Abstract
We describe the software package borderbasix dedicated to the computation of border bases and the solutions of polynomial equations. We present the main ingredients of the border basis algorithm and the other methods implemented in this package: numerical solutions from multiplication matrices, real radical computation, polynomial optimization. The implementation parameterized by the coefficient type and the choice function provides a versatile family of tools for polynomial computation with modular arithmetic, floating point arithmetic or rational arithmetic. It relies on linear algebra solvers for dense and sparse matrices for these various types of coefficients. A connection with SDP solvers has been integrated for the combination of relaxation approaches with border basis computation. Extensive benchmarks on typical polynomial systems are reported, which show the very good performance of the tool.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
counted with cloc.
References
Bucero, M.A., Mourrain, B.: Exact relaxation for polynomial optimization on semi-algebraic sets (2013). http://hal.inria.fr/hal-00846977
Bucero, M.A., Mourrain, B.: Border basis relaxation for polynomial optimization. J. Symb. Comput. 74, 378–399 (2015)
Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D.: LAPACK Users’ Guide. SIAM, Philadelphia (1992). http://www.netlib.org/lapack/
Bosma, W., Cannon, J., Playoust, C.: The magma algebra system I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
Corless, R.M., Gianni, P.M., Trager, B.M.: A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In: Küchlin, W.W. (ed.) Proceedings of ISSAC, pp. 133–140 (1997)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2 – A computer algebra system for polynomial computations (2015). www.singular.uni-kl.de
Demmel, J.W., Eisenstat, S.C., Gilbert, J.R., Liu, J.W.H., Li, X.S.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20, 720–755 (1999)
Faugère, J.-C.: FGb: a library for computing Gröbner bases. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 84–87. Springer, Heidelberg (2010)
Fujisawa, K., Fukuda, M., Kobayashi, K., Kojima, M., Nakata, K., Nakata, M., Yamashita, M.: SDPA (SemiDefinite Programming Algorithm) (2008)
Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999)
Graillat, S., Trébuchet, P.: A new algorithm for computing certified numerical approximations of the roots of a zero-dimensional system. In: ISSAC 2009, pp. 167–173 (2009)
Huot, L.: Polynomial systems solving and elliptic curve cryptography. Ph.D. thesis, Université Pierre et Marie Curie (UPMC) (2013)
Lasserre, J.-B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2009)
Lasserre, J.-B., Laurent, M., Mourrain, B., Rostalski, P., Trébuchet, P.: Moment matrices, border bases and real radical computation. J. Symb. Comput. 51, 63–85 (2012)
Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)
Mourrain, B., Trébuchet, P.: Generalized normal forms and polynomials system solving. In: Kauers, M. (ed.) ISSAC 2005, pp. 253–260 (2005)
Mourrain, B., Trébuchet, P.: Border basis representation of a general quotient algebra. In: van der Hoeven, J. (ed.) ISSAC 2012, pp. 265–272 (2012)
MOSEK ApS. The MOSEK optimization library (2015). www.mosek.com
Ottaviani, G., Spaenlehauer, P.-J., Sturmfels, B.: Exact solutions in structured low-rank approximation. SIAM J. Matrix Anal. Appl. 35(4), 1521–1542 (2014)
Parisse, B.: Giac/XCas, a free computer algebra system. Technical report, University of Grenoble (2008)
Trébuchet, P.: A new certified numerical algorithm for solving polynomial systems. In: SCAN 2010, pp. 1–8 (2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Trébuchet, P., Mourrain, B., Bucero, M.A. (2016). Border Basis for Polynomial System Solving and Optimization. In: Greuel, GM., Koch, T., Paule, P., Sommese, A. (eds) Mathematical Software – ICMS 2016. ICMS 2016. Lecture Notes in Computer Science(), vol 9725. Springer, Cham. https://doi.org/10.1007/978-3-319-42432-3_27
Download citation
DOI: https://doi.org/10.1007/978-3-319-42432-3_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42431-6
Online ISBN: 978-3-319-42432-3
eBook Packages: Computer ScienceComputer Science (R0)