|
ABSTRACT
We present a method based on symbolic-numeric reduction to geometric involutive form to compute the primary component and the differential operators f solution of a polynomial ideal. The singular solution can be exact or approximate. If the singular solution is known with limited accuracy, then we propose a new method to refine it to high accuracy.
REFERENCES
Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.
| |
1
|
Auzinger, W., and Stetter, H. An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. Intern. Series in Numer. Math. 86 (1988), 11--30.
|
| |
2
|
|
| |
3
|
Bayer, D., and Stillman, M. A criterion for detecting m-regularity. Inventiones Mathematicae 87, 1 (1987), 1--11.
|
| |
4
|
Bonasia, J., Lemaire, F., Reid, G., Scott, R., and Zhi, L. Determination of approximate symmetries of differential equations. In CRM Proceedings and Lecture Notes (2004), pp. 233--249.
|
 |
5
|
Robert M. Corless , Patrizia M. Gianni , Barry M. Trager, A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, p.133-140, July 21-23, 1997, Kihei, Maui, Hawaii, United States
[doi> 10.1145/258726.258767]
|
| |
6
|
David A. Cox , John Little , Donal O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics), Springer-Verlag New York, Inc., Secaucus, NJ, 2007
|
| |
7
|
Damiano, A., Sabadini, I., and Struppa, D. Computational methods for the construction of a class of Noetherian operators. Experiment. Math. 16 (2007), 41--55.
|
 |
8
|
|
 |
9
|
|
 |
10
|
|
| |
11
|
|
| |
12
|
|
| |
13
|
Kauers, M., Ed. ISSAC '05 Proc. 2005 Internat. Symp. Symbolic Algebraic Comput. (New York, 2005), ACM Press.
|
| |
14
|
|
| |
15
|
Kuranishi, M. On E.Cartan's prolongation theorem of exterior differential systems. Amer. J. Math 79 (1957), 1--47.
|
| |
16
|
Lakshman, Y. A single exponential bound on the complexity of computing Grobner bases of zero dimensional ideals. Effective Methods in Algebraic Geometry, Progress in Mathematics (1994), 227--234.
|
| |
17
|
|
| |
18
|
Lecerf, G. Quadratic Newton iteration for systems with multiplicity. Foundations of Computational Mathematics 2, 3 (2002), 247--293.
|
| |
19
|
|
| |
20
|
Leykin, A., Verschelde, J., and Zhao, A. Higher-order deflation for polynomial systems with isolated singular solutions. Manuscript, 19 pages, Jan 2007.
|
| |
21
|
Malgrange, B. Cartan involutiveness = Mumford regularity. Contemp. Math. 331 (2003), 193--205.
|
 |
22
|
Maria Grazia Marinari , Teo Mora , Hans Michael Möller, Gröbner duality and multiplicities in polynomial system solving, Proceedings of the 1995 international symposium on Symbolic and algebraic computation, p.167-179, July 10-12, 1995, Montreal, Quebec, Canada
[doi> 10.1145/220346.220368]
|
| |
23
|
Marinari, M., Mora, T., and M¨oller, H. On multiplicities in polynomial system solving. Trans. Amer. Math. Soc. 348 (1996), 3283--3321.
|
| |
24
|
Moller, H., and Sauer, T. H-bases for polynomial interpolation and system solving. Advances Comput. Math. 12 (2000), 23--35.
|
| |
25
|
|
| |
26
|
|
| |
27
|
Mourrain, B. Isolated points, duality and residues. J. of Pure and Applied Algebra 117 & 118 (1996), 469--493.
|
| |
28
|
|
 |
29
|
|
| |
30
|
Mourrain, B., and Trebuchet, P. Algebraic methods for numerical solving. In Proceedings of the 3rd International Workshop on Symbolic and Numeric Algorithms for Scientific Computing (2002), pp. 42--47.
|
 |
31
|
|
| |
32
|
Ojika, T. Modified deflation algorithm for the solution of singular problems. J. Math. Anal. Appl. 123 (1987), 199--221.
|
| |
33
|
Ojika, T., Watanabe, S., and Mitsui, T. Deflation algorithm for the multiple roots of a system of nonlinear equations. J. Math. Anal. Appl. 96 (1983), 463--479.
|
| |
34
|
Pommaret, J. Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach Science Publishers, 1978.
|
| |
35
|
Reid, G., Scott, R., Wu, W., and Zhi, L. Algebraic and geometric properties of nearby projectively involutive polynomial systems. Manuscript, 2005.
|
 |
36
|
Greg Reid , Jianliang Tang , Lihong Zhi, A complete symbolic-numeric linear method for camera pose determination, Proceedings of the 2003 international symposium on Symbolic and algebraic computation, p.215-223, August 03-06, 2003, Philadelphia, PA, USA
[doi> 10.1145/860854.860900]
|
| |
37
|
Reid, G., and Zhi, L. Solving polynomial systems via symbolic-numeric elimination method. J. Symbolic Comput. (2008). To appear.
|
| |
38
|
Seiler, W. Involution - The formal theory of differential equations and its applications in computer algebra and numerical analysis. Habilitation thesis, Univ. of Mannheim, Germany, 2002.
|
| |
39
|
|
| |
40
|
Trebuchet, P. Vers une R´esolution Stable et Rapide des Equations Algebriques. Dissertation, Universite Pierre et Marie Curie, France, 2002.
|
| |
41
|
van der Waerden B. L. Algebra. Frederick Ungar Pub. Co., 1970.
|
| |
42
|
Wittkopf, A., and Reid, G. Fast differential elimination in C: The CDiffelim Environment. Comp. Phys. Comm. 139(2) (2001), 192--217.
|
| |
43
|
Zhi, L., and Reid, G. Solving nonlinear polynomial system via symbolic-numeric conference on polynomial system solving (2004), J. Faugere and F. Rouillier, Eds., pp. 50--53.
|
|