Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study

We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditi...

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Detalles Bibliográficos
Autores principales:	De Leo, Mariano Fernando, Dratman, Ezequiel, Matera, Guillermo
Publicado:	2005
Materias:	Complexity Conditioning Homotopy algorithms Polynomial system solving Semi-linear parabolic problems Stationary solutions Algorithms Boundary value problems Mathematical models Matrix algebra Partial differential equations Polynomials Set theory Semi linear parabolic problems Computational complexity
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p502_DeLeo http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p502_DeLeo
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_0885064X_v21_n4_p502_DeLeo
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spelling	paper:paper_0885064X_v21_n4_p502_DeLeo2025-07-30T18:22:59Z Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study De Leo, Mariano Fernando Dratman, Ezequiel Matera, Guillermo Complexity Conditioning Homotopy algorithms Polynomial system solving Semi-linear parabolic problems Stationary solutions Algorithms Boundary value problems Mathematical models Matrix algebra Partial differential equations Polynomials Set theory Homotopy algorithms Polynomial system solving Semi linear parabolic problems Stationary solutions Computational complexity We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditioned from the symbolic point of view. We exhibit a polynomial-time numeric algorithm solving any member of this family, which significantly contrasts the exponential behavior of all known symbolic algorithms solving a generic instance of this family of systems. © 2005 Elsevier Inc. All rights reserved. Fil:De Leo, M. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Dratman, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Matera, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2005 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p502_DeLeo http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p502_DeLeo
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Complexity Conditioning Homotopy algorithms Polynomial system solving Semi-linear parabolic problems Stationary solutions Algorithms Boundary value problems Mathematical models Matrix algebra Partial differential equations Polynomials Set theory Homotopy algorithms Polynomial system solving Semi linear parabolic problems Stationary solutions Computational complexity
spellingShingle	Complexity Conditioning Homotopy algorithms Polynomial system solving Semi-linear parabolic problems Stationary solutions Algorithms Boundary value problems Mathematical models Matrix algebra Partial differential equations Polynomials Set theory Homotopy algorithms Polynomial system solving Semi linear parabolic problems Stationary solutions Computational complexity De Leo, Mariano Fernando Dratman, Ezequiel Matera, Guillermo Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
topic_facet	Complexity Conditioning Homotopy algorithms Polynomial system solving Semi-linear parabolic problems Stationary solutions Algorithms Boundary value problems Mathematical models Matrix algebra Partial differential equations Polynomials Set theory Homotopy algorithms Polynomial system solving Semi linear parabolic problems Stationary solutions Computational complexity
description	We consider a family of polynomial systems which arises in the analysis of the stationary solutions of a standard discretization of certain semi-linear second-order parabolic partial differential equations. We prove that this family is well-conditioned from the numeric point of view, and ill-conditioned from the symbolic point of view. We exhibit a polynomial-time numeric algorithm solving any member of this family, which significantly contrasts the exponential behavior of all known symbolic algorithms solving a generic instance of this family of systems. © 2005 Elsevier Inc. All rights reserved.
author	De Leo, Mariano Fernando Dratman, Ezequiel Matera, Guillermo
author_facet	De Leo, Mariano Fernando Dratman, Ezequiel Matera, Guillermo
author_sort	De Leo, Mariano Fernando
title	Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
title_short	Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
title_full	Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
title_fullStr	Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
title_full_unstemmed	Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study
title_sort	numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study
publishDate	2005
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p502_DeLeo http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p502_DeLeo
work_keys_str_mv	AT deleomarianofernando numericvssymbolichomotopyalgorithmsinpolynomialsystemsolvingacasestudy AT dratmanezequiel numericvssymbolichomotopyalgorithmsinpolynomialsystemsolvingacasestudy AT materaguillermo numericvssymbolichomotopyalgorithmsinpolynomialsystemsolvingacasestudy
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Numeric vs. symbolic homotopy algorithms in polynomial system solving: A case study

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