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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Published in:	Foundations of computational mathematics Vol. 13; no. 2; pp. 253 - 295
Main Authors:	Beltrán, Carlos, Leykin, Anton
Format:	Journal Article
Language:	English
Published:	New York Springer-Verlag 01.04.2013 Springer Nature B.V
Subjects:	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
Author_xml	– sequence: 1 givenname: Carlos surname: Beltrán fullname: Beltrán, Carlos email: beltranc@unican.es organization: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria – sequence: 2 givenname: Anton surname: Leykin fullname: Leykin, Anton organization: School of Mathematics, Georgia Tech
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Cites_doi	10.1080/10586458.2011.606184 10.1007/s10208-005-0211-0 10.1007/s00211-010-0334-3 10.1016/j.jpaa.2010.10.016 10.1007/978-1-4612-2752-6_19 10.1007/978-1-4612-4984-9_13 10.1090/S0273-0979-1981-14858-8 10.1006/jcom.2000.0552 10.1016/0304-3975(94)00065-4 10.1007/978-1-4612-0701-6 10.1007/978-0-387-75155-9_3 10.1007/s10208-010-9078-9 10.1090/S0894-0347-08-00630-9 10.1090/S0273-0979-1989-15750-9 10.1007/BF01459082 10.4007/annals.2011.174.3.8 10.1016/0304-3975(83)90110-X 10.1006/jcom.2000.0572 10.1137/0731048 10.1007/978-1-4612-2740-3_40 10.1007/s10208-007-9017-6 10.1145/317275.317286 10.1090/S0025-5718-09-02239-X 10.2140/jsag.2011.3.5 10.1007/s002080010017
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References	Billey, Vakil, Dickenstein, Schreyer, Sommese (CR9) 2008 Blum, Cucker, Shub, Smale (CR11) 1998 Leykin (CR23) 2011; 3 Beltrán, Leykin (CR5) 2012; 21 CR19 Beltrán (CR4) 2011; 117 CR18 Papadimitriou (CR29) 1994 Dixon (CR17) 1982; 40 CR37 Cormen, Leiserson, Rivest (CR15) 1990 Smale (CR36) 1986 Dedieu, Malajovich, Shub (CR16) 2010 Shub, Smale, Eyssette, Galligo (CR32) 1993 Smale (CR35) 1981; 4 Bates, Peterson, Sommese, Wampler (CR1) 2011; 215 Malajovich (CR26) 1994; 133 Beltrán, Pardo (CR8) 2011; 11 CR2 Castro, Montaña, Pardo, San Martín (CR14) 2002; 2 Verschelde (CR38) 1999; 25 Shub (CR31) 2009; 9 CR28 Shub, Hirsch, Marsden, Shub (CR30) 1993 CR25 Baur, Strassen (CR3) 1983; 22 Bürguisser, Cucker (CR12) 2011; 174 Kearfott, Xing (CR20) 1994; 31 Leykin, Sottile (CR24) 2009; 78 Blum, Shub, Smale (CR10) 1989; 21 CR22 Shub, Smale (CR33) 1993; 6 CR21 Malajovich (CR27) 2000; 16 Beltrán, Pardo (CR6) 2008; 8 Castro, Hägele, Morais, Pardo (CR13) 2001; 17 Beltrán, Pardo (CR7) 2009; 22 Shub, Smale (CR34) 1994 9143_CR37 C. Beltrán (9143_CR4) 2011; 117 9143_CR19 C. Beltrán (9143_CR7) 2009; 22 D. Castro (9143_CR13) 2001; 17 9143_CR18 M. Shub (9143_CR32) 1993 A. Leykin (9143_CR24) 2009; 78 M. Shub (9143_CR34) 1994 J. Verschelde (9143_CR38) 1999; 25 C. Beltrán (9143_CR5) 2012; 21 D. Castro (9143_CR14) 2002; 2 T.H. Cormen (9143_CR15) 1990 9143_CR2 L. Blum (9143_CR11) 1998 L. Blum (9143_CR10) 1989; 21 G. Malajovich (9143_CR26) 1994; 133 D.J. Bates (9143_CR1) 2011; 215 M. Shub (9143_CR33) 1993; 6 C. Beltrán (9143_CR6) 2008; 8 G. Malajovich (9143_CR27) 2000; 16 J.D. Dixon (9143_CR17) 1982; 40 R.B. Kearfott (9143_CR20) 1994; 31 S. Billey (9143_CR9) 2008 J.-P. Dedieu (9143_CR16) 2010 9143_CR25 P. Bürguisser (9143_CR12) 2011; 174 S. Smale (9143_CR36) 1986 9143_CR28 9143_CR22 9143_CR21 S. Smale (9143_CR35) 1981; 4 M. Shub (9143_CR30) 1993 C. Beltrán (9143_CR8) 2011; 11 A. Leykin (9143_CR23) 2011; 3 C.H. Papadimitriou (9143_CR29) 1994 W. Baur (9143_CR3) 1983; 22 M. Shub (9143_CR31) 2009; 9
References_xml	– volume: 21 start-page: 69 issue: 1 year: 2012 end-page: 83 ident: CR5 article-title: Certified numerical homotopy tracking publication-title: Exp. Math. doi: 10.1080/10586458.2011.606184 – volume: 8 start-page: 1 issue: 1 year: 2008 end-page: 43 ident: CR6 article-title: On Smale’s 17th problem: a probabilistic positive solution publication-title: Found. Comput. Math. doi: 10.1007/s10208-005-0211-0 – ident: CR22 – ident: CR18 – volume: 117 start-page: 89 issue: 1 year: 2011 end-page: 113 ident: CR4 article-title: A continuation method to solve polynomial systems, and its complexity publication-title: Numer. Math. doi: 10.1007/s00211-010-0334-3 – volume: 215 start-page: 1844 issue: 8 year: 2011 end-page: 1851 ident: CR1 article-title: Numerical computation of the genus of an irreducible curve within an algebraic set publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2010.10.016 – start-page: 267 year: 1993 end-page: 285 ident: CR32 article-title: Complexity of Bézout’s theorem. II. Volumes and probabilities publication-title: Computational Algebraic Geometry doi: 10.1007/978-1-4612-2752-6_19 – start-page: 185 year: 1986 end-page: 196 ident: CR36 article-title: Newton’s method estimates from data at one point publication-title: The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics doi: 10.1007/978-1-4612-4984-9_13 – volume: 4 start-page: 1 issue: 1 year: 1981 end-page: 36 ident: CR35 article-title: The fundamental theorem of algebra and complexity theory publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0273-0979-1981-14858-8 – volume: 16 start-page: 529 issue: 3 year: 2000 end-page: 551 ident: CR27 article-title: Condition number bounds for problems with integer coefficients publication-title: J. Complex. doi: 10.1006/jcom.2000.0552 – ident: CR2 – volume: 133 start-page: 65 year: 1994 end-page: 84 ident: CR26 article-title: On generalized Newton algorithms: quadratic convergence, path-following and error analysis publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(94)00065-4 – ident: CR37 – year: 1998 ident: CR11 publication-title: Complexity and Real Computation doi: 10.1007/978-1-4612-0701-6 – volume: 3 start-page: 5 year: 2011 end-page: 10 ident: CR23 article-title: Numerical algebraic geometry for Macaulay2 publication-title: J. Softw. Algebr. Geom. – start-page: 21 year: 2008 end-page: 54 ident: CR9 article-title: Intersections of Schubert varieties and other permutation array schemes publication-title: Algorithms in Algebraic Geometry doi: 10.1007/978-0-387-75155-9_3 – volume: 11 start-page: 95 issue: 1 year: 2011 end-page: 129 ident: CR8 article-title: Fast linear homotopy to find approximate zeros of polynomial systems publication-title: Found. Comput. Math. doi: 10.1007/s10208-010-9078-9 – volume: 22 start-page: 363 year: 2009 end-page: 385 ident: CR7 article-title: Smale’s 17th problem: average polynomial time to compute affine and projective solutions publication-title: J. Am. Math. Soc. doi: 10.1090/S0894-0347-08-00630-9 – volume: 21 start-page: 1 year: 1989 end-page: 46 ident: CR10 article-title: On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0273-0979-1989-15750-9 – volume: 40 start-page: 137 issue: 1 year: 1982 end-page: 141 ident: CR17 article-title: Exact solution of linear equations using -adic expansions publication-title: Numer. Math. doi: 10.1007/BF01459082 – volume: 174 start-page: 1785 year: 2011 end-page: 1836 ident: CR12 article-title: On a problem posed by Steve Smale publication-title: Ann. Math. doi: 10.4007/annals.2011.174.3.8 – volume: 22 start-page: 317 issue: 3 year: 1983 end-page: 330 ident: CR3 article-title: The complexity of partial derivatives publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(83)90110-X – year: 1990 ident: CR15 publication-title: Introduction to Algorithms – start-page: 141 year: 1994 end-page: 164 ident: CR34 article-title: Complexity of Bezout’s theorem. V. Polynomial time publication-title: Selected Papers of the Workshop on Continuous Algorithms and Complexity – ident: CR25 – year: 1994 ident: CR29 publication-title: Computational Complexity – ident: CR21 – ident: CR19 – volume: 6 start-page: 459 issue: 2 year: 1993 end-page: 501 ident: CR33 article-title: Complexity of Bézout’s theorem. I. Geometric aspects publication-title: J. Am. Math. Soc. – volume: 17 start-page: 212 issue: 1 year: 2001 end-page: 303 ident: CR13 article-title: Kronecker’s and Newton’s approaches to solving: a first comparison publication-title: J. Complex. doi: 10.1006/jcom.2000.0572 – volume: 2 start-page: 1 issue: 1 year: 2002 end-page: 52 ident: CR14 article-title: The distribution of condition numbers of rational data of bounded bit length publication-title: Found. Comput. Math. – volume: 31 start-page: 892 issue: 3 year: 1994 end-page: 914 ident: CR20 article-title: An interval step control for continuation methods publication-title: SIAM J. Numer. Anal. doi: 10.1137/0731048 – start-page: 443 year: 1993 end-page: 455 ident: CR30 article-title: Some remarks on Bezout’s theorem and complexity theory publication-title: From Topology to Computation: Proceedings of the Smalefest doi: 10.1007/978-1-4612-2740-3_40 – year: 2010 ident: CR16 article-title: Adaptative step size selection for homotopy methods to solve polynomial equations publication-title: IMA J. Numer. Anal. – volume: 9 start-page: 171 issue: 2 year: 2009 end-page: 178 ident: CR31 article-title: Complexity of Bézout’s theorem. VI: Geodesics in the condition (number) metric publication-title: Found. Comput. Math. doi: 10.1007/s10208-007-9017-6 – volume: 25 start-page: 251 issue: 2 year: 1999 end-page: 276 ident: CR38 article-title: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation publication-title: ACM Trans. Math. Softw. doi: 10.1145/317275.317286 – ident: CR28 – volume: 78 start-page: 1749 issue: 267 year: 2009 end-page: 1765 ident: CR24 article-title: Galois groups of Schubert problems via homotopy computation publication-title: Math. Comput. doi: 10.1090/S0025-5718-09-02239-X – volume: 31 start-page: 892 issue: 3 year: 1994 ident: 9143_CR20 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0731048 – ident: 9143_CR22 – volume: 40 start-page: 137 issue: 1 year: 1982 ident: 9143_CR17 publication-title: Numer. Math. doi: 10.1007/BF01459082 – ident: 9143_CR18 – volume: 16 start-page: 529 issue: 3 year: 2000 ident: 9143_CR27 publication-title: J. Complex. doi: 10.1006/jcom.2000.0552 – ident: 9143_CR28 – volume-title: Introduction to Algorithms year: 1990 ident: 9143_CR15 – volume: 21 start-page: 1 year: 1989 ident: 9143_CR10 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0273-0979-1989-15750-9 – volume: 11 start-page: 95 issue: 1 year: 2011 ident: 9143_CR8 publication-title: Found. Comput. Math. doi: 10.1007/s10208-010-9078-9 – volume: 4 start-page: 1 issue: 1 year: 1981 ident: 9143_CR35 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0273-0979-1981-14858-8 – volume-title: Computational Complexity year: 1994 ident: 9143_CR29 – start-page: 267 volume-title: Computational Algebraic Geometry year: 1993 ident: 9143_CR32 doi: 10.1007/978-1-4612-2752-6_19 – start-page: 141 volume-title: Selected Papers of the Workshop on Continuous Algorithms and Complexity year: 1994 ident: 9143_CR34 – volume: 78 start-page: 1749 issue: 267 year: 2009 ident: 9143_CR24 publication-title: Math. Comput. doi: 10.1090/S0025-5718-09-02239-X – start-page: 185 volume-title: The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics year: 1986 ident: 9143_CR36 doi: 10.1007/978-1-4612-4984-9_13 – volume: 174 start-page: 1785 year: 2011 ident: 9143_CR12 publication-title: Ann. Math. doi: 10.4007/annals.2011.174.3.8 – year: 2010 ident: 9143_CR16 publication-title: IMA J. Numer. Anal. – ident: 9143_CR37 – volume: 3 start-page: 5 year: 2011 ident: 9143_CR23 publication-title: J. Softw. Algebr. Geom. doi: 10.2140/jsag.2011.3.5 – volume: 21 start-page: 69 issue: 1 year: 2012 ident: 9143_CR5 publication-title: Exp. Math. doi: 10.1080/10586458.2011.606184 – volume: 215 start-page: 1844 issue: 8 year: 2011 ident: 9143_CR1 publication-title: J. Pure Appl. Algebra doi: 10.1016/j.jpaa.2010.10.016 – volume: 22 start-page: 317 issue: 3 year: 1983 ident: 9143_CR3 publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(83)90110-X – volume: 8 start-page: 1 issue: 1 year: 2008 ident: 9143_CR6 publication-title: Found. Comput. Math. doi: 10.1007/s10208-005-0211-0 – ident: 9143_CR21 – volume-title: Complexity and Real Computation year: 1998 ident: 9143_CR11 doi: 10.1007/978-1-4612-0701-6 – volume: 22 start-page: 363 year: 2009 ident: 9143_CR7 publication-title: J. Am. Math. Soc. doi: 10.1090/S0894-0347-08-00630-9 – ident: 9143_CR25 – ident: 9143_CR19 – volume: 117 start-page: 89 issue: 1 year: 2011 ident: 9143_CR4 publication-title: Numer. Math. doi: 10.1007/s00211-010-0334-3 – volume: 17 start-page: 212 issue: 1 year: 2001 ident: 9143_CR13 publication-title: J. Complex. doi: 10.1006/jcom.2000.0572 – start-page: 443 volume-title: From Topology to Computation: Proceedings of the Smalefest year: 1993 ident: 9143_CR30 doi: 10.1007/978-1-4612-2740-3_40 – volume: 2 start-page: 1 issue: 1 year: 2002 ident: 9143_CR14 publication-title: Found. Comput. Math. doi: 10.1007/s002080010017 – volume: 133 start-page: 65 year: 1994 ident: 9143_CR26 publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(94)00065-4 – ident: 9143_CR2 – start-page: 21 volume-title: Algorithms in Algebraic Geometry year: 2008 ident: 9143_CR9 doi: 10.1007/978-0-387-75155-9_3 – volume: 6 start-page: 459 issue: 2 year: 1993 ident: 9143_CR33 publication-title: J. Am. Math. Soc. – volume: 25 start-page: 251 issue: 2 year: 1999 ident: 9143_CR38 publication-title: ACM Trans. Math. Softw. doi: 10.1145/317275.317286 – volume: 9 start-page: 171 issue: 2 year: 2009 ident: 9143_CR31 publication-title: Found. Comput. Math. doi: 10.1007/s10208-007-9017-6
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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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