Efficient path tracking methods
Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particu...
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| Published in | Numerical algorithms Vol. 58; no. 4; pp. 451 - 459 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.12.2011
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1017-1398 1572-9265 |
| DOI | 10.1007/s11075-011-9463-8 |
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| Abstract | Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described. |
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| AbstractList | Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for using Runge-Kutta methods in conjunction with adaptive precision are provided. These methods have been implemented in the Bertini software package, and several examples are described. |
| Author | Sommese, Andrew J. Bates, Daniel J. Hauenstein, Jonathan D. |
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| Cites_doi | 10.1145/7921.7923 10.1016/0771-050X(81)90010-3 10.1142/9789812567727 10.1137/1.9780898719031 10.1016/0096-3003(87)90064-6 10.1137/0715051 10.1145/79505.79507 10.1137/060658862 10.1115/1.2916909 10.1090/conm/496/09717 10.1016/S1570-8659(02)11004-0 |
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| Keywords | 65L06 Polynomial systems Path tracking Euler’s method Precision 65H20 Ordinary differential equations Homotopy continuation 65E05 65H10 Runge-Kutta methods Adaptive precision Numerical algebraic geometry |
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| References | Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Cucker, F. (ed.) Handbook of Numerical Analysis, Volume XI, Special Volume: Foundations of Computational Mathematics, North-Holland, pp. 209–304 (2003) BatesDJHauensteinJDSommeseAJWamplerCWStepsize control for adaptive multiprecision path trackingContemp. Math.200949621312555948 Morgan, A.P.: Solving polynomial systems using continuation for engineering and scientific problems. Prentice Hall Inc., Englewood Cliffs, NJ (1987) Reprinted as Classics in Applied Mathematics (2009) 57, SIAM EnrightWHJacksonKRNørsettSPThomsenPGInterpolants for Runge-Kutta formulasACM Trans. Math. Software19861231932188890660617.6506810.1145/7921.7923 Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks/Cole Publishing Co., Pacific Grove, CA (2002) FehlbergEKlassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-KontrolleComputing (Arch. Elektron. Rechnen)19694931062601790185.41302 PrincePJDormandJRHigh order embedded Runge-Kutta formulaeJ. Comput. Appl. Math.19817167756119530449.6504810.1016/0771-050X(81)90010-3 CashJRKarpAHA variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sidesACM Trans. Math. Software199016320122210707980900.6523410.1145/79505.79507 AllgowerELGeorgKNumerical continuation methods, An introduction. Springer Series in Computational Mathematics, vol. 131990BerlinSpringer VernerJHExplicit Runge-Kutta methods with estimates of the local truncation errorSIAM J. Numer. Anal.19781547727904834710403.6502910.1137/0715051 ShampineLFNumerical Solution of Ordinary Differential Equations1994New YorkChapman & Hall0832.65063 BatesDJHauensteinJDSommeseAJWamplerCWAdaptive multiprecision path trackingSIAM J. Numer. Anal.20084672274623832091162.6502610.1137/060658862 FehlbergEKlassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf WärmeleitungsproblemeComputing (Arch. Elektron. Rechnen)197066571280007 SommeseAJWamplerCWThe Numerical Solution of Systems of Polynomials Arising in Engineering and Science2005SingaporeWorld Scientific1091.6504910.1142/9789812567727 Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for Numerical Algebraic Geometry. Available at http://www.nd.edu/∼sommese/bertini MorganAPSommeseAJComputing all solutions to polynomial systems using homotopy continuationAppl. Math. Comput.19872421151389148070635.6505810.1016/0096-3003(87)90064-6Errata: Appl. Math. Comput., 51, 209 (1992) WamplerCWMorganASommeseAJComplete solution of the nine-point path synthesis problem for four-bar linkagesASME J. Mech. Des.1992114115315910.1115/1.2916909 DJ Bates (9463_CR4) 2009; 496 9463_CR9 JR Cash (9463_CR5) 1990; 16 AJ Sommese (9463_CR15) 2005 E Fehlberg (9463_CR8) 1970; 6 E Fehlberg (9463_CR7) 1969; 4 WH Enright (9463_CR6) 1986; 12 JH Verner (9463_CR16) 1978; 15 PJ Prince (9463_CR13) 1981; 7 LF Shampine (9463_CR14) 1994 AP Morgan (9463_CR12) 1987; 24 EL Allgower (9463_CR1) 1990 9463_CR2 CW Wampler (9463_CR17) 1992; 114 9463_CR11 9463_CR10 DJ Bates (9463_CR3) 2008; 46 |
| References_xml | – reference: CashJRKarpAHA variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sidesACM Trans. Math. Software199016320122210707980900.6523410.1145/79505.79507 – reference: Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Cucker, F. (ed.) Handbook of Numerical Analysis, Volume XI, Special Volume: Foundations of Computational Mathematics, North-Holland, pp. 209–304 (2003) – reference: Bates, D.J., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Bertini: Software for Numerical Algebraic Geometry. Available at http://www.nd.edu/∼sommese/bertini – reference: Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Brooks/Cole Publishing Co., Pacific Grove, CA (2002) – reference: SommeseAJWamplerCWThe Numerical Solution of Systems of Polynomials Arising in Engineering and Science2005SingaporeWorld Scientific1091.6504910.1142/9789812567727 – reference: FehlbergEKlassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-KontrolleComputing (Arch. Elektron. Rechnen)19694931062601790185.41302 – reference: EnrightWHJacksonKRNørsettSPThomsenPGInterpolants for Runge-Kutta formulasACM Trans. Math. Software19861231932188890660617.6506810.1145/7921.7923 – reference: BatesDJHauensteinJDSommeseAJWamplerCWAdaptive multiprecision path trackingSIAM J. Numer. Anal.20084672274623832091162.6502610.1137/060658862 – reference: VernerJHExplicit Runge-Kutta methods with estimates of the local truncation errorSIAM J. Numer. Anal.19781547727904834710403.6502910.1137/0715051 – reference: MorganAPSommeseAJComputing all solutions to polynomial systems using homotopy continuationAppl. Math. Comput.19872421151389148070635.6505810.1016/0096-3003(87)90064-6Errata: Appl. Math. Comput., 51, 209 (1992) – reference: BatesDJHauensteinJDSommeseAJWamplerCWStepsize control for adaptive multiprecision path trackingContemp. Math.200949621312555948 – reference: PrincePJDormandJRHigh order embedded Runge-Kutta formulaeJ. Comput. Appl. Math.19817167756119530449.6504810.1016/0771-050X(81)90010-3 – reference: AllgowerELGeorgKNumerical continuation methods, An introduction. Springer Series in Computational Mathematics, vol. 131990BerlinSpringer – reference: FehlbergEKlassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf WärmeleitungsproblemeComputing (Arch. Elektron. Rechnen)197066571280007 – reference: ShampineLFNumerical Solution of Ordinary Differential Equations1994New YorkChapman & Hall0832.65063 – reference: WamplerCWMorganASommeseAJComplete solution of the nine-point path synthesis problem for four-bar linkagesASME J. Mech. Des.1992114115315910.1115/1.2916909 – reference: Morgan, A.P.: Solving polynomial systems using continuation for engineering and scientific problems. Prentice Hall Inc., Englewood Cliffs, NJ (1987) Reprinted as Classics in Applied Mathematics (2009) 57, SIAM – volume: 12 start-page: 193 issue: 3 year: 1986 ident: 9463_CR6 publication-title: ACM Trans. Math. Software doi: 10.1145/7921.7923 – volume-title: Numerical Solution of Ordinary Differential Equations year: 1994 ident: 9463_CR14 – ident: 9463_CR9 – volume: 7 start-page: 67 issue: 1 year: 1981 ident: 9463_CR13 publication-title: J. Comput. Appl. Math. doi: 10.1016/0771-050X(81)90010-3 – volume-title: Numerical continuation methods, An introduction. Springer Series in Computational Mathematics, vol. 13 year: 1990 ident: 9463_CR1 – volume-title: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science year: 2005 ident: 9463_CR15 doi: 10.1142/9789812567727 – volume: 6 start-page: 65 year: 1970 ident: 9463_CR8 publication-title: Computing (Arch. Elektron. Rechnen) – ident: 9463_CR11 doi: 10.1137/1.9780898719031 – volume: 24 start-page: 115 issue: 2 year: 1987 ident: 9463_CR12 publication-title: Appl. Math. Comput. doi: 10.1016/0096-3003(87)90064-6 – volume: 15 start-page: 772 issue: 4 year: 1978 ident: 9463_CR16 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0715051 – volume: 16 start-page: 201 issue: 3 year: 1990 ident: 9463_CR5 publication-title: ACM Trans. Math. Software doi: 10.1145/79505.79507 – volume: 46 start-page: 722 year: 2008 ident: 9463_CR3 publication-title: SIAM J. Numer. Anal. doi: 10.1137/060658862 – volume: 4 start-page: 93 year: 1969 ident: 9463_CR7 publication-title: Computing (Arch. Elektron. Rechnen) – volume: 114 start-page: 153 issue: 1 year: 1992 ident: 9463_CR17 publication-title: ASME J. Mech. Des. doi: 10.1115/1.2916909 – volume: 496 start-page: 21 year: 2009 ident: 9463_CR4 publication-title: Contemp. Math. doi: 10.1090/conm/496/09717 – ident: 9463_CR10 doi: 10.1016/S1570-8659(02)11004-0 – ident: 9463_CR2 |
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| SubjectTerms | Algebra Algorithms Computer Science Numeric Computing Numerical Analysis Original Paper Path tracking Polynomials Predictor-corrector methods Runge-Kutta method Software Theory of Computation |
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