Numerical factorization of multivariate complex polynomials

Theoretical Computer Science

Volumn 315, Issue 2-3, 2004, Pages 651-669

  Sommese, Andrew J ^a   Verschelde, Jan ^b   Wampler, Charles W ^c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c GENERAL MOTORS CORPORATION   (United States)

Author keywords

Approximate factorization; Divided differences; Generic points; Homotopy continuation; Irreducible decomposition; Monodromy; Multiple roots; Newton interpolation; Numerical algebraic geometry; Polynomial; Stewart Gough platform; Symbolic numeric computation

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMPUTATIONAL METHODS; ERROR ANALYSIS; INTERPOLATION; PROBLEM SOLVING; SET THEORY;

APPROXIMATE FACTORIZATION; DIVIDED DIFFERENCES; GENERIC POINTS; HOMOTOPY CONTINUATION; IRREDUCIBLE DECOMPOSITION; MONODROMY; MULTIPLE ROOTS; NEWTON INTERPOLATION; NUMERICAL ALGEBRAIC GEOMETRY; WITNESS POINTS;

POLYNOMIALS;

EID: 2042445705     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2004.01.011     Document Type: Article

References (32)

1. Bajaj C., Canny J., Garrity T., Warren J. Factoring rational polynomials over the complex numbers. SIAM J. Comput. 22(2):1993;318-331.
2. Corless R.M., Galligo A., Kotsireas I.S., Watt S.M. A geometric-numeric algorithm for factoring multivariate polynomials. Mora T. Proc. 2002 Internat. Symp. on Symbolic and Algebraic Computation (ISSAC 2002). 2002;ACM, New York.
3. Corless R.M., Giesbrecht M.W., van Hoeij M., Kotsireas I.S., Watt S.M. Towards factoring bivariate approximate polynomials. Mourrain B. Proc. 2001 Internat. Symp. on Symbolic and Algebraic Computation (ISSAC 2001). 2001;85-92 ACM, New York.
4. Corless R.M., Kaltofen E., Watt S.M. Hybrid methods. Grabmeier J., Kaltofen E., Weispfenning V. Computer Algebra Handbook. 2002;112-125 Springer, Berlin.
5. Galligo A., Rupprecht D. Semi-numerical determination of irreducible branches of a reduced space curve. Mourrain B. Proc. 2001 Internat. Symp. on Symbolic and Algebraic Computation (ISSAC 2001). 2001;137-142 ACM, New York.
6. Galligo A., Rupprecht D. Irreducible decomposition of curves. J. Symbolic Comput. 33(5):2002;661-677.
7. Higham N.J. Accuracy and Stability of Numerical Algorithms. 1996;SIAM, Philadelphia, PA.
8. Huang Y., Wu W., Stetter H.J., Zhi L. Pseudofactors of multivariate polynomials. Traverso C. Proc. 2000 Internat. Symp. on Symbolic and Algebraic Computation (ISSAC 2000). 2000;161-168.
9. Isaacson E., Keller H.B. Analysis of Numerical Methods. 1994;Dover, New York. (Dover Reprint of the 1966 Wiley edition).
10. Kaltofen E. Challenges of symbolic computation. my favorite open problems J. Symbolic Comput. 29(6):2000;891-919.
11. Marden M. The geometry of the zeroes of a polynomial in a complex variable. Mathematical Surveys. Vol. 3:1949;American Mathematical Society, New York.
12. Mayer St-Onge B., Gosselin C.M. Singularity analysis and representation of the general Gough-Stewart platform. Internat. J. Robotics Res. 19(3):2000;271-288.
13. Merlet J.P. Singular configurations of parallel manipulators and Grassmann geometry. Internat. J. Robotics Res. 8(5):1999;45-56.
14. Merlet J.P. Parallel Robots. 2000;Kluwer Academic Publishers, Dordrecht, The Netherlands.
15. Pan V. Solving a polynomial equation. some history and recent progress SIAM Rev. 39(2):1997;187-220.
16. Pan V. Computation of approximate polynomial GCDs and an extension. Inform. Comput. 167:2001;71-85.
17. Pasquini L., Trigiante D. A globally convergent method for simultaneously finding polynomial roots. Math. Comput. 44(169):1985;135-149.
18. F. Pernkopf, M.L. Husty, Singularity analysis of spatial Stewart-Gough platforms with planar base and platform, Proc. ASME Design Eng. Tech. Conf. Montreal, Canada, September 30-October 2, 2002.
19. D. Rupprecht, Semi-numerical absolute factorization of polynomials with integer coefficients, manuscript.
20. Sasaki T. Approximate multivariate polynomial factorization based on zero-sum relations. Mourrain B. Proc. 2001 Internat. Symp. on Symbolic and Algebraic Computation (ISSAC 2001). 2001;284-291 ACM, New York.
21. Sommese A.J., Verschelde J. Numerical homotopies to compute generic points on positive dimensional algebraic sets. J. Complexity. 16(3):2000;572-602.
22. Sommese A.J., Verschelde J., Wampler C.W. Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6):2001;2022-2046.
23. Sommese A.J., Verschelde J., Wampler C.W. Numerical irreducible decomposition using projections from points on the components. Green E.L., Hoşten S., Laubenbacher R.C., Powers V. Journal of Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering. Contemporary Mathematics. Vol. 286:2001;37-51 American Mathematical Society, Providence, RI.
24. Sommese A.J., Verschelde J., Wampler C.W. Using monodromy to decompose solution sets of polynomial systems into irreducible components. Ciliberto C., Hirzebruch F., Miranda R., Teicher M. Application of Algebraic Geometry to Coding Theory, Physics and Computation, Proc. NATO Conf., Eilat, Israel. February 25-March 1, 2001;297-315 Kluwer Academic Publishers, Dordrecht.
25. Sommese A.J., Verschelde J., Wampler C.W. Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal. 40(6):2002;2026-2046.
26. Sommese A.J., Verschelde J., Wampler C.W. A method for tracking singular paths with application to the numerical irreducible decomposition. Beltrametti M.C., Catanese F., Ciliberto C., Lanteri A., Pedrini. W. de Gruyter C. Algebraic Geometry, a Volume in Memory of Paolo Francia. 2002;329-345 W. de Gruyter, Belmont, CA.
27. A.J. Sommese, J. Verschelde, C.W. Wampler, Advances in polynomial continuation for solving problems in kinematics, in: Proc. ASME Design Engineering Tech. Conf. (CDROM), Paper DETC2002/MECH-34254, Montreal, Quebec, September 29-October 2, 2002 (A revised version will appear in the ASME Journal of Mechanical Design).
28. Sommese A.J., Verschelde J., Wampler C.W. Numerical irreducible decomposition using PHCpack. Joswig M., Takayama N. Algebra, Geometry, and Software Systems. 2003;109-130 Springer, Berlin.
29. A.J. Sommese, J. Verschelde, C.W. Wampler, Introduction to numerical algebraic geometry, in: Graduate School on Systems of Polynomial Equations: From Algebraic Geometry to Industrial Applications, 2003, pp. 229-247 (Organized by A. Dickenstein, I. Emiris, 14-25 July 2003, Buenos Aires, Argentina. Notes published by INRIA).
30. A.J. Sommese, C.W. Wampler, Numerical algebraic geometry, in: J. Renegar, M. Shub, S. Smale (Eds.), The Mathematics of Numerical Analysis, Lectures in Applied Mathematics, Vol. 32, Proc. AMS-SIAM Summer Seminar in Applied Mathematics, Park City, Utah, July 17-August 11, 1995, Park City, UT, 1996, pp. 749-763.
31. Verschelde J. Algorithm 795: PHCpack. a general-purpose solver for polynomial systems by homotopy continuation ACM Transactions on Mathematical Software. 25(2):1999;251-276. Software available at http://www.math.uic.edu/ ~jan.
32. Yakoubsohn J.-C. Finding a cluster of zeros of univariate polynomials. J. Complexity. 16(3):2000;603-638.