors Duff, Timothy, Leykin, Anton, Rodriguez, Jose Israel
Format Journal Article
LanguageEnglish
Published New York Springer US 01.02.2024
Subjects
Algebra
Algorithms
Computer Science
Numeric Computing
Numerical Analysis
Original Paper
Theory of Computation
Regeneration
68W30
Solving polynomials
65H20
14Q65
Numerical algebraic geometry
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ISSN1017-1398
1572-9265
DOI10.1007/s11075-023-01590-1

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Summary:We develop a new method that improves the efficiency of equation-by-equation homotopy continuation methods for solving polynomial systems. Our method is based on a novel geometric construction and reduces the total number of homotopy paths that must be numerically continued. These improvements may be applied to the basic algorithms of numerical algebraic geometry in the settings of both projective and multiprojective varieties. Our computational experiments demonstrate significant savings obtained on several benchmark systems. We also present an extended case study on maximum likelihood estimation for rank-constrained symmetric n × n matrices, in which multiprojective u -generation allows us to complete the list of ML degrees for n ≤ 6 .
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-023-01590-1