Robust Certified Numerical Homotopy Tracking

We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational comput...

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Vydáno v:Foundations of computational mathematics Ročník 13; číslo 2; s. 253 - 295
Hlavní autoři: Beltrán, Carlos, Leykin, Anton
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer-Verlag 01.04.2013
Springer Nature B.V
Témata:
Algorithms
Applications of Mathematics
Approximation
Computation
Computational mathematics
Computer Science
Economics
Foundations
Linear and Multilinear Algebras
Logarithms
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Polynomials
Tracking
Homotopy method
68W30
Rational computation
65H20
Condition metric
Computer proof
14Q20
Polynomial systems
Symbolic–numeric methods
Complexity
ISSN:1615-3375, 1615-3383
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Abstract We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.
AbstractList We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.[PUBLICATION ABSTRACT]
We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.
Author Leykin, Anton
Beltrán, Carlos
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  surname: Beltrán
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  givenname: Anton
  surname: Leykin
  fullname: Leykin, Anton
  organization: School of Mathematics, Georgia Tech
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Keywords Homotopy method
68W30
Rational computation
65H20
Condition metric
Computer proof
14Q20
Polynomial systems
Symbolic–numeric methods
Complexity
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Snippet We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems...
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SubjectTerms Algorithms
Applications of Mathematics
Approximation
Computation
Computational mathematics
Computer Science
Economics
Foundations
Linear and Multilinear Algebras
Logarithms
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Polynomials
Tracking
Title Robust Certified Numerical Homotopy Tracking
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Volume 13
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