article

Geometric completion of differential systems using numeric-symbolic continuation

Authors:

Chris Smith, and

Jan VerscheldeAuthors Info & Claims

ACM SIGSAM Bulletin, Volume 36, Issue 2

Pages 1 - 17

https://doi.org/10.1145/581316.581317

Published: 02 June 2002 Publication History

Abstract

Symbolic algorithms using a finite number of exact differentiations and eliminations are able to reduce over and under-determined systems of polynomially nonlinear differential equations to involutive form. The output involutive form enables the identification of consistent initial values, and eases the application of exact or numerical integration methods.Motivated to avoid expression swell of pure symbolic approaches and with the desire to handle systems with approximate coefficients, we propose the use of homotopy continuation methods to perform the differential-elimination process on such non-square systems. Examples such as the classic index 3 Pendulum illustrate the new procedure. Our approach uses slicing by random linear subspaces to intersect its jet components in finitely many points. Generation of enough generic points enables irreducible jet components of the differential system to be interpolated.

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Cited By

Reid GZhi L(2009)Solving polynomial systems via symbolic-numeric reduction to geometric involutive formJournal of Symbolic Computation10.1016/j.jsc.2007.10.01344:3(280-291)Online publication date: 1-Mar-2009
https://dl.acm.org/doi/10.1016/j.jsc.2007.10.013
Wu WReid GGolubitsky O(2009)Towards Geometric Completion of Differential Systems by PointsApproximate Commutative Algebra10.1007/978-3-211-99314-9_3(79-97)Online publication date: 21-Aug-2009
https://doi.org/10.1007/978-3-211-99314-9_3
Wu WReid GWang D(2007)Symbolic-numeric computation of implicit riquier bases for PDEProceedings of the 2007 international symposium on Symbolic and algebraic computation10.1145/1277548.1277599(377-386)Online publication date: 29-Jul-2007
https://dl.acm.org/doi/10.1145/1277548.1277599
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Index Terms

Geometric completion of differential systems using numeric-symbolic continuation
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Partial differential equations
    2. Numerical analysis
      1. Computations on polynomials
      2. Numerical differentiation
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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Published In

cover image ACM SIGSAM Bulletin

ACM SIGSAM Bulletin Volume 36, Issue 2

June 2002

30 pages

ISSN:0163-5824

DOI:10.1145/581316

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Copyright © 2002 Authors.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 02 June 2002

Published in SIGSAM Volume 36, Issue 2

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Cited By

Reid GZhi L(2009)Solving polynomial systems via symbolic-numeric reduction to geometric involutive formJournal of Symbolic Computation10.1016/j.jsc.2007.10.01344:3(280-291)Online publication date: 1-Mar-2009
https://dl.acm.org/doi/10.1016/j.jsc.2007.10.013
Wu WReid GGolubitsky O(2009)Towards Geometric Completion of Differential Systems by PointsApproximate Commutative Algebra10.1007/978-3-211-99314-9_3(79-97)Online publication date: 21-Aug-2009
https://doi.org/10.1007/978-3-211-99314-9_3
Wu WReid GWang D(2007)Symbolic-numeric computation of implicit riquier bases for PDEProceedings of the 2007 international symposium on Symbolic and algebraic computation10.1145/1277548.1277599(377-386)Online publication date: 29-Jul-2007
https://dl.acm.org/doi/10.1145/1277548.1277599
Wu WReid GTrager B(2006)Application of numerical algebraic geometry and numerical linear algebra to PDEProceedings of the 2006 international symposium on Symbolic and algebraic computation10.1145/1145768.1145824(345-352)Online publication date: 9-Jul-2006
https://dl.acm.org/doi/10.1145/1145768.1145824
Reid GVerschelde JWittkopf AWu WGao XLabahn G(2005)Symbolic-numeric completion of differential systems by homotopy continuationProceedings of the 2005 international symposium on Symbolic and algebraic computation10.1145/1073884.1073922(269-276)Online publication date: 24-Jul-2005
https://dl.acm.org/doi/10.1145/1073884.1073922
Reid GTang JYu JZhi L(2004)Hybrid method for solving new pose estimation equation systemProceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications10.1007/11499251_5(44-55)Online publication date: 19-May-2004
https://dl.acm.org/doi/10.1007/11499251_5
Reid GTang JZhi LHong H(2003)A complete symbolic-numeric linear method for camera pose determinationProceedings of the 2003 international symposium on Symbolic and algebraic computation10.1145/860854.860900(215-223)Online publication date: 3-Aug-2003
https://dl.acm.org/doi/10.1145/860854.860900
Matera GSedoglavic A(2003)Fast computation of discrete invariants associated to a differential rational mappingJournal of Symbolic Computation10.1016/S0747-7171(03)00091-936:3-4(473-499)Online publication date: 1-Sep-2003
https://dl.acm.org/doi/10.1016/S0747-7171%2803%2900091-9

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